# CS计算机代考程序代写 cache algorithm scheme arm database compiler chain assembly flex discrete mathematics data structure information theory data mining AI Java Bioinformatics computational biology Excel distributed system DNA This page intentionally left blank

This page intentionally left blank

Acquisitions Editor: Matt Goldstein

Project Editor: Maite Suarez-Rivas Production Supervisor: Marilyn Lloyd Marketing Manager: Michelle Brown Marketing Coordinator: Jake Zavracky Project Management: Windfall Software Composition: Windfall Software, using ZzTEX Copyeditor: Carol Leyba

Technical Illustration: Dartmouth Publishing Proofreader: Jennifer McClain

Indexer: Ted Laux

Cover Design: Joyce Cosentino Wells

Cover Photo: © 2005 Tim Laman / National Geographic. A pair of weaverbirds work together on their nest in Africa.

Prepress and Manufacturing: Caroline Fell Printer: Courier Westford

Access the latest information about Addison-Wesley titles from our World Wide Web site: http://www.aw-bc.com/computing

Many of the designations used by manufacturers and sellers to distinguish their products are claimed as trademarks. Where those designations appear in this book, and Addison-Wesley was aware of a trademark claim, the designations have been printed in initial caps or all caps.

The programs and applications presented in this book have been included for their instructional value. They have been tested with care, but are not guaranteed for any particular purpose. The publisher does not offer any warranties or representations, nor does it accept any liabilities with respect to the programs or applications.

Library of Congress Cataloging-in-Publication Data

Kleinberg, Jon.

Algorithm design / Jon Kleinberg, E ́va Tardos.—1st ed. p. cm.

Includes bibliographical references and index.

ISBN 0-321-29535-8 (alk. paper)

1. Computer algorithms. 2. Data structures (Computer science) I. Tardos, E ́va. II. Title.

QA76.9.A43K54 2005

005.1—dc22 2005000401

Copyright © 2006 by Pearson Education, Inc.

For information on obtaining permission for use of material in this work, please submit a written request to Pearson Education, Inc., Rights and Contract Department, 75 Arlington Street, Suite 300, Boston, MA 02116 or fax your request to (617) 848-7047.

All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or any toher media embodiments now known or hereafter to become known, without the prior written permission of the publisher. Printed in the United States of America.

ISBN 0-321-29535-8

1 2 3 4 5 6 7 8 9 10-CRW-08 07 06 05

About the Authors

Jon Kleinberg is a professor of Computer Science at Cornell University. He received his Ph.D. from M.I.T. in 1996. He is the recipient of an NSF Career Award, an ONR Young Investigator Award, an IBM Outstand- ing Innovation Award, the National Academy of Sci- ences Award for Initiatives in Research, research fel- lowships from the Packard and Sloan Foundations, and teaching awards from the Cornell Engineering College and Computer Science Department.

Kleinberg’s research is centered around algorithms, particularly those con- cerned with the structure of networks and information, and with applications to information science, optimization, data mining, and computational biol- ogy. His work on network analysis using hubs and authorities helped form the foundation for the current generation of Internet search engines.

E ́vaTardosisaprofessorofComputerScienceatCor- nell University. She received her Ph.D. from Eo ̈tvo ̈s University in Budapest, Hungary in 1984. She is a member of the American Academy of Arts and Sci- ences, and an ACM Fellow; she is the recipient of an NSF Presidential Young Investigator Award, the Fulk- erson Prize, research fellowships from the Guggen- heim, Packard, and Sloan Foundations, and teach- ing awards from the Cornell Engineering College and

Computer Science Department.

Tardos’s research interests are focused on the design and analysis of algorithms for problems on graphs or networks. She is most known for her work on network-flow algorithms and approximation algorithms for network problems. Her recent work focuses on algorithmic game theory, an emerging area concerned with designing systems and algorithms for selfish users.

This page intentionally left blank

Contents

About the Authors v Preface xiii

1 Introduction: Some Representative Problems 1 1.1 A First Problem: Stable Matching 1

1.2 Five Representative Problems Solved Exercises 19

Exercises 22

Notes and Further Reading

2 Basics of Algorithm Analysis

2.1 Computational Tractability

2.2 Asymptotic Order of Growth

2.3 Implementing the Stable Matching Algorithm Using Lists and

Arrays 42

2.4 A Survey of Common Running Times 47

2.5 A More Complex Data Structure: Priority Queues 57

Solved Exercises 65

Exercises 67

Notes and Further Reading 70

3 Graphs

3.1 Basic Definitions and Applications 73

3.2 Graph Connectivity and Graph Traversal 78

3.3 Implementing Graph Traversal Using Queues and Stacks 87

3.4 Testing Bipartiteness: An Application of Breadth-First Search 94

3.5 Connectivity in Directed Graphs 97

12

28

29 35

29

73

viii

Contents

3.6 Directed Acyclic Graphs and Topological Ordering Solved Exercises 104

Exercises 107

Notes and Further Reading 112

4 Greedy Algorithms

99

5

Algorithm 177

Solved Exercises 183

Exercises 188

Notes and Further Reading 205

Divide and Conquer

5.1 A First Recurrence: The Mergesort Algorithm

5.2 Further Recurrence Relations 214

5.3 Counting Inversions 221

5.4 Finding the Closest Pair of Points 225

5.5 Integer Multiplication 231

5.6 Convolutions and the Fast Fourier Transform

Solved Exercises 242

Exercises 246

Notes and Further Reading 249

Dynamic Programming

209

6

251

4.1 Interval Scheduling: The Greedy Algorithm Stays Ahead 116 4.2 Scheduling to Minimize Lateness: An Exchange Argument 125 4.3 Optimal Caching: A More Complex Exchange Argument 131 4.4 Shortest Paths in a Graph 137

4.5 The Minimum Spanning Tree Problem 142

4.6 Implementing Kruskal’s Algorithm: The Union-Find Data

Structure 151 4.7 Clustering 157

4.8 Huffman Codes and Data Compression 161

∗ 4.9 Minimum-Cost Arborescences: A Multi-Phase Greedy

6.1 Weighted Interval Scheduling: A Recursive Procedure

6.2 Principles of Dynamic Programming: Memoization or Iteration

over Subproblems 258

6.3 Segmented Least Squares: Multi-way Choices 261

210

234

252

115

∗ The star indicates an optional section. (See the Preface for more information about the relationships among the chapters and sections.)

8

Exercises 415

Notes and Further Reading 448

NP and Computational Intractability

8.1 Polynomial-Time Reductions 452

8.2 Reductions via “Gadgets”: The Satisfiability Problem

8.3 Efficient Certification and the Definition of NP 463

8.4 NP-Complete Problems 466

8.5 Sequencing Problems 473

8.6 Partitioning Problems 481

8.7 Graph Coloring 485

451

6.4 Subset Sums and Knapsacks: Adding a Variable 266

6.5 RNA Secondary Structure: Dynamic Programming over

Intervals 272

6.6 Sequence Alignment 278

6.7 Sequence Alignment in Linear Space via Divide and

Conquer 284

6.8 Shortest Paths in a Graph 290

6.9 Shortest Paths and Distance Vector Protocols

297

∗ 6.10

Negative Cycles in a Graph Solved Exercises 307 Exercises 312

Notes and Further Reading

301

335

7 Network Flow

337

7.1 The Maximum-Flow Problem and the Ford-Fulkerson Algorithm 338

7.2 Maximum Flows and Minimum Cuts in a Network 346

7.3 Choosing Good Augmenting Paths 352

∗ 7.4 The Preflow-Push Maximum-Flow Algorithm 357

7.5 A First Application: The Bipartite Matching Problem 367 7.6 Disjoint Paths in Directed and Undirected Graphs 373 7.7 Extensions to the Maximum-Flow Problem 378

7.8 Survey Design 384

7.9 Airline Scheduling 387

7.10 Image Segmentation

7.11 Project Selection 396 7.12 Baseball Elimination 400

391

∗ 7.13 A Further Direction: Adding Costs to the Matching Problem 404 Solved Exercises 411

Contents

ix

459

x

Contents

8.8 Numerical Problems 490

8.9 Co-NP and the Asymmetry of NP 495

8.10 A Partial Taxonomy of Hard Problems 497

Solved Exercises 500

Exercises 505

Notes and Further Reading 529

9 PSPACE: A Class of Problems beyond NP

9.1 PSPACE 531

9.2 Some Hard Problems in PSPACE 533

9.3 Solving Quantified Problems and Games in Polynomial

531

Space 536

9.4 Solving the Planning Problem in Polynomial Space

538

9.5 Proving Problems PSPACE-Complete Solved Exercises 547

Exercises 550

Notes and Further Reading 551

10 Extending the Limits of Tractability

10.1 Finding Small Vertex Covers 554

10.2 Solving NP-Hard Problems on Trees

10.3 Coloring a Set of Circular Arcs 563

543

558

584

553

∗ 10.4 ∗ 10.5

Tree Decompositions of Graphs 572 Constructing a Tree Decomposition Solved Exercises 591

Exercises 594

Notes and Further Reading 598

11 Approximation Algorithms

599

11.1 Greedy Algorithms and Bounds on the Optimum: A Load Balancing Problem 600

11.2 The Center Selection Problem 606

11.3 Set Cover: A General Greedy Heuristic 612

11.4 The Pricing Method: Vertex Cover 618

11.5 Maximization via the Pricing Method: The Disjoint Paths

Problem 624

11.6 Linear Programming and Rounding: An Application to Vertex

∗ 11.7

Cover 630

Load Balancing Revisited: A More Advanced LP Application 637

11.8 Arbitrarily Good Approximations: The Knapsack Problem 644 Solved Exercises 649

Exercises 651

Notes and Further Reading 659

12 Local Search

661

12.1 The Landscape of an Optimization Problem 662

12.2 The Metropolis Algorithm and Simulated Annealing 666

12.3 An Application of Local Search to Hopfield Neural Networks 671

12.4 Maximum-Cut Approximation via Local Search 676

12.5 Choosing a Neighbor Relation 679

∗ 12.6

12.7 Best-Response Dynamics and Nash Equilibria

Solved Exercises 700

Exercises 702

Notes and Further Reading 705

13 Randomized Algorithms

Classification via Local Search 681

13.1 A First Application: Contention Resolution

13.2 Finding the Global Minimum Cut 714

13.3 Random Variables and Their Expectations 719

13.4 A Randomized Approximation Algorithm for MAX 3-SAT 724

13.5 Randomized Divide and Conquer: Median-Finding and

Quicksort 727

13.6 Hashing: A Randomized Implementation of Dictionaries 734

13.7 Finding the Closest Pair of Points: A Randomized Approach 741

13.8 Randomized Caching 750

13.9 Chernoff Bounds 758

13.10 Load Balancing 760

13.11 Packet Routing 762

13.12 Background: Some Basic Probability Definitions

Solved Exercises 776 Exercises 782

Notes and Further Reading

Epilogue: Algorithms That Run Forever References

Index

793

708

690

769

Contents

xi

707

795 805 815

This page intentionally left blank

Preface

Algorithmic ideas are pervasive, and their reach is apparent in examples both within computer science and beyond. Some of the major shifts in Internet routing standards can be viewed as debates over the deficiencies of one shortest-path algorithm and the relative advantages of another. The basic notions used by biologists to express similarities among genes and genomes have algorithmic definitions. The concerns voiced by economists over the feasibility of combinatorial auctions in practice are rooted partly in the fact that these auctions contain computationally intractable search problems as special cases. And algorithmic notions aren’t just restricted to well-known and long- standing problems; one sees the reflections of these ideas on a regular basis, in novel issues arising across a wide range of areas. The scientist from Yahoo! who told us over lunch one day about their system for serving ads to users was describing a set of issues that, deep down, could be modeled as a network flow problem. So was the former student, now a management consultant working on staffing protocols for large hospitals, whom we happened to meet on a trip to New York City.

The point is not simply that algorithms have many applications. The deeper issue is that the subject of algorithms is a powerful lens through which to view the field of computer science in general. Algorithmic problems form the heart of computer science, but they rarely arrive as cleanly packaged, mathematically precise questions. Rather, they tend to come bundled together with lots of messy, application-specific detail, some of it essential, some of it extraneous. As a result, the algorithmic enterprise consists of two fundamental components: the task of getting to the mathematically clean core of a problem, and then the task of identifying the appropriate algorithm design techniques, based on the structure of the problem. These two components interact: the more comfortable one is with the full array of possible design techniques, the more one starts to recognize the clean formulations that lie within messy

xiv

Preface

problems out in the world. At their most effective, then, algorithmic ideas do not just provide solutions to well-posed problems; they form the language that lets you cleanly express the underlying questions.

The goal of our book is to convey this approach to algorithms, as a design process that begins with problems arising across the full range of computing applications, builds on an understanding of algorithm design techniques, and results in the development of efficient solutions to these problems. We seek to explore the role of algorithmic ideas in computer science generally, and relate these ideas to the range of precisely formulated problems for which we can design and analyze algorithms. In other words, what are the underlying issues that motivate these problems, and how did we choose these particular ways of formulating them? How did we recognize which design principles were appropriate in different situations?

In keeping with this, our goal is to offer advice on how to identify clean algorithmic problem formulations in complex issues from different areas of computing and, from this, how to design efficient algorithms for the resulting problems. Sophisticated algorithms are often best understood by reconstruct- ing the sequence of ideas—including false starts and dead ends—that led from simpler initial approaches to the eventual solution. The result is a style of ex- position that does not take the most direct route from problem statement to algorithm, but we feel it better reflects the way that we and our colleagues genuinely think about these questions.

Overview

The book is intended for students who have completed a programming- based two-semester introductory computer science sequence (the standard “CS1/CS2” courses) in which they have written programs that implement basic algorithms, manipulate discrete structures such as trees and graphs, and apply basic data structures such as arrays, lists, queues, and stacks. Since the interface between CS1/CS2 and a first algorithms course is not entirely standard, we begin the book with self-contained coverage of topics that at some institutions are familiar to students from CS1/CS2, but which at other institutions are included in the syllabi of the first algorithms course. This material can thus be treated either as a review or as new material; by including it, we hope the book can be used in a broader array of courses, and with more flexibility in the prerequisite knowledge that is assumed.

In keeping with the approach outlined above, we develop the basic algo- rithm design techniques by drawing on problems from across many areas of computer science and related fields. To mention a few representative examples here, we include fairly detailed discussions of applications from systems and networks (caching, switching, interdomain routing on the Internet), artificial

intelligence (planning, game playing, Hopfield networks), computer vision (image segmentation), data mining (change-point detection, clustering), op- erations research (airline scheduling), and computational biology (sequence alignment, RNA secondary structure).

The notion of computational intractability, and NP-completeness in par- ticular, plays a large role in the book. This is consistent with how we think about the overall process of algorithm design. Some of the time, an interest- ing problem arising in an application area will be amenable to an efficient solution, and some of the time it will be provably NP-complete; in order to fully address a new algorithmic problem, one should be able to explore both of these options with equal familiarity. Since so many natural problems in computer science are NP-complete, the development of methods to deal with intractable problems has become a crucial issue in the study of algorithms, and our book heavily reflects this theme. The discovery that a problem is NP- complete should not be taken as the end of the story, but as an invitation to begin looking for approximation algorithms, heuristic local search techniques, or tractable special cases. We include extensive coverage of each of these three approaches.

Problems and Solved Exercises

An important feature of the book is the collection of problems. Across all chapters, the book includes over 200 problems, almost all of them developed and class-tested in homework or exams as part of our teaching of the course at Cornell. We view the problems as a crucial component of the book, and they are structured in keeping with our overall approach to the material. Most of them consist of extended verbal descriptions of a problem arising in an application area in computer science or elsewhere out in the world, and part of the problem is to practice what we discuss in the text: setting up the necessary notation and formalization, designing an algorithm, and then analyzing it and proving it correct. (We view a complete answer to one of these problems as consisting of all these components: a fully explained algorithm, an analysis of the running time, and a proof of correctness.) The ideas for these problems come in large part from discussions we have had over the years with people working in different areas, and in some cases they serve the dual purpose of recording an interesting (though manageable) application of algorithms that we haven’t seen written down anywhere else.

To help with the process of working on these problems, we include in each chapter a section entitled “Solved Exercises,” where we take one or more problems and describe how to go about formulating a solution. The discussion devoted to each solved exercise is therefore significantly longer than what would be needed simply to write a complete, correct solution (in other words,

Preface

xv

xvi

Preface

significantly longer than what it would take to receive full credit if these were being assigned as homework problems). Rather, as with the rest of the text, the discussions in these sections should be viewed as trying to give a sense of the larger process by which one might think about problems of this type, culminating in the specification of a precise solution.

It is worth mentioning two points concerning the use of these problems as homework in a course. First, the problems are sequenced roughly in order of increasing difficulty, but this is only an approximate guide and we advise against placing too much weight on it: since the bulk of the problems were designed as homework for our undergraduate class, large subsets of the problems in each chapter are really closely comparable in terms of difficulty. Second, aside from the lowest-numbered ones, the problems are designed to involve some investment of time, both to relate the problem description to the algorithmic techniques in the chapter, and then to actually design the necessary algorithm. In our undergraduate class, we have tended to assign roughly three of these problems per week.

Pedagogical Features and Supplements

In addition to the problems and solved exercises, the book has a number of further pedagogical features, as well as additional supplements to facilitate its use for teaching.

As noted earlier, a large number of the sections in the book are devoted to the formulation of an algorithmic problem—including its background and underlying motivation—and the design and analysis of an algorithm for this problem. To reflect this style, these sections are consistently structured around a sequence of subsections: “The Problem,” where the problem is described and a precise formulation is worked out; “Designing the Algorithm,” where the appropriate design technique is employed to develop an algorithm; and “Analyzing the Algorithm,” which proves properties of the algorithm and analyzes its efficiency. These subsections are highlighted in the text with an icon depicting a feather. In cases where extensions to the problem or further analysis of the algorithm is pursued, there are additional subsections devoted to these issues. The goal of this structure is to offer a relatively uniform style of presentation that moves from the initial discussion of a problem arising in a computing application through to the detailed analysis of a method to solve it.

A number of supplements are available in support of the book itself. An instructor’s manual works through all the problems, providing full solutions to each. A set of lecture slides, developed by Kevin Wayne of Princeton University, is also available; these slides follow the order of the book’s sections and can thus be used as the foundation for lectures in a course based on the book. These files are available at www.aw.com. For instructions on obtaining a professor

login and password, search the site for either “Kleinberg” or “Tardos” or contact your local Addison-Wesley representative.

Finally, we would appreciate receiving feedback on the book. In particular, as in any book of this length, there are undoubtedly errors that have remained in the final version. Comments and reports of errors can be sent to us by e-mail, at the address algbook@cs.cornell.edu; please include the word “feedback” in the subject line of the message.

Chapter-by-Chapter Synopsis

Chapter 1 starts by introducing some representative algorithmic problems. We begin immediately with the Stable Matching Problem, since we feel it sets up the basic issues in algorithm design more concretely and more elegantly than any abstract discussion could: stable matching is motivated by a natural though complex real-world issue, from which one can abstract an interesting problem statement and a surprisingly effective algorithm to solve this problem. The remainder of Chapter 1 discusses a list of five “representative problems” that foreshadow topics from the remainder of the course. These five problems are interrelated in the sense that they are all variations and/or special cases of the Independent Set Problem; but one is solvable by a greedy algorithm, one by dynamic programming, one by network flow, one (the Independent Set Problem itself) is NP-complete, and one is PSPACE-complete. The fact that closely related problems can vary greatly in complexity is an important theme of the book, and these five problems serve as milestones that reappear as the book progresses.

Chapters 2 and 3 cover the interface to the CS1/CS2 course sequence mentioned earlier. Chapter 2 introduces the key mathematical definitions and notations used for analyzing algorithms, as well as the motivating principles behind them. It begins with an informal overview of what it means for a prob- lem to be computationally tractable, together with the concept of polynomial time as a formal notion of efficiency. It then discusses growth rates of func- tions and asymptotic analysis more formally, and offers a guide to commonly occurring functions in algorithm analysis, together with standard applications in which they arise. Chapter 3 covers the basic definitions and algorithmic primitives needed for working with graphs, which are central to so many of the problems in the book. A number of basic graph algorithms are often im- plemented by students late in the CS1/CS2 course sequence, but it is valuable to present the material here in a broader algorithm design context. In par- ticular, we discuss basic graph definitions, graph traversal techniques such as breadth-first search and depth-first search, and directed graph concepts including strong connectivity and topological ordering.

Preface

xvii

xviii

Preface

Chapters 2 and 3 also present many of the basic data structures that will be used for implementing algorithms throughout the book; more advanced data structures are presented in subsequent chapters. Our approach to data structures is to introduce them as they are needed for the implementation of the algorithms being developed in the book. Thus, although many of the data structures covered here will be familiar to students from the CS1/CS2 sequence, our focus is on these data structures in the broader context of algorithm design and analysis.

Chapters 4 through 7 cover four major algorithm design techniques: greedy algorithms, divide and conquer, dynamic programming, and network flow. With greedy algorithms, the challenge is to recognize when they work and when they don’t; our coverage of this topic is centered around a way of clas- sifying the kinds of arguments used to prove greedy algorithms correct. This chapter concludes with some of the main applications of greedy algorithms, for shortest paths, undirected and directed spanning trees, clustering, and compression. For divide and conquer, we begin with a discussion of strategies for solving recurrence relations as bounds on running times; we then show how familiarity with these recurrences can guide the design of algorithms that improve over straightforward approaches to a number of basic problems, in- cluding the comparison of rankings, the computation of closest pairs of points in the plane, and the Fast Fourier Transform. Next we develop dynamic pro- gramming by starting with the recursive intuition behind it, and subsequently building up more and more expressive recurrence formulations through appli- cations in which they naturally arise. This chapter concludes with extended discussions of the dynamic programming approach to two fundamental prob- lems: sequence alignment, with applications in computational biology; and shortest paths in graphs, with connections to Internet routing protocols. Fi- nally, we cover algorithms for network flow problems, devoting much of our focus in this chapter to discussing a large array of different flow applications. To the extent that network flow is covered in algorithms courses, students are often left without an appreciation for the wide range of problems to which it can be applied; we try to do justice to its versatility by presenting applications to load balancing, scheduling, image segmentation, and a number of other problems.

Chapters 8 and 9 cover computational intractability. We devote most of our attention to NP-completeness, organizing the basic NP-complete problems thematically to help students recognize candidates for reductions when they encounter new problems. We build up to some fairly complex proofs of NP- completeness, with guidance on how one goes about constructing a difficult reduction. We also consider types of computational hardness beyond NP- completeness, particularly through the topic of PSPACE-completeness. We

find this is a valuable way to emphasize that intractability doesn’t end at NP-completeness, and PSPACE-completeness also forms the underpinning for some central notions from artificial intelligence—planning and game playing— that would otherwise not find a place in the algorithmic landscape we are surveying.

Chapters 10 through 12 cover three major techniques for dealing with com- putationally intractable problems: identification of structured special cases, approximation algorithms, and local search heuristics. Our chapter on tractable special cases emphasizes that instances of NP-complete problems arising in practice may not be nearly as hard as worst-case instances, because they often contain some structure that can be exploited in the design of an efficient algo- rithm. We illustrate how NP-complete problems are often efficiently solvable when restricted to tree-structured inputs, and we conclude with an extended discussion of tree decompositions of graphs. While this topic is more suit- able for a graduate course than for an undergraduate one, it is a technique with considerable practical utility for which it is hard to find an existing accessible reference for students. Our chapter on approximation algorithms discusses both the process of designing effective algorithms and the task of understanding the optimal solution well enough to obtain good bounds on it. As design techniques for approximation algorithms, we focus on greedy algo- rithms, linear programming, and a third method we refer to as “pricing,” which incorporates ideas from each of the first two. Finally, we discuss local search heuristics, including the Metropolis algorithm and simulated annealing. This topic is often missing from undergraduate algorithms courses, because very little is known in the way of provable guarantees for these algorithms; how- ever, given their widespread use in practice, we feel it is valuable for students to know something about them, and we also include some cases in which guarantees can be proved.

Chapter 13 covers the use of randomization in the design of algorithms. This is a topic on which several nice graduate-level books have been written. Our goal here is to provide a more compact introduction to some of the ways in which students can apply randomized techniques using the kind of background in probability one typically gains from an undergraduate discrete math course.

Use of the Book

The book is primarily designed for use in a first undergraduate course on algorithms, but it can also be used as the basis for an introductory graduate course.

When we use the book at the undergraduate level, we spend roughly one lecture per numbered section; in cases where there is more than one

Preface

xix

xx

Preface

lecture’s worth of material in a section (for example, when a section provides further applications as additional examples), we treat this extra material as a supplement that students can read about outside of lecture. We skip the starred sections; while these sections contain important topics, they are less central to the development of the subject, and in some cases they are harder as well. We also tend to skip one or two other sections per chapter in the first half of the book (for example, we tend to skip Sections 4.3, 4.7–4.8, 5.5–5.6, 6.5, 7.6, and 7.11). We cover roughly half of each of Chapters 11–13.

This last point is worth emphasizing: rather than viewing the later chapters as “advanced,” and hence off-limits to undergraduate algorithms courses, we have designed them with the goal that the first few sections of each should be accessible to an undergraduate audience. Our own undergraduate course involves material from all these chapters, as we feel that all of these topics have an important place at the undergraduate level.

Finally, we treat Chapters 2 and 3 primarily as a review of material from earlier courses; but, as discussed above, the use of these two chapters depends heavily on the relationship of each specific course to its prerequisites.

The resulting syllabus looks roughly as follows: Chapter 1; Chapters 4–8 (excluding 4.3, 4.7–4.9, 5.5–5.6, 6.5, 6.10, 7.4, 7.6, 7.11, and 7.13); Chapter 9 (briefly); Chapter 10, Sections.10.1 and 10.2; Chapter 11, Sections 11.1, 11.2, 11.6, and 11.8; Chapter 12, Sections 12.1–12.3; and Chapter 13, Sections 13.1– 13.5.

The book also naturally supports an introductory graduate course on algorithms. Our view of such a course is that it should introduce students destined for research in all different areas to the important current themes in algorithm design. Here we find the emphasis on formulating problems to be useful as well, since students will soon be trying to define their own research problems in many different subfields. For this type of course, we cover the later topics in Chapters 4 and 6 (Sections 4.5–4.9 and 6.5–6.10), cover all of Chapter 7 (moving more rapidly through the early sections), quickly cover NP- completeness in Chapter 8 (since many beginning graduate students will have seen this topic as undergraduates), and then spend the remainder of the time on Chapters 10–13. Although our focus in an introductory graduate course is on the more advanced sections, we find it useful for the students to have the full book to consult for reviewing or filling in background knowledge, given the range of different undergraduate backgrounds among the students in such a course.

Finally, the book can be used to support self-study by graduate students, researchers, or computer professionals who want to get a sense for how they

might be able to use particular algorithm design techniques in the context of their own work. A number of graduate students and colleagues have used portions of the book in this way.

Acknowledgments

This book grew out of the sequence of algorithms courses that we have taught at Cornell. These courses have grown, as the field has grown, over a number of years, and they reflect the influence of the Cornell faculty who helped to shape them during this time, including Juris Hartmanis, Monika Henzinger, John Hopcroft, Dexter Kozen, Ronitt Rubinfeld, and Sam Toueg. More generally, we would like to thank all our colleagues at Cornell for countless discussions both on the material here and on broader issues about the nature of the field.

The course staffs we’ve had in teaching the subject have been tremen- dously helpful in the formulation of this material. We thank our undergradu- ate and graduate teaching assistants, Siddharth Alexander, Rie Ando, Elliot Anshelevich, Lars Backstrom, Steve Baker, Ralph Benzinger, John Bicket, Doug Burdick, Mike Connor, Vladimir Dizhoor, Shaddin Doghmi, Alexan- der Druyan, Bowei Du, Sasha Evfimievski, Ariful Gani, Vadim Grinshpun, Ara Hayrapetyan, Chris Jeuell, Igor Kats, Omar Khan, Mikhail Kobyakov, Alexei Kopylov, Brian Kulis, Amit Kumar, Yeongwee Lee, Henry Lin, Ash- win Machanavajjhala, Ayan Mandal, Bill McCloskey, Leonid Meyerguz, Evan Moran, Niranjan Nagarajan, Tina Nolte, Travis Ortogero, Martin Pa ́l, Jon Peress, Matt Piotrowski, Joe Polastre, Mike Priscott, Xin Qi, Venu Ramasubra- manian, Aditya Rao, David Richardson, Brian Sabino, Rachit Siamwalla, Se- bastian Silgardo, Alex Slivkins, Chaitanya Swamy, Perry Tam, Nadya Travinin, Sergei Vassilvitskii, Matthew Wachs, Tom Wexler, Shan-Leung Maverick Woo, Justin Yang, and Misha Zatsman. Many of them have provided valuable in- sights, suggestions, and comments on the text. We also thank all the students in these classes who have provided comments and feedback on early drafts of the book over the years.

For the past several years, the development of the book has benefited greatly from the feedback and advice of colleagues who have used prepubli- cation drafts for teaching. Anna Karlin fearlessly adopted a draft as her course textbook at the University of Washington when it was still in an early stage of development; she was followed by a number of people who have used it either as a course textbook or as a resource for teaching: Paul Beame, Allan Borodin, Devdatt Dubhashi, David Kempe, Gene Kleinberg, Dexter Kozen, Amit Kumar, Mike Molloy, Yuval Rabani, Tim Roughgarden, Alexa Sharp, Shanghua Teng, Aravind Srinivasan, Dieter van Melkebeek, Kevin Wayne, Tom Wexler, and

Preface

xxi

xxii

Preface

Sue Whitesides. We deeply appreciate their input and advice, which has in- formed many of our revisions to the content. We would like to additionally thank Kevin Wayne for producing supplementary material associated with the book, which promises to greatly extend its utility to future instructors.

In a number of other cases, our approach to particular topics in the book reflects the infuence of specific colleagues. Many of these contributions have undoubtedly escaped our notice, but we especially thank Yuri Boykov, Ron Elber, Dan Huttenlocher, Bobby Kleinberg, Evie Kleinberg, Lillian Lee, David McAllester, Mark Newman, Prabhakar Raghavan, Bart Selman, David Shmoys, Steve Strogatz, Olga Veksler, Duncan Watts, and Ramin Zabih.

It has been a pleasure working with Addison Wesley over the past year. First and foremost, we thank Matt Goldstein for all his advice and guidance in this process, and for helping us to synthesize a vast amount of review material into a concrete plan that improved the book. Our early conversations about the book with Susan Hartman were extremely valuable as well. We thank Matt and Susan, together with Michelle Brown, Marilyn Lloyd, Patty Mahtani, and Maite Suarez-Rivas at Addison Wesley, and Paul Anagnostopoulos and Jacqui Scarlott at Windfall Software, for all their work on the editing, production, and management of the project. We further thank Paul and Jacqui for their expert composition of the book. We thank Joyce Wells for the cover design, Nancy Murphy of Dartmouth Publishing for her work on the figures, Ted Laux for the indexing, and Carol Leyba and Jennifer McClain for the copyediting and proofreading.

We thank Anselm Blumer (Tufts University), Richard Chang (University of Maryland, Baltimore County), Kevin Compton (University of Michigan), Diane Cook (University of Texas, Arlington), Sariel Har-Peled (University of Illinois, Urbana-Champaign), Sanjeev Khanna (University of Pennsylvania), Philip Klein (Brown University), David Matthias (Ohio State University), Adam Mey- erson (UCLA), Michael Mitzenmacher (Harvard University), Stephan Olariu (Old Dominion University), Mohan Paturi (UC San Diego), Edgar Ramos (Uni- versity of Illinois, Urbana-Champaign), Sanjay Ranka (University of Florida, Gainesville), Leon Reznik (Rochester Institute of Technology), Subhash Suri (UC Santa Barbara), Dieter van Melkebeek (University of Wisconsin, Madi- son), and Bulent Yener (Rensselaer Polytechnic Institute) who generously contributed their time to provide detailed and thoughtful reviews of the man- uscript; their comments led to numerous improvements, both large and small, in the final version of the text.

Finally, we thank our families—Lillian and Alice, and David, Rebecca, and Amy. We appreciate their support, patience, and many other contributions more than we can express in any acknowledgments here.

This book was begun amid the irrational exuberance of the late nineties, when the arc of computing technology seemed, to many of us, briefly to pass through a place traditionally occupied by celebrities and other inhabitants of the pop-cultural firmament. (It was probably just in our imaginations.) Now, several years after the hype and stock prices have come back to earth, one can appreciate that in some ways computer science was forever changed by this period, and in other ways it has remained the same: the driving excitement that has characterized the field since its early days is as strong and enticing as ever, the public’s fascination with information technology is still vibrant, and the reach of computing continues to extend into new disciplines. And so to all students of the subject, drawn to it for so many different reasons, we hope you find this book an enjoyable and useful guide wherever your computational pursuits may take you.

Jon Kleinberg E ́va Tardos Ithaca, 2005

Preface

xxiii

This page intentionally left blank

Chapter 1

Introduction: Some Representative Problems

1.1 A First Problem: Stable Matching

As an opening topic, we look at an algorithmic problem that nicely illustrates many of the themes we will be emphasizing. It is motivated by some very natural and practical concerns, and from these we formulate a clean and simple statement of a problem. The algorithm to solve the problem is very clean as well, and most of our work will be spent in proving that it is correct and giving an acceptable bound on the amount of time it takes to terminate with an answer. The problem itself—the Stable Matching Problem—has several origins.

The Problem

The Stable Matching Problem originated, in part, in 1962, when David Gale and Lloyd Shapley, two mathematical economists, asked the question: Could one design a college admissions process, or a job recruiting process, that was self-enforcing? What did they mean by this?

To set up the question, let’s first think informally about the kind of situation that might arise as a group of friends, all juniors in college majoring in computer science, begin applying to companies for summer internships. The crux of the application process is the interplay between two different types of parties: companies (the employers) and students (the applicants). Each applicant has a preference ordering on companies, and each company—once the applications come in—forms a preference ordering on its applicants. Based on these preferences, companies extend offers to some of their applicants, applicants choose which of their offers to accept, and people begin heading off to their summer internships.

2

Chapter 1 Introduction: Some Representative Problems

Gale and Shapley considered the sorts of things that could start going wrong with this process, in the absence of any mechanism to enforce the status quo. Suppose, for example, that your friend Raj has just accepted a summer job at the large telecommunications company CluNet. A few days later, the small start-up company WebExodus, which had been dragging its feet on making a few final decisions, calls up Raj and offers him a summer job as well. Now, Raj actually prefers WebExodus to CluNet—won over perhaps by the laid-back, anything-can-happen atmosphere—and so this new development may well cause him to retract his acceptance of the CluNet offer and go to WebExodus instead. Suddenly down one summer intern, CluNet offers a job to one of its wait-listed applicants, who promptly retracts his previous acceptance of an offer from the software giant Babelsoft, and the situation begins to spiral out of control.

Things look just as bad, if not worse, from the other direction. Suppose that Raj’s friend Chelsea, destined to go to Babelsoft but having just heard Raj’s story, calls up the people at WebExodus and says, “You know, I’d really rather spend the summer with you guys than at Babelsoft.” They find this very easy to believe; and furthermore, on looking at Chelsea’s application, they realize that they would have rather hired her than some other student who actually is scheduled to spend the summer at WebExodus. In this case, if WebExodus were a slightly less scrupulous company, it might well find some way to retract its offer to this other student and hire Chelsea instead.

Situations like this can rapidly generate a lot of chaos, and many people— both applicants and employers—can end up unhappy with the process as well as the outcome. What has gone wrong? One basic problem is that the process is not self-enforcing—if people are allowed to act in their self-interest, then it risks breaking down.

We might well prefer the following, more stable situation, in which self- interest itself prevents offers from being retracted and redirected. Consider another student, who has arranged to spend the summer at CluNet but calls up WebExodus and reveals that he, too, would rather work for them. But in this case, based on the offers already accepted, they are able to reply, “No, it turns out that we prefer each of the students we’ve accepted to you, so we’re afraid there’s nothing we can do.” Or consider an employer, earnestly following up with its top applicants who went elsewhere, being told by each of them, “No, I’m happy where I am.” In such a case, all the outcomes are stable—there are no further outside deals that can be made.

So this is the question Gale and Shapley asked: Given a set of preferences among employers and applicants, can we assign applicants to employers so that for every employer E, and every applicant A who is not scheduled to work for E, at least one of the following two things is the case?

(i) E prefers every one of its accepted applicants to A; or

(ii) A prefers her current situation over working for employer E.

If this holds, the outcome is stable: individual self-interest will prevent any applicant/employer deal from being made behind the scenes.

Gale and Shapley proceeded to develop a striking algorithmic solution to this problem, which we will discuss presently. Before doing this, let’s note that this is not the only origin of the Stable Matching Problem. It turns out that for a decade before the work of Gale and Shapley, unbeknownst to them, the National Resident Matching Program had been using a very similar procedure, with the same underlying motivation, to match residents to hospitals. Indeed, this system, with relatively little change, is still in use today.

This is one testament to the problem’s fundamental appeal. And from the point of view of this book, it provides us with a nice first domain in which to reason about some basic combinatorial definitions and the algorithms that build on them.

Formulating the Problem To get at the essence of this concept, it helps to make the problem as clean as possible. The world of companies and applicants contains some distracting asymmetries. Each applicant is looking for a single company, but each company is looking for many applicants; moreover, there may be more (or, as is sometimes the case, fewer) applicants than there are available slots for summer jobs. Finally, each applicant does not typically apply to every company.

It is useful, at least initially, to eliminate these complications and arrive at a more “bare-bones” version of the problem: each of n applicants applies to each of n companies, and each company wants to accept a single applicant. We will see that doing this preserves the fundamental issues inherent in the problem; in particular, our solution to this simplified version will extend directly to the more general case as well.

Following Gale and Shapley, we observe that this special case can be viewed as the problem of devising a system by which each of n men and n women can end up getting married: our problem naturally has the analogue of two “genders”—the applicants and the companies—and in the case we are considering, everyone is seeking to be paired with exactly one individual of the opposite gender.1

1 Gale and Shapley considered the same-sex Stable Matching Problem as well, where there is only a single gender. This is motivated by related applications, but it turns out to be fairly different at a technical level. Given the applicant-employer application we’re considering here, we’ll be focusing on the version with two genders.

1.1 A First Problem: Stable Matching

3

4

Chapter 1 Introduction: Some Representative Problems

An instability: m and w each prefer the other to their current partners.

mw m w

Figure 1.1 Perfect matching S with instability (m, w′).

So consider a set M = {m1, . . . , mn} of n men, and a set W = {w1, . . . , wn} of n women. Let M × W denote the set of all possible ordered pairs of the form (m, w), where m ∈ M and w ∈ W. A matching S is a set of ordered pairs, each from M × W, with the property that each member of M and each member of W appears in at most one pair in S. A perfect matching S′ is a matching with the property that each member of M and each member of W appears in exactly one pair in S′.

Matchings and perfect matchings are objects that will recur frequently throughout the book; they arise naturally in modeling a wide range of algo- rithmic problems. In the present situation, a perfect matching corresponds simply to a way of pairing off the men with the women, in such a way that everyone ends up married to somebody, and nobody is married to more than one person—there is neither singlehood nor polygamy.

Now we can add the notion of preferences to this setting. Each man m ∈ M ranks all the women; we will say that m prefers w to w′ if m ranks w higher than w′. We will refer to the ordered ranking of m as his preference list. We will not allow ties in the ranking. Each woman, analogously, ranks all the men.

Given a perfect matching S, what can go wrong? Guided by our initial motivation in terms of employers and applicants, we should be worried about the following situation: There are two pairs (m,w) and (m′,w′) in S (as depicted in Figure 1.1) with the property that m prefers w′ to w, and w′ prefers m to m′. In this case, there’s nothing to stop m and w′ from abandoning their current partners and heading off together; the set of marriages is not self- enforcing. We’ll say that such a pair (m, w′) is an instability with respect to S: (m, w′) does not belong to S, but each of m and w′ prefers the other to their partner in S.

Our goal, then, is a set of marriages with no instabilities. We’ll say that a matching S is stable if (i) it is perfect, and (ii) there is no instability with respect to S. Two questions spring immediately to mind:

. Does there exist a stable matching for every set of preference lists?

. Given a set of preference lists, can we efficiently construct a stable

matching if there is one?

Some Examples To illustrate these definitions, consider the following two very simple instances of the Stable Matching Problem.

First, suppose we have a set of two men, {m, m′}, and a set of two women, {w, w′}. The preference lists are as follows:

m prefers w to w′. m′ prefers w to w′.

w prefers m to m′. w′ prefers m to m′.

If we think about this set of preference lists intuitively, it represents complete agreement: the men agree on the order of the women, and the women agree on the order of the men. There is a unique stable matching here, consisting of the pairs (m, w) and (m′, w′). The other perfect matching, consisting of the pairs (m′, w) and (m, w′), would not be a stable matching, because the pair (m, w) would form an instability with respect to this matching. (Both m and w would want to leave their respective partners and pair up.)

Next, here’s an example where things are a bit more intricate. Suppose the preferences are

m prefers w to w′. m′ prefers w′ to w. w prefers m′ to m. w′ prefers m to m′.

What’s going on in this case? The two men’s preferences mesh perfectly with each other (they rank different women first), and the two women’s preferences likewise mesh perfectly with each other. But the men’s preferences clash completely with the women’s preferences.

In this second example, there are two different stable matchings. The matching consisting of the pairs (m, w) and (m′, w′) is stable, because both men are as happy as possible, so neither would leave their matched partner. But the matching consisting of the pairs (m′, w) and (m, w′) is also stable, for the complementary reason that both women are as happy as possible. This is an important point to remember as we go forward—it’s possible for an instance to have more than one stable matching.

Designing the Algorithm

We now show that there exists a stable matching for every set of preference lists among the men and women. Moreover, our means of showing this will also answer the second question that we asked above: we will give an efficient algorithm that takes the preference lists and constructs a stable matching.

Let us consider some of the basic ideas that motivate the algorithm.

. Initially, everyone is unmarried. Suppose an unmarried man m chooses the woman w who ranks highest on his preference list and proposes to her. Can we declare immediately that (m, w) will be one of the pairs in our final stable matching? Not necessarily: at some point in the future, a man m′ whom w prefers may propose to her. On the other hand, it would be

1.1 A First Problem: Stable Matching

5

6

Chapter 1 Introduction: Some Representative Problems

Woman w will become engaged to m if she prefers him to m.

dangerous for w to reject m right away; she may never receive a proposal from someone she ranks as highly as m. So a natural idea would be to have the pair (m, w) enter an intermediate state—engagement.

. Suppose we are now at a state in which some men and women are free— not engaged—and some are engaged. The next step could look like this. An arbitrary free man m chooses the highest-ranked woman w to whom he has not yet proposed, and he proposes to her. If w is also free, then m and w become engaged. Otherwise, w is already engaged to some other man m′. In this case, she determines which of m or m′ ranks higher on her preference list; this man becomes engaged to w and the other becomes free.

. Finally, the algorithm will terminate when no one is free; at this moment, all engagements are declared final, and the resulting perfect matching is returned.

Here is a concrete description of the Gale-Shapley algorithm, with Fig- ure 1.2 depicting a state of the algorithm.

Initially all m∈M and w∈W are free

While there is a man m who is free and hasn’t proposed to every woman

Choose such a man m

Let w be the highest-ranked woman in m’s preference list

to whom m has not yet proposed If w is free then

(m, w) become engaged

Else w is currently engaged to m′

If w prefers m′ to m then m remains free

Else w prefers m to m′ (m, w) become engaged m′ becomes free

Endif Endif

Endwhile

Return the set S of engaged pairs

An intriguing thing is that, although the G-S algorithm is quite simple to state, it is not immediately obvious that it returns a stable matching, or even a perfect matching. We proceed to prove this now, through a sequence of intermediate facts.

mw m

Figure 1.2 An intermediate state of the G-S algorithm when a free man m is propos- ing to a woman w.

Analyzing the Algorithm

First consider the view of a woman w during the execution of the algorithm. For a while, no one has proposed to her, and she is free. Then a man m may propose to her, and she becomes engaged. As time goes on, she may receive additional proposals, accepting those that increase the rank of her partner. So we discover the following.

(1.1) w remains engaged from the point at which she receives her first proposal; and the sequence of partners to which she is engaged gets better and better (in terms of her preference list).

The view of a man m during the execution of the algorithm is rather different. He is free until he proposes to the highest-ranked woman on his list; at this point he may or may not become engaged. As time goes on, he may alternate between being free and being engaged; however, the following property does hold.

(1.2) The sequence of women to whom m proposes gets worse and worse (in terms of his preference list).

Now we show that the algorithm terminates, and give a bound on the maximum number of iterations needed for termination.

(1.3) The G-S algorithm terminates after at most n2 iterations of the While loop.

Proof. Ausefulstrategyforupper-boundingtherunningtimeofanalgorithm, as we are trying to do here, is to find a measure of progress. Namely, we seek some precise way of saying that each step taken by the algorithm brings it closer to termination.

In the case of the present algorithm, each iteration consists of some man proposing (for the only time) to a woman he has never proposed to before. So if we let P(t) denote the set of pairs (m, w) such that m has proposed to w by the end of iteration t, we see that for all t, the size of P(t + 1) is strictly greater than the size of P(t). But there are only n2 possible pairs of men and women in total, so the value of P(·) can increase at most n2 times over the course of the algorithm. It follows that there can be at most n2 iterations.

Two points are worth noting about the previous fact and its proof. First, there are executions of the algorithm (with certain preference lists) that can involve close to n2 iterations, so this analysis is not far from the best possible. Second, there are many quantities that would not have worked well as a progress measure for the algorithm, since they need not strictly increase in each

1.1 A First Problem: Stable Matching

7

8

Chapter 1 Introduction: Some Representative Problems

iteration. For example, the number of free individuals could remain constant from one iteration to the next, as could the number of engaged pairs. Thus, these quantities could not be used directly in giving an upper bound on the maximum possible number of iterations, in the style of the previous paragraph.

Let us now establish that the set S returned at the termination of the algorithm is in fact a perfect matching. Why is this not immediately obvious? Essentially, we have to show that no man can “fall off” the end of his preference list; the only way for the While loop to exit is for there to be no free man. In this case, the set of engaged couples would indeed be a perfect matching.

So the main thing we need to show is the following.

(1.4) If m is free at some point in the execution of the algorithm, then there is a woman to whom he has not yet proposed.

Proof. Suppose there comes a point when m is free but has already proposed to every woman. Then by (1.1), each of the n women is engaged at this point in time. Since the set of engaged pairs forms a matching, there must also be n engaged men at this point in time. But there are only n men total, and m is not engaged, so this is a contradiction.

(1.5) The set S returned at termination is a perfect matching.

Proof. The set of engaged pairs always forms a matching. Let us suppose that the algorithm terminates with a free man m. At termination, it must be the case that m had already proposed to every woman, for otherwise the While loop would not have exited. But this contradicts (1.4), which says that there cannot be a free man who has proposed to every woman.

Finally, we prove the main property of the algorithm—namely, that it results in a stable matching.

(1.6) Consider an execution of the G-S algorithm that returns a set of pairs S. The set S is a stable matching.

Proof. We have already seen, in (1.5), that S is a perfect matching. Thus, to prove S is a stable matching, we will assume that there is an instability with respect to S and obtain a contradiction. As defined earlier, such an instability would involve two pairs, (m, w) and (m′, w′), in S with the properties that

. m prefers w′ to w, and

. w′ prefers m to m′.

In the execution of the algorithm that produced S, m’s last proposal was, by definition, to w. Now we ask: Did m propose to w′ at some earlier point in

this execution? If he didn’t, then w must occur higher on m’s preference list than w′, contradicting our assumption that m prefers w′ to w. If he did, then he was rejected by w′ in favor of some other man m′′, whom w′ prefers to m. m′ is the final partner of w′, so either m′′ = m′ or, by (1.1), w′ prefers her final partner m′ to m′′; either way this contradicts our assumption that w′ prefers m to m′.

It follows that S is a stable matching. Extensions

We began by defining the notion of a stable matching; we have just proven that the G-S algorithm actually constructs one. We now consider some further questions about the behavior of the G-S algorithm and its relation to the properties of different stable matchings.

To begin with, recall that we saw an example earlier in which there could be multiple stable matchings. To recap, the preference lists in this example were as follows:

m prefers w to w′. m′ prefers w′ to w. w prefers m′ to m. w′ prefers m to m′.

Now, in any execution of the Gale-Shapley algorithm, m will become engaged to w, m′ will become engaged to w′ (perhaps in the other order), and things will stop there. Thus, the other stable matching, consisting of the pairs (m′, w) and (m, w′), is not attainable from an execution of the G-S algorithm in which the men propose. On the other hand, it would be reached if we ran a version of the algorithm in which the women propose. And in larger examples, with more than two people on each side, we can have an even larger collection of possible stable matchings, many of them not achievable by any natural algorithm.

This example shows a certain “unfairness” in the G-S algorithm, favoring men. If the men’s preferences mesh perfectly (they all list different women as their first choice), then in all runs of the G-S algorithm all men end up matched with their first choice, independent of the preferences of the women. If the women’s preferences clash completely with the men’s preferences (as was the case in this example), then the resulting stable matching is as bad as possible for the women. So this simple set of preference lists compactly summarizes a world in which someone is destined to end up unhappy: women are unhappy if men propose, and men are unhappy if women propose.

Let’s now analyze the G-S algorithm in more detail and try to understand how general this “unfairness” phenomenon is.

1.1 A First Problem: Stable Matching

9

10

Chapter 1 Introduction: Some Representative Problems

To begin with, our example reinforces the point that the G-S algorithm is actually underspecified: as long as there is a free man, we are allowed to choose any free man to make the next proposal. Different choices specify different executions of the algorithm; this is why, to be careful, we stated (1.6) as “Consider an execution of the G-S algorithm that returns a set of pairs S,” instead of “Consider the set S returned by the G-S algorithm.”

Thus, we encounter another very natural question: Do all executions of the G-S algorithm yield the same matching? This is a genre of question that arises in many settings in computer science: we have an algorithm that runs asynchronously, with different independent components performing actions that can be interleaved in complex ways, and we want to know how much variability this asynchrony causes in the final outcome. To consider a very different kind of example, the independent components may not be men and women but electronic components activating parts of an airplane wing; the effect of asynchrony in their behavior can be a big deal.

In the present context, we will see that the answer to our question is surprisingly clean: all executions of the G-S algorithm yield the same matching. We proceed to prove this now.

All Executions Yield the Same Matching There are a number of possible ways to prove a statement such as this, many of which would result in quite complicated arguments. It turns out that the easiest and most informative ap- proach for us will be to uniquely characterize the matching that is obtained and then show that all executions result in the matching with this characterization.

What is the characterization? We’ll show that each man ends up with the “best possible partner” in a concrete sense. (Recall that this is true if all men prefer different women.) First, we will say that a woman w is a valid partner of a man m if there is a stable matching that contains the pair (m, w). We will say that w is the best valid partner of m if w is a valid partner of m, and no woman whom m ranks higher than w is a valid partner of his. We will use best(m) to denote the best valid partner of m.

Now, let S∗ denote the set of pairs {(m, best(m)) : m ∈ M}. We will prove the following fact.

(1.7) Every execution of the G-S algorithm results in the set S∗.

This statement is surprising at a number of levels. First of all, as defined, there is no reason to believe that S∗ is a matching at all, let alone a stable matching. After all, why couldn’t it happen that two men have the same best valid partner? Second, the result shows that the G-S algorithm gives the best possible outcome for every man simultaneously; there is no stable matching in which any of the men could have hoped to do better. And finally, it answers

our question above by showing that the order of proposals in the G-S algorithm has absolutely no effect on the final outcome.

Despite all this, the proof is not so difficult.

Proof. Let us suppose, by way of contradiction, that some execution E of the G-S algorithm results in a matching S in which some man is paired with a woman who is not his best valid partner. Since men propose in decreasing order of preference, this means that some man is rejected by a valid partner during the execution E of the algorithm. So consider the first moment during the execution E in which some man, say m, is rejected by a valid partner w. Again, since men propose in decreasing order of preference, and since this is the first time such a rejection has occurred, it must be that w is m’s best valid partner best(m).

The rejection of m by w may have happened either because m proposed and was turned down in favor of w’s existing engagement, or because w broke her engagement to m in favor of a better proposal. But either way, at this moment w forms or continues an engagement with a man m′ whom she prefers to m.

Since w is a valid partner of m, there exists a stable matching S′ containing the pair (m, w). Now we ask: Who is m′ paired with in this matching? Suppose it is a woman w′ ̸=w.

Since the rejection of m by w was the first rejection of a man by a valid partner in the execution E, it must be that m′ had not been rejected by any valid partner at the point in E when he became engaged to w. Since he proposed in decreasing order of preference, and since w′ is clearly a valid partner of m′, it must be that m′ prefers w to w′. But we have already seen that w prefers m′ to m, for in execution E she rejected m in favor of m′. Since (m′, w) ̸∈ S′, it follows that (m′, w) is an instability in S′.

This contradicts our claim that S′ is stable and hence contradicts our initial assumption.

So for the men, the G-S algorithm is ideal. Unfortunately, the same cannot be said for the women. For a woman w, we say that m is a valid partner if there is a stable matching that contains the pair (m, w). We say that m is the worst valid partner of w if m is a valid partner of w, and no man whom w ranks lower than m is a valid partner of hers.

(1.8) In the stable matching S∗, each woman is paired with her worst valid partner.

Proof. Suppose there were a pair (m, w) in S∗ such that m is not the worst valid partner of w. Then there is a stable matching S′ in which w is paired

1.1 A First Problem: Stable Matching

11

12

Chapter 1 Introduction: Some Representative Problems

with a man m′ whom she likes less than m. In S′, m is paired with a woman w′ ̸= w; since w is the best valid partner of m, and w′ is a valid partner of m, we see that m prefers w to w′.

But from this it follows that (m, w) is an instability in S′, contradicting the claim that S′ is stable and hence contradicting our initial assumption.

Thus, we find that our simple example above, in which the men’s pref- erences clashed with the women’s, hinted at a very general phenomenon: for any input, the side that does the proposing in the G-S algorithm ends up with the best possible stable matching (from their perspective), while the side that does not do the proposing correspondingly ends up with the worst possible stable matching.

1.2 Five Representative Problems

The Stable Matching Problem provides us with a rich example of the process of algorithm design. For many problems, this process involves a few significant steps: formulating the problem with enough mathematical precision that we can ask a concrete question and start thinking about algorithms to solve it; designing an algorithm for the problem; and analyzing the algorithm by proving it is correct and giving a bound on the running time so as to establish the algorithm’s efficiency.

This high-level strategy is carried out in practice with the help of a few fundamental design techniques, which are very useful in assessing the inherent complexity of a problem and in formulating an algorithm to solve it. As in any area, becoming familiar with these design techniques is a gradual process; but with experience one can start recognizing problems as belonging to identifiable genres and appreciating how subtle changes in the statement of a problem can have an enormous effect on its computational difficulty.

To get this discussion started, then, it helps to pick out a few representa- tive milestones that we’ll be encountering in our study of algorithms: cleanly formulated problems, all resembling one another at a general level, but differ- ing greatly in their difficulty and in the kinds of approaches that one brings to bear on them. The first three will be solvable efficiently by a sequence of increasingly subtle algorithmic techniques; the fourth marks a major turning point in our discussion, serving as an example of a problem believed to be un- solvable by any efficient algorithm; and the fifth hints at a class of problems believed to be harder still.

The problems are self-contained and are all motivated by computing applications. To talk about some of them, though, it will help to use the terminology of graphs. While graphs are a common topic in earlier computer

science courses, we’ll be introducing them in a fair amount of depth in Chapter 3; due to their enormous expressive power, we’ll also be using them extensively throughout the book. For the discussion here, it’s enough to think of a graph G as simply a way of encoding pairwise relationships among a set of objects. Thus, G consists of a pair of sets (V,E)—a collection V of nodes and a collection E of edges, each of which “joins” two of the nodes. We thus represent an edge e ∈ E as a two-element subset of V: e = {u, v} for some u, v ∈ V, where we call u and v the ends of e. We typically draw graphs as in Figure 1.3, with each node as a small circle and each edge as a line segment joining its two ends.

Let’s now turn to a discussion of the five representative problems.

Interval Scheduling

Consider the following very simple scheduling problem. You have a resource— it may be a lecture room, a supercomputer, or an electron microscope—and many people request to use the resource for periods of time. A request takes the form: Can I reserve the resource starting at time s, until time f? We will assume that the resource can be used by at most one person at a time. A scheduler wants to accept a subset of these requests, rejecting all others, so that the accepted requests do not overlap in time. The goal is to maximize the number of requests accepted.

More formally, there will be n requests labeled 1, . . . , n, with each request i specifying a start time si and a finish time fi. Naturally, we have si < fi for all i. Two requests i and j are compatible if the requested intervals do not overlap: that is, either request i is for an earlier time interval than request j (fi ≤ sj), or request i is for a later time than request j (fj ≤ si). We’ll say more generally that a subset A of requests is compatible if all pairs of requests i, j ∈ A, i ̸= j are compatible. The goal is to select a compatible subset of requests of maximum possible size.
We illustrate an instance of this Interval Scheduling Problem in Figure 1.4. Note that there is a single compatible set of size 4, and this is the largest compatible set.
(a)
(b)
Figure 1.3 Each of (a) and (b) depicts a graph on four nodes.
1.2 Five Representative Problems
13
Figure 1.4 An instance of the Interval Scheduling Problem.
14
Chapter 1 Introduction: Some Representative Problems
We will see shortly that this problem can be solved by a very natural algorithm that orders the set of requests according to a certain heuristic and then “greedily” processes them in one pass, selecting as large a compatible subset as it can. This will be typical of a class of greedy algorithms that we will consider for various problems—myopic rules that process the input one piece at a time with no apparent look-ahead. When a greedy algorithm can be shown to find an optimal solution for all instances of a problem, it’s often fairly surprising. We typically learn something about the structure of the underlying problem from the fact that such a simple approach can be optimal.
Weighted Interval Scheduling
In the Interval Scheduling Problem, we sought to maximize the number of requests that could be accommodated simultaneously. Now, suppose more generally that each request interval i has an associated value, or weight, vi > 0; we could picture this as the amount of money we will make from the ith individual if we schedule his or her request. Our goal will be to find a compatible subset of intervals of maximum total value.

The case in which vi = 1 for each i is simply the basic Interval Scheduling Problem; but the appearance of arbitrary values changes the nature of the maximization problem quite a bit. Consider, for example, that if v1 exceeds the sum of all other vi, then the optimal solution must include interval 1 regardless of the configuration of the full set of intervals. So any algorithm for this problem must be very sensitive to the values, and yet degenerate to a method for solving (unweighted) interval scheduling when all the values are equal to 1.

There appears to be no simple greedy rule that walks through the intervals one at a time, making the correct decision in the presence of arbitrary values. Instead, we employ a technique, dynamic programming, that builds up the optimal value over all possible solutions in a compact, tabular way that leads to a very efficient algorithm.

Bipartite Matching

When we considered the Stable Matching Problem, we defined a matching to be a set of ordered pairs of men and women with the property that each man and each woman belong to at most one of the ordered pairs. We then defined a perfect matching to be a matching in which every man and every woman belong to some pair.

We can express these concepts more generally in terms of graphs, and in order to do this it is useful to define the notion of a bipartite graph. We say that a graph G = (V , E) is bipartite if its node set V can be partitioned into sets X

andY insuchawaythateveryedgehasoneendinX andtheotherendinY. A bipartite graph is pictured in Figure 1.5; often, when we want to emphasize a graph’s “bipartiteness,” we will draw it this way, with the nodes in X and Y in two parallel columns. But notice, for example, that the two graphs in Figure 1.3 are also bipartite.

Now, in the problem of finding a stable matching, matchings were built from pairs of men and women. In the case of bipartite graphs, the edges are pairsofnodes,sowesaythatamatchinginagraphG=(V,E)isasetofedges M ⊆ E with the property that each node appears in at most one edge of M. M is a perfect matching if every node appears in exactly one edge of M.

To see that this does capture the same notion we encountered in the Stable Matching Problem, consider a bipartite graph G′ with a set X of n men, a set Y of n women, and an edge from every node in X to every node in Y. Then the matchings and perfect matchings in G′ are precisely the matchings and perfect matchings among the set of men and women.

In the Stable Matching Problem, we added preferences to this picture. Here, we do not consider preferences; but the nature of the problem in arbitrary bipartite graphs adds a different source of complexity: there is not necessarily an edge from every x ∈ X to every y ∈ Y, so the set of possible matchings has quite a complicated structure. In other words, it is as though only certain pairs of men and women are willing to be paired off, and we want to figure out how to pair off many people in a way that is consistent with this. Consider, for example, the bipartite graph G in Figure 1.5: there are many matchings in G, but there is only one perfect matching. (Do you see it?)

Matchings in bipartite graphs can model situations in which objects are being assigned to other objects. Thus, the nodes in X can represent jobs, the nodes in Y can represent machines, and an edge (xi,yj) can indicate that machine yj is capable of processing job xi. A perfect matching is then a way of assigning each job to a machine that can process it, with the property that each machine is assigned exactly one job. In the spring, computer science departments across the country are often seen pondering a bipartite graph in which X is the set of professors in the department, Y is the set of offered courses, and an edge (xi, yj) indicates that professor xi is capable of teaching course yj. A perfect matching in this graph consists of an assignment of each professor to a course that he or she can teach, in such a way that every course is covered.

Thus the Bipartite Matching Problem is the following: Given an arbitrary bipartite graph G, find a matching of maximum size. If |X| = |Y| = n, then there is a perfect matching if and only if the maximum matching has size n. We will find that the algorithmic techniques discussed earlier do not seem adequate

x1 y1 x2 y2 x3 y3 x4 y4 x5 y5

Figure 1.5 A bipartite graph.

1.2 Five Representative Problems

15

16

Chapter 1 Introduction: Some Representative Problems

12

345

67

Figure 1.6 A graph whose largest independent set has size 4.

for providing an efficient algorithm for this problem. There is, however, a very elegant and efficient algorithm to find a maximum matching; it inductively builds up larger and larger matchings, selectively backtracking along the way. This process is called augmentation, and it forms the central component in a large class of efficiently solvable problems called network flow problems.

Independent Set

Now let’s talk about an extremely general problem, which includes most of these earlier problems as special cases. Given a graph G=(V,E), we say a set of nodes S ⊆ V is independent if no two nodes in S are joined by an edge. The Independent Set Problem is, then, the following: Given G, find an independent set that is as large as possible. For example, the maximum size of an independent set in the graph in Figure 1.6 is four, achieved by the four-node independent set {1, 4, 5, 6}.

The Independent Set Problem encodes any situation in which you are trying to choose from among a collection of objects and there are pairwise conflicts among some of the objects. Say you have n friends, and some pairs of them don’t get along. How large a group of your friends can you invite to dinner if you don’t want any interpersonal tensions? This is simply the largest independent set in the graph whose nodes are your friends, with an edge between each conflicting pair.

Interval Scheduling and Bipartite Matching can both be encoded as special cases of the Independent Set Problem. For Interval Scheduling, define a graph G = (V , E) in which the nodes are the intervals and there is an edge between each pair of them that overlap; the independent sets in G are then just the compatible subsets of intervals. Encoding Bipartite Matching as a special case of Independent Set is a little trickier to see. Given a bipartite graph G′ = (V′, E′), the objects being chosen are edges, and the conflicts arise between two edges that share an end. (These, indeed, are the pairs of edges that cannot belong to a common matching.) So we define a graph G = (V , E) in which the node set V is equal to the edge set E′ of G′. We define an edge between each pair of elements in V that correspond to edges of G′ with a common end. We can now check that the independent sets of G are precisely the matchings of G′. While it is not complicated to check this, it takes a little concentration to deal with this type of “edges-to-nodes, nodes-to-edges” transformation.2

2 For those who are curious, we note that not every instance of the Independent Set Problem can arise in this way from Interval Scheduling or from Bipartite Matching; the full Independent Set Problem really is more general. The graph in Figure 1.3(a) cannot arise as the “conflict graph” in an instance of

Given the generality of the Independent Set Problem, an efficient algorithm to solve it would be quite impressive. It would have to implicitly contain algorithms for Interval Scheduling, Bipartite Matching, and a host of other natural optimization problems.

The current status of Independent Set is this: no efficient algorithm is known for the problem, and it is conjectured that no such algorithm exists. The obvious brute-force algorithm would try all subsets of the nodes, checking each to see if it is independent, and then recording the largest one encountered. It is possible that this is close to the best we can do on this problem. We will see later in the book that Independent Set is one of a large class of problems that are termed NP-complete. No efficient algorithm is known for any of them; but they are all equivalent in the sense that a solution to any one of them would imply, in a precise sense, a solution to all of them.

Here’s a natural question: Is there anything good we can say about the complexity of the Independent Set Problem? One positive thing is the following: If we have a graph G on 1,000 nodes, and we want to convince you that it contains an independent set S of size 100, then it’s quite easy. We simply show you the graph G, circle the nodes of S in red, and let you check that no two of them are joined by an edge. So there really seems to be a great difference in difficulty between checking that something is a large independent set and actually finding a large independent set. This may look like a very basic observation—and it is—but it turns out to be crucial in understanding this class of problems. Furthermore, as we’ll see next, it’s possible for a problem to be so hard that there isn’t even an easy way to “check” solutions in this sense.

Competitive Facility Location

Finally, we come to our fifth problem, which is based on the following two- player game. Consider two large companies that operate cafe ́ franchises across the country—let’s call them JavaPlanet and Queequeg’s Coffee—and they are currently competing for market share in a geographic area. First JavaPlanet opens a franchise; then Queequeg’s Coffee opens a franchise; then JavaPlanet; then Queequeg’s; and so on. Suppose they must deal with zoning regulations that require no two franchises be located too close together, and each is trying to make its locations as convenient as possible. Who will win?

Let’s make the rules of this “game” more concrete. The geographic region in question is divided into n zones, labeled 1, 2, . . . , n. Each zone i has a

Interval Scheduling, and the graph in Figure 1.3(b) cannot arise as the “conflict graph” in an instance of Bipartite Matching.

1.2 Five Representative Problems

17

18

Chapter 1 Introduction: Some Representative Problems

10 1 5 15 5 1 5 1 15 10

Figure 1.7 An instance of the Competitive Facility Location Problem.

value bi, which is the revenue obtained by either of the companies if it opens a franchise there. Finally, certain pairs of zones (i, j) are adjacent, and local zoning laws prevent two adjacent zones from each containing a franchise, regardless of which company owns them. (They also prevent two franchises from being opened in the same zone.) We model these conflicts via a graph G=(V,E), where V is the set of zones, and (i,j) is an edge in E if the zones i and j are adjacent. The zoning requirement then says that the full set of franchises opened must form an independent set in G.

Thus our game consists of two players, P1 and P2, alternately selecting nodes in G, with P1 moving first. At all times, the set of all selected nodes must form an independent set in G. Suppose that player P2 has a target bound B, and we want to know: is there a strategy for P2 so that no matter how P1 plays, P2 will be able to select a set of nodes with a total value of at least B? We will call this an instance of the Competitive Facility Location Problem.

Consider, for example, the instance pictured in Figure 1.7, and suppose that P2’s target bound is B = 20. Then P2 does have a winning strategy. On the other hand, if B = 25, then P2 does not.

One can work this out by looking at the figure for a while; but it requires some amount of case-checking of the form, “If P1 goes here, then P2 will go there; but if P1 goes over there, then P2 will go here. . . . ” And this appears to be intrinsic to the problem: not only is it computationally difficult to determine whether P2 has a winning strategy; on a reasonably sized graph, it would even be hard for us to convince you that P2 has a winning strategy. There does not seem to be a short proof we could present; rather, we’d have to lead you on a lengthy case-by-case analysis of the set of possible moves.

This is in contrast to the Independent Set Problem, where we believe that finding a large solution is hard but checking a proposed large solution is easy. This contrast can be formalized in the class of PSPACE-complete problems, of which Competitive Facility Location is an example. PSPACE-complete prob- lems are believed to be strictly harder than NP-complete problems, and this conjectured lack of short “proofs” for their solutions is one indication of this greater hardness. The notion of PSPACE-completeness turns out to capture a large collection of problems involving game-playing and planning; many of these are fundamental issues in the area of artificial intelligence.

Solved Exercises

Solved Exercise 1

Consider a town with n men and n women seeking to get married to one another. Each man has a preference list that ranks all the women, and each woman has a preference list that ranks all the men.

The set of all 2n people is divided into two categories: good people and bad people. Suppose that for some number k, 1 ≤ k ≤ n − 1, there are k good men and k good women; thus there are n − k bad men and n − k bad women.

Everyone would rather marry any good person than any bad person. Formally, each preference list has the property that it ranks each good person of the opposite gender higher than each bad person of the opposite gender: its first k entries are the good people (of the opposite gender) in some order, and its next n − k are the bad people (of the opposite gender) in some order.

Show that in every stable matching, every good man is married to a good woman.

Solution A natural way to get started thinking about this problem is to assume the claim is false and try to work toward obtaining a contradiction. What would it mean for the claim to be false? There would exist some stable matching M in which a good man m was married to a bad woman w.

Now, let’s consider what the other pairs in M look like. There are k good men and k good women. Could it be the case that every good woman is married to a good man in this matching M? No: one of the good men (namely, m) is already married to a bad woman, and that leaves only k − 1 other good men. So even if all of them were married to good women, that would still leave some good woman who is married to a bad man.

Let w′ be such a good woman, who is married to a bad man. It is now easy to identify an instability in M: consider the pair (m, w′). Each is good, but is married to a bad partner. Thus, each of m and w′ prefers the other to their current partner, and hence (m, w′) is an instability. This contradicts our assumption that M is stable, and hence concludes the proof.

Solved Exercise 2

We can think about a generalization of the Stable Matching Problem in which certain man-woman pairs are explicitly forbidden. In the case of employers and applicants, we could imagine that certain applicants simply lack the necessary qualifications or certifications, and so they cannot be employed at certain companies, however desirable they may seem. Using the analogy to marriage between men and women, we have a set M of n men, a set W of n women,

Solved Exercises

19

20

Chapter 1 Introduction: Some Representative Problems

and a set F ⊆ M × W of pairs who are simply not allowed to get married. Each man m ranks all the women w for which (m, w) ̸∈ F, and each woman w′ ranks all the men m′ for which (m′, w′) ̸∈ F.

In this more general setting, we say that a matching S is stable if it does not exhibit any of the following types of instability.

(i) There are two pairs (m,w) and (m′,w′) in S with the property that (m, w′) ̸∈ F, m prefers w′ to w, and w′ prefers m to m′. (The usual kind of instability.)

(ii) There is a pair (m,w)∈S, and a man m′, so that m′ is not part of any pair in the matching, (m′ , w) ̸∈ F , and w prefers m′ to m. (A single man is more desirable and not forbidden.)

(iii) There is a pair (m,w)∈S, and a woman w′, so that w′ is not part of any pair in the matching, (m, w′) ̸∈ F, and m prefers w′ to w. (A single woman is more desirable and not forbidden.)

(iv) There is a man m and a woman w, neither of whom is part of any pair in the matching, so that (m, w) ̸∈ F. (There are two single people with nothing preventing them from getting married to each other.)

Note that under these more general definitions, a stable matching need not be a perfect matching.

Now we can ask: For every set of preference lists and every set of forbidden pairs, is there always a stable matching? Resolve this question by doing one of the following two things: (a) give an algorithm that, for any set of preference lists and forbidden pairs, produces a stable matching; or (b) give an example of a set of preference lists and forbidden pairs for which there is no stable matching.

Solution The Gale-Shapley algorithm is remarkably robust to variations on the Stable Matching Problem. So, if you’re faced with a new variation of the problem and can’t find a counterexample to stability, it’s often a good idea to check whether a direct adaptation of the G-S algorithm will in fact produce stable matchings.

That turns out to be the case here. We will show that there is always a stable matching, even in this more general model with forbidden pairs, and we will do this by adapting the G-S algorithm. To do this, let’s consider why the original G-S algorithm can’t be used directly. The difficulty, of course, is that the G-S algorithm doesn’t know anything about forbidden pairs, and so the condition in the While loop,

While there is a man m who is free and hasn’t proposed to every woman,

won’t work: we don’t want m to propose to a woman w for which the pair (m, w) is forbidden.

Thus, let’s consider a variation of the G-S algorithm in which we make only one change: we modify the While loop to say,

While there is a man m who is free and hasn’t proposed to every woman w for which (m, w) ̸∈ F.

Here is the algorithm in full.

Initially all m∈M and w∈W are free

While there is a man m who is free and hasn’t proposed to every woman w for which (m, w) ̸∈ F

Choose such a man m

Let w be the highest-ranked woman in m’s preference list

to which m has not yet proposed If w is free then

(m, w) become engaged

Else w is currently engaged to m′

If w prefers m′ to m then m remains free

Else w prefers m to m′ (m, w) become engaged m′ becomes free

Endif Endif

Endwhile

Return the set S of engaged pairs

We now prove that this yields a stable matching, under our new definition of stability.

To begin with, facts (1.1), (1.2), and (1.3) from the text remain true (in particular, the algorithm will terminate in at most n2 iterations). Also, we don’t have to worry about establishing that the resulting matching S is perfect (indeed, it may not be). We also notice an additional pairs of facts. If m is a man who is not part of a pair in S, then m must have proposed to every nonforbidden woman; and if w is a woman who is not part of a pair in S, then it must be that no man ever proposed to w.

Finally, we need only show

Solved Exercises

21

(1.9) There is no instability with respect to the returned matching S.

22

Chapter 1 Introduction: Some Representative Problems

Proof. Our general definition of instability has four parts: This means that we have to make sure that none of the four bad things happens.

First, suppose there is an instability of type (i), consisting of pairs (m, w) and (m′, w′) in S with the property that (m, w′) ̸∈ F, m prefers w′ to w, and w′ prefers m to m′. It follows that m must have proposed to w′; so w′ rejected m, and thus she prefers her final partner to m—a contradiction.

Next, suppose there is an instability of type (ii), consisting of a pair (m, w) ∈ S, and a man m′, so that m′ is not part of any pair in the matching, (m′, w) ̸∈ F, and w prefers m′ to m. Then m′ must have proposed to w and been rejected; again, it follows that w prefers her final partner to m′—a contradiction.

Third, suppose there is an instability of type (iii), consisting of a pair (m, w) ∈ S, and a woman w′, so that w′ is not part of any pair in the matching, (m, w′) ̸∈ F, and m prefers w′ to w. Then no man proposed to w′ at all; in particular, m never proposed to w′, and so he must prefer w to w′—a contradiction.

Finally, suppose there is an instability of type (iv), consisting of a man m and a woman w, neither of which is part of any pair in the matching, so that (m, w) ̸∈ F. But for m to be single, he must have proposed to every nonforbidden woman; in particular, he must have proposed to w, which means she would no longer be single—a contradiction.

Exercises

1. Decidewhetheryouthinkthefollowingstatementistrueorfalse.Ifitis true, give a short explanation. If it is false, give a counterexample.

True or false? In every instance of the Stable Matching Problem, there is a stable matching containing a pair (m, w) such that m is ranked first on the preference list of w and w is ranked first on the preference list of m.

2. Decidewhetheryouthinkthefollowingstatementistrueorfalse.Ifitis true, give a short explanation. If it is false, give a counterexample.

True or false? Consider an instance of the Stable Matching Problem in which there exists a man m and a woman w such that m is ranked first on the preference list of w and w is ranked first on the preference list of m. Then in every stable matching S for this instance, the pair (m, w) belongs to S.

3. There are many other settings in which we can ask questions related to some type of “stability” principle. Here’s one, involving competition between two enterprises.

Suppose we have two television networks, whom we’ll call A and B. There are n prime-time programming slots, and each network has n TV shows. Each network wants to devise a schedule—an assignment of each show to a distinct slot—so as to attract as much market share as possible.

Here is the way we determine how well the two networks perform relative to each other, given their schedules. Each show has a fixed rating, which is based on the number of people who watched it last year; we’ll assume that no two shows have exactly the same rating. A network wins a given time slot if the show that it schedules for the time slot has a larger rating than the show the other network schedules for that time slot. The goal of each network is to win as many time slots as possible.

Suppose in the opening week of the fall season, Network A reveals a schedule S and Network B reveals a schedule T. On the basis of this pair of schedules, each network wins certain time slots, according to the rule above. We’ll say that the pair of schedules (S , T ) is stable if neither network can unilaterally change its own schedule and win more time slots. That is, there is no schedule S′ such that Network A wins more slots with the pair (S′, T) than it did with the pair (S, T); and symmetrically, there is no schedule T′ such that Network B wins more slots with the pair (S, T′) than it did with the pair (S, T).

The analogue of Gale and Shapley’s question for this kind of stability is the following: For every set of TV shows and ratings, is there always a stable pair of schedules? Resolve this question by doing one of the following two things:

(a) give an algorithm that, for any set of TV shows and associated ratings, produces a stable pair of schedules; or

(b) give an example of a set of TV shows and associated ratings for which there is no stable pair of schedules.

4. Gale and Shapley published their paper on the Stable Matching Problem in 1962; but a version of their algorithm had already been in use for ten years by the National Resident Matching Program, for the problem of assigning medical residents to hospitals.

Basically, the situation was the following. There were m hospitals, each with a certain number of available positions for hiring residents. There were n medical students graduating in a given year, each interested in joining one of the hospitals. Each hospital had a ranking of the students in order of preference, and each student had a ranking of the hospitals in order of preference. We will assume that there were more students graduating than there were slots available in the m hospitals.

Exercises

23

24

Chapter 1 Introduction: Some Representative Problems

The interest, naturally, was in finding a way of assigning each student to at most one hospital, in such a way that all available positions in all hospitals were filled. (Since we are assuming a surplus of students, there would be some students who do not get assigned to any hospital.)

We say that an assignment of students to hospitals is stable if neither of the following situations arises.

. First type of instability: There are students s and s′, and a hospital h, so that

– sisassignedtoh,and

– s′ is assigned to no hospital, and – h prefers s′ to s.

. Second type of instability: There are students s and s′, and hospitals h and h′, so that

– sisassignedtoh,and – s′ is assigned to h′, and – h prefers s′ to s, and

– s′ prefers h to h′.

So we basically have the Stable Matching Problem, except that (i) hospitals generally want more than one resident, and (ii) there is a surplus of medical students.

Show that there is always a stable assignment of students to hospi- tals, and give an algorithm to find one.

5. The Stable Matching Problem, as discussed in the text, assumes that all men and women have a fully ordered list of preferences. In this problem we will consider a version of the problem in which men and women can be indifferent between certain options. As before we have a set M of n men and a set W of n women. Assume each man and each woman ranks the members of the opposite gender, but now we allow ties in the ranking. For example (with n = 4), a woman could say that m1 is ranked in first place; second place is a tie between m2 and m3 (she has no preference between them); and m4 is in last place. We will say that w prefers m to m′ if m is ranked higher than m′ on her preference list (they are not tied).

With indifferences in the rankings, there could be two natural notions for stability. And for each, we can ask about the existence of stable matchings, as follows.

(a) A strong instability in a perfect matching S consists of a man m and a woman w, such that each of m and w prefers the other to their partner in S. Does there always exist a perfect matching with no

strong instability? Either give an example of a set of men and women with preference lists for which every perfect matching has a strong instability; or give an algorithm that is guaranteed to find a perfect matching with no strong instability.

(b) A weak instability in a perfect matching S consists of a man m and a woman w, such that their partners in S are w′ and m′, respectively, and one of the following holds:

– m prefers w to w′, and w either prefers m to m′ or is indifferent between these two choices; or

– w prefers m to m′, and m either prefers w to w′ or is indifferent between these two choices.

In other words, the pairing between m and w is either preferred by both, or preferred by one while the other is indifferent. Does there always exist a perfect matching with no weak instability? Either give an example of a set of men and women with preference lists for which every perfect matching has a weak instability; or give an algorithm that is guaranteed to find a perfect matching with no weak instability.

6. PeripateticShippingLines,Inc.,isashippingcompanythatownsnships and provides service to n ports. Each of its ships has a schedule that says, for each day of the month, which of the ports it’s currently visiting, or whether it’s out at sea. (You can assume the “month” here has m days, for some m > n.) Each ship visits each port for exactly one day during the month. For safety reasons, PSL Inc. has the following strict requirement:

(†) No two ships can be in the same port on the same day.

The company wants to perform maintenance on all the ships this month, via the following scheme. They want to truncate each ship’s schedule: for each ship Si, there will be some day when it arrives in its scheduled port and simply remains there for the rest of the month (for maintenance). This means that Si will not visit the remaining ports on its schedule (if any) that month, but this is okay. So the truncation of Si’s schedule will simply consist of its original schedule up to a certain specified day on which it is in a port P; the remainder of the truncated schedule simply has it remain in port P.

Now the company’s question to you is the following: Given the sched- ule for each ship, find a truncation of each so that condition (†) continues to hold: no two ships are ever in the same port on the same day.

Show that such a set of truncations can always be found, and give an algorithm to find them.

Exercises

25

26

Chapter 1 Introduction: Some Representative Problems

Example. Suppose we have two ships and two ports, and the “month” has four days. Suppose the first ship’s schedule is

port P1; at sea; port P2; at sea and the second ship’s schedule is at sea; port P1; at sea; port P2

Then the (only) way to choose truncations would be to have the first ship remain in port P2 starting on day 3, and have the second ship remain in port P1 starting on day 2.

7. Some of your friends are working for CluNet, a builder of large commu- nication networks, and they are looking at algorithms for switching in a particular type of input/output crossbar.

Here is the setup. There are n input wires and n output wires, each directed from a source to a terminus. Each input wire meets each output wire in exactly one distinct point, at a special piece of hardware called a junction box. Points on the wire are naturally ordered in the direction from source to terminus; for two distinct points x and y on the same wire, we say that x is upstream from y if x is closer to the source than y, and otherwise we say x is downstream from y. The order in which one input wire meets the output wires is not necessarily the same as the order in which another input wire meets the output wires. (And similarly for the orders in which output wires meet input wires.) Figure 1.8 gives an example of such a collection of input and output wires.

Now, here’s the switching component of this situation. Each input wire is carrying a distinct data stream, and this data stream must be switched onto one of the output wires. If the stream of Input i is switched onto Output j, at junction box B, then this stream passes through all junction boxes upstream from B on Input i, then through B, then through all junction boxes downstream from B on Output j. It does not matter which input data stream gets switched onto which output wire, but each input data stream must be switched onto a different output wire. Furthermore—and this is the tricky constraint—no two data streams can pass through the same junction box following the switching operation.

Finally, here’s the problem. Show that for any specified pattern in which the input wires and output wires meet each other (each pair meet- ing exactly once), a valid switching of the data streams can always be found—one in which each input data stream is switched onto a different output, and no two of the resulting streams pass through the same junc- tion box. Additionally, give an algorithm to find such a valid switching.

Junction Junction

Input 1 (meets Output 2 before Output 1)

Input 2 (meets Output 1 before Output 2)

Figure 1.8 An example with two input wires and two output wires. Input 1 has its junction with Output 2 upstream from its junction with Output 1; Input 2 has its junction with Output 1 upstream from its junction with Output 2. A valid solution is to switch the data stream of Input 1 onto Output 2, and the data stream of Input 2 onto Output 1. On the other hand, if the stream of Input 1 were switched onto Output 1, and the stream of Input 2 were switched onto Output 2, then both streams would pass through the junction box at the meeting of Input 1 and Output 2—and this is not allowed.

8. For this problem, we will explore the issue of truthfulness in the Stable Matching Problem and specifically in the Gale-Shapley algorithm. The basic question is: Can a man or a woman end up better off by lying about his or her preferences? More concretely, we suppose each participant has a true preference order. Now consider a woman w. Suppose w prefers man m to m′, but both m and m′ are low on her list of preferences. Can it be the case that by switching the order of m and m′ on her list of preferences (i.e., by falsely claiming that she prefers m′ to m) and running the algorithm with this false preference list, w will end up with a man m′′ that she truly prefers to both m and m′? (We can ask the same question for men, but will focus on the case of women for purposes of this question.)

Resolve this question by doing one of the following two things:

(a) Give a proof that, for any set of preference lists, switching the order of a pair on the list cannot improve a woman’s partner in the Gale- Shapley algorithm; or

Junction

Exercises

27

Junction

Output 1 (meets Input 2 before Input 1)

Output 2 (meets Input 2 before Input 1)

28

Chapter 1 Introduction: Some Representative Problems

(b) Give an example of a set of preference lists for which there is a switch that would improve the partner of a woman who switched preferences.

Notes and Further Reading

The Stable Matching Problem was first defined and analyzed by Gale and Shapley (1962); according to David Gale, their motivation for the problem came from a story they had recently read in the New Yorker about the intricacies of the college admissions process (Gale, 2001). Stable matching has grown into an area of study in its own right, covered in books by Gusfield and Irving (1989) and Knuth (1997c). Gusfield and Irving also provide a nice survey of the “parallel” history of the Stable Matching Problem as a technique invented for matching applicants with employers in medicine and other professions.

As discussed in the chapter, our five representative problems will be central to the book’s discussions, respectively, of greedy algorithms, dynamic programming, network flow, NP-completeness, and PSPACE-completeness. We will discuss the problems in these contexts later in the book.

Chapter 2

Basics of Algorithm Analysis

Analyzing algorithms involves thinking about how their resource require- ments—the amount of time and space they use—will scale with increasing input size. We begin this chapter by talking about how to put this notion on a concrete footing, as making it concrete opens the door to a rich understanding of computational tractability. Having done this, we develop the mathematical machinery needed to talk about the way in which different functions scale with increasing input size, making precise what it means for one function to grow faster than another.

We then develop running-time bounds for some basic algorithms, begin- ning with an implementation of the Gale-Shapley algorithm from Chapter 1 and continuing to a survey of many different running times and certain char- acteristic types of algorithms that achieve these running times. In some cases, obtaining a good running-time bound relies on the use of more sophisticated data structures, and we conclude this chapter with a very useful example of such a data structure: priority queues and their implementation using heaps.

2.1 Computational Tractability

A major focus of this book is to find efficient algorithms for computational problems. At this level of generality, our topic seems to encompass the whole of computer science; so what is specific to our approach here?

First, we will try to identify broad themes and design principles in the development of algorithms. We will look for paradigmatic problems and ap- proaches that illustrate, with a minimum of irrelevant detail, the basic ap- proaches to designing efficient algorithms. At the same time, it would be pointless to pursue these design principles in a vacuum, so the problems and

30

Chapter 2 Basics of Algorithm Analysis

approaches we consider are drawn from fundamental issues that arise through- out computer science, and a general study of algorithms turns out to serve as a nice survey of computational ideas that arise in many areas.

Another property shared by many of the problems we study is their fundamentally discrete nature. That is, like the Stable Matching Problem, they will involve an implicit search over a large set of combinatorial possibilities; and the goal will be to efficiently find a solution that satisfies certain clearly delineated conditions.

As we seek to understand the general notion of computational efficiency, we will focus primarily on efficiency in running time: we want algorithms that run quickly. But it is important that algorithms be efficient in their use of other resources as well. In particular, the amount of space (or memory) used by an algorithm is an issue that will also arise at a number of points in the book, and we will see techniques for reducing the amount of space needed to perform a computation.

Some Initial Attempts at Defining Efficiency

The first major question we need to answer is the following: How should we turn the fuzzy notion of an “efficient” algorithm into something more concrete?

A first attempt at a working definition of efficiency is the following. Proposed Definition of Efficiency (1): An algorithm is efficient if, when

implemented, it runs quickly on real input instances.

Let’s spend a little time considering this definition. At a certain level, it’s hard to argue with: one of the goals at the bedrock of our study of algorithms is solving real problems quickly. And indeed, there is a significant area of research devoted to the careful implementation and profiling of different algorithms for discrete computational problems.

But there are some crucial things missing from this definition, even if our main goal is to solve real problem instances quickly on real computers. The first is the omission of where, and how well, we implement an algorithm. Even bad algorithms can run quickly when applied to small test cases on extremely fast processors; even good algorithms can run slowly when they are coded sloppily. Also, what is a “real” input instance? We don’t know the full range of input instances that will be encountered in practice, and some input instances can be much harder than others. Finally, this proposed definition above does not consider how well, or badly, an algorithm may scale as problem sizes grow to unexpected levels. A common situation is that two very different algorithms will perform comparably on inputs of size 100; multiply the input size tenfold, and one will still run quickly while the other consumes a huge amount of time.

So what we could ask for is a concrete definition of efficiency that is platform-independent, instance-independent, and of predictive value with respect to increasing input sizes. Before focusing on any specific consequences of this claim, we can at least explore its implicit, high-level suggestion: that we need to take a more mathematical view of the situation.

We can use the Stable Matching Problem as an example to guide us. The input has a natural “size” parameter N ; we could take this to be the total size of the representation of all preference lists, since this is what any algorithm for the problem will receive as input. N is closely related to the other natural parameter in this problem: n, the number of men and the number of women. Since there are 2n preference lists, each of length n, we can view N = 2n2, suppressing more fine-grained details of how the data is represented. In considering the problem, we will seek to describe an algorithm at a high level, and then analyze its running time mathematically as a function of this input size N.

Worst-Case Running Times and Brute-Force Search

To begin with, we will focus on analyzing the worst-case running time: we will look for a bound on the largest possible running time the algorithm could have over all inputs of a given size N, and see how this scales with N. The focus on worst-case performance initially seems quite draconian: what if an algorithm performs well on most instances and just has a few pathological inputs on which it is very slow? This certainly is an issue in some cases, but in general the worst-case analysis of an algorithm has been found to do a reasonable job of capturing its efficiency in practice. Moreover, once we have decided to go the route of mathematical analysis, it is hard to find an effective alternative to worst-case analysis. Average-case analysis—the obvious appealing alternative, in which one studies the performance of an algorithm averaged over “random” instances—can sometimes provide considerable insight, but very often it can also become a quagmire. As we observed earlier, it’s very hard to express the full range of input instances that arise in practice, and so attempts to study an algorithm’s performance on “random” input instances can quickly devolve into debates over how a random input should be generated: the same algorithm can perform very well on one class of random inputs and very poorly on another. After all, real inputs to an algorithm are generally not being produced from a random distribution, and so average-case analysis risks telling us more about the means by which the random inputs were generated than about the algorithm itself.

So in general we will think about the worst-case analysis of an algorithm’s running time. But what is a reasonable analytical benchmark that can tell us whether a running-time bound is impressive or weak? A first simple guide

2.1 Computational Tractability

31

32

Chapter 2 Basics of Algorithm Analysis

is by comparison with brute-force search over the search space of possible solutions.

Let’s return to the example of the Stable Matching Problem. Even when the size of a Stable Matching input instance is relatively small, the search space it defines is enormous (there are n! possible perfect matchings between n men and n women), and we need to find a matching that is stable. The natural “brute-force” algorithm for this problem would plow through all perfect matchings by enumeration, checking each to see if it is stable. The surprising punchline, in a sense, to our solution of the Stable Matching Problem is that we needed to spend time proportional only to N in finding a stable matching from among this stupendously large space of possibilities. This was a conclusion we reached at an analytical level. We did not implement the algorithm and try it out on sample preference lists; we reasoned about it mathematically. Yet, at the same time, our analysis indicated how the algorithm could be implemented in practice and gave fairly conclusive evidence that it would be a big improvement over exhaustive enumeration.

This will be a common theme in most of the problems we study: a compact representation, implicitly specifying a giant search space. For most of these problems, there will be an obvious brute-force solution: try all possibilities and see if any one of them works. Not only is this approach almost always too slow to be useful, it is an intellectual cop-out; it provides us with absolutely no insight into the structure of the problem we are studying. And so if there is a common thread in the algorithms we emphasize in this book, it would be the following alternative definition of efficiency.

Proposed Definition of Efficiency (2): An algorithm is efficient if it achieves qualitatively better worst-case performance, at an analytical level, than brute-force search.

This will turn out to be a very useful working definition for us. Algorithms that improve substantially on brute-force search nearly always contain a valuable heuristic idea that makes them work; and they tell us something about the intrinsic structure, and computational tractability, of the underlying problem itself.

But if there is a problem with our second working definition, it is vague- ness. What do we mean by “qualitatively better performance?” This suggests that we consider the actual running time of algorithms more carefully, and try to quantify what a reasonable running time would be.

Polynomial Time as a Definition of Efficiency

When people first began analyzing discrete algorithms mathematically—a thread of research that began gathering momentum through the 1960s—

a consensus began to emerge on how to quantify the notion of a “reasonable” running time. Search spaces for natural combinatorial problems tend to grow exponentially in the size N of the input; if the input size increases by one, the number of possibilities increases multiplicatively. We’d like a good algorithm for such a problem to have a better scaling property: when the input size increases by a constant factor—say, a factor of 2—the algorithm should only slow down by some constant factor C.

Arithmetically, we can formulate this scaling behavior as follows. Suppose an algorithm has the following property: There are absolute constants c > 0 and d>0 so that on every input instance of size N, its running time is bounded by cNd primitive computational steps. (In other words, its running time is at most proportional to Nd.) For now, we will remain deliberately vague on what we mean by the notion of a “primitive computational step”— but it can be easily formalized in a model where each step corresponds to a single assembly-language instruction on a standard processor, or one line of a standard programming language such as C or Java. In any case, if this running-time bound holds, for some c and d, then we say that the algorithm has a polynomial running time, or that it is a polynomial-time algorithm. Note that any polynomial-time bound has the scaling property we’re looking for. If the input size increases from N to 2N, the bound on the running time increases from cNd to c(2N)d = c · 2dNd, which is a slow-down by a factor of 2d. Since d is a constant, so is 2d; of course, as one might expect, lower-degree polynomials exhibit better scaling behavior than higher-degree polynomials.

From this notion, and the intuition expressed above, emerges our third attempt at a working definition of efficiency.

Proposed Definition of Efficiency (3): An algorithm is efficient if it has a polynomial running time.

Where our previous definition seemed overly vague, this one seems much too prescriptive. Wouldn’t an algorithm with running time proportional to n100—and hence polynomial—be hopelessly inefficient? Wouldn’t we be rel- atively pleased with a nonpolynomial running time of n1+.02(log n)? The an- swers are, of course, “yes” and “yes.” And indeed, however much one may try to abstractly motivate the definition of efficiency in terms of polynomial time, a primary justification for it is this: It really works. Problems for which polynomial-time algorithms exist almost invariably turn out to have algorithms with running times proportional to very moderately growing polynomials like n, n log n, n2, or n3. Conversely, problems for which no polynomial-time al- gorithm is known tend to be very difficult in practice. There are certainly exceptions to this principle in both directions: there are cases, for example, in

2.1 Computational Tractability

33

34

Chapter 2 Basics of Algorithm Analysis

Table 2.1 The running times (rounded up) of different algorithms on inputs of increasing size, for a processor performing a million high-level instructions per second. In cases where the running time exceeds 1025 years, we simply record the algorithm as taking a very long time.

n

< 1 sec < 1 sec < 1 sec < 1 sec < 1 sec < 1 sec < 1 sec
1 sec
n log2 n <1sec
<1sec <1sec <1sec <1sec <1sec
2 sec 20 sec
n2 n3 1.5n 2n n!
n = 10
n = 30
n = 50
n = 100
n = 1,000
n = 10,000 n = 100,000 n = 1,000,000
< 1 sec < 1 sec < 1 sec < 1 sec
1 sec 2 min 3 hours 12 days
<1sec <1sec <1sec
1 sec 18 min 12 days 32 years 31,710 years
< 1 sec < 1 sec 11 min 12,892 years very long very long very long very long
< 1 sec 18 min
36 years 1017 years very long
very long very long very long
4 sec 1025 years very long very long very long very long very long very long
which an algorithm with exponential worst-case behavior generally runs well on the kinds of instances that arise in practice; and there are also cases where the best polynomial-time algorithm for a problem is completely impractical due to large constants or a high exponent on the polynomial bound. All this serves to reinforce the point that our emphasis on worst-case, polynomial-time bounds is only an abstraction of practical situations. But overwhelmingly, the concrete mathematical definition of polynomial time has turned out to corre- spond surprisingly well in practice to what we observe about the efficiency of algorithms, and the tractability of problems, in real life.
One further reason why the mathematical formalism and the empirical evidence seem to line up well in the case of polynomial-time solvability is that the gulf between the growth rates of polynomial and exponential functions is enormous. Suppose, for example, that we have a processor that executes a million high-level instructions per second, and we have algorithms with running-time bounds of n, n log2 n, n2, n3, 1.5n, 2n, and n!. In Table 2.1, we show the running times of these algorithms (in seconds, minutes, days, or years) for inputs of size n = 10, 30, 50, 100, 1,000, 10,000, 100,000, and 1,000,000.
There is a final, fundamental benefit to making our definition of efficiency so specific: it becomes negatable. It becomes possible to express the notion that there is no efficient algorithm for a particular problem. In a sense, being able to do this is a prerequisite for turning our study of algorithms into good science, for it allows us to ask about the existence or nonexistence of efficient algorithms as a well-defined question. In contrast, both of our
previous definitions were completely subjective, and hence limited the extent to which we could discuss certain issues in concrete terms.
In particular, the first of our definitions, which was tied to the specific implementation of an algorithm, turned efficiency into a moving target: as processor speeds increase, more and more algorithms fall under this notion of efficiency. Our definition in terms of polynomial time is much more an absolute notion; it is closely connected with the idea that each problem has an intrinsic level of computational tractability: some admit efficient solutions, and others do not.
2.2 Asymptotic Order of Growth
Our discussion of computational tractability has turned out to be intrinsically based on our ability to express the notion that an algorithm’s worst-case running time on inputs of size n grows at a rate that is at most proportional to some function f (n). The function f (n) then becomes a bound on the running time of the algorithm. We now discuss a framework for talking about this concept.
We will mainly express algorithms in the pseudo-code style that we used for the Gale-Shapley algorithm. At times we will need to become more formal, but this style of specifying algorithms will be completely adequate for most purposes. When we provide a bound on the running time of an algorithm, we will generally be counting the number of such pseudo-code steps that are executed; in this context, one step will consist of assigning a value to a variable, looking up an entry in an array, following a pointer, or performing an arithmetic operation on a fixed-size integer.
When we seek to say something about the running time of an algorithm on inputs of size n, one thing we could aim for would be a very concrete statement such as, “On any input of size n, the algorithm runs for at most 1.62n2 + 3.5n + 8 steps.” This may be an interesting statement in some contexts, but as a general goal there are several things wrong with it. First, getting such a precise bound may be an exhausting activity, and more detail than we wanted anyway. Second, because our ultimate goal is to identify broad classes of algorithms that have similar behavior, we’d actually like to classify running times at a coarser level of granularity so that similarities among different algorithms, and among different problems, show up more clearly. And finally, extremely detailed statements about the number of steps an algorithm executes are often—in a strong sense—meaningless. As just discussed, we will generally be counting steps in a pseudo-code specification of an algorithm that resembles a high- level programming language. Each one of these steps will typically unfold into some fixed number of primitive steps when the program is compiled into
2.2 Asymptotic Order of Growth
35
36
Chapter 2 Basics of Algorithm Analysis
an intermediate representation, and then into some further number of steps depending on the particular architecture being used to do the computing. So the most we can safely say is that as we look at different levels of computational abstraction, the notion of a “step” may grow or shrink by a constant factor— for example, if it takes 25 low-level machine instructions to perform one operation in our high-level language, then our algorithm that took at most 1.62n2 + 3.5n + 8 steps can also be viewed as taking 40.5n2 + 87.5n + 200 steps when we analyze it at a level that is closer to the actual hardware.
O, , and
For all these reasons, we want to express the growth rate of running times and other functions in a way that is insensitive to constant factors and low- order terms. In other words, we’d like to be able to take a running time like the one we discussed above, 1.62n2 + 3.5n + 8, and say that it grows like n2, up to constant factors. We now discuss a precise way to do this.
Asymptotic Upper Bounds Let T(n) be a function—say, the worst-case run- ning time of a certain algorithm on an input of size n. (We will assume that all the functions we talk about here take nonnegative values.) Given another function f(n), we say that T(n) is O(f(n)) (read as “T(n) is order f(n)”) if, for sufficiently large n, the function T(n) is bounded above by a constant multiple of f (n). We will also sometimes write this as T (n) = O(f (n)). More precisely, T(n) is O(f(n)) if there exist constants c > 0 and n0 ≥ 0 so that for all n ≥ n0, we have T (n) ≤ c · f (n). In this case, we will say that T is asymptotically upper- bounded by f . It is important to note that this definition requires a constant c to exist that works for all n; in particular, c cannot depend on n.

As an example of how this definition lets us express upper bounds on running times, consider an algorithm whose running time (as in the earlier discussion) has the form T(n) = pn2 + qn + r for positive constants p, q, and r. We’d like to claim that any such function is O(n2). To see why, we notice that for all n≥1, we have qn≤qn2, and r ≤rn2. So we can write

T(n)=pn2 +qn+r ≤pn2 +qn2 +rn2 =(p+q+r)n2

for all n ≥ 1. This inequality is exactly what the definition of O(·) requires:

T(n) ≤ cn2, where c = p + q + r.

Note that O(·) expresses only an upper bound, not the exact growth rate of the function. For example, just as we claimed that the function T(n)= pn2 + qn + r is O(n2), it’s also correct to say that it’s O(n3). Indeed, we just argued that T(n)≤(p+q+r)n2, and since we also have n2≤n3, we can conclude that T(n) ≤ (p + q + r)n3 as the definition of O(n3) requires. The fact that a function can have many upper bounds is not just a trick of the notation; it shows up in the analysis of running times as well. There are cases

where an algorithm has been proved to have running time O(n3); some years pass, people analyze the same algorithm more carefully, and they show that in fact its running time is O(n2). There was nothing wrong with the first result; it was a correct upper bound. It’s simply that it wasn’t the “tightest” possible running time.

Asymptotic Lower Bounds There is a complementary notation for lower bounds. Often when we analyze an algorithm—say we have just proven that its worst-case running time T(n) is O(n2)—we want to show that this upper bound is the best one possible. To do this, we want to express the notion that for arbitrarily large input sizes n, the function T(n) is at least a constant multiple of some specific function f (n). (In this example, f (n) happens to be n2.) Thus, we say that T(n) is (f(n)) (also written T(n) = (f(n))) if there exist constants ε > 0 and n0 ≥ 0 so that for all n ≥ n0, we have T (n) ≥ ε · f (n). By analogy with O(·) notation, we will refer to T in this case as being asymptotically lower- bounded by f. Again, note that the constant ε must be fixed, independent of n.

This definition works just like O(·), except that we are bounding the function T(n) from below, rather than from above. For example, returning to the function T(n) = pn2 + qn + r, where p, q, and r are positive constants, let’s claim that T(n) = (n2). Whereas establishing the upper bound involved “inflating” the terms in T(n) until it looked like a constant times n2, now we need to do the opposite: we need to reduce the size of T(n) until it looks like a constant times n2. It is not hard to do this; for all n ≥ 0, we have

T(n) = pn2 + qn + r ≥ pn2,

which meets what is required by the definition of (·) with ε = p > 0.

Just as we discussed the notion of “tighter” and “weaker” upper bounds, the same issue arises for lower bounds. For example, it is correct to say that ourfunctionT(n)=pn2+qn+r is(n),sinceT(n)≥pn2≥pn.

Asymptotically Tight Bounds If we can show that a running time T(n) is both O(f (n)) and also (f (n)), then in a natural sense we’ve found the “right” bound: T(n) grows exactly like f(n) to within a constant factor. This, for example, is the conclusion we can draw from the fact that T(n) = pn2 + qn + r is both O(n2) and (n2).

There is a notation to express this: if a function T(n) is both O(f(n)) and (f(n)), we say that T(n) is (f(n)). In this case, we say that f(n) is an asymptotically tight bound for T(n). So, for example, our analysis above shows that T(n)=pn2 +qn+r is (n2).

Asymptotically tight bounds on worst-case running times are nice things to find, since they characterize the worst-case performance of an algorithm

2.2 Asymptotic Order of Growth

37

38

Chapter 2 Basics of Algorithm Analysis

precisely up to constant factors. And as the definition of (·) shows, one can obtain such bounds by closing the gap between an upper bound and a lower bound. For example, sometimes you will read a (slightly informally phrased) sentence such as “An upper bound of O(n3) has been shown on the worst-case running time of the algorithm, but there is no example known on which the algorithm runs for more than (n2) steps.” This is implicitly an invitation to search for an asymptotically tight bound on the algorithm’s worst-case running time.

Sometimes one can also obtain an asymptotically tight bound directly by computing a limit as n goes to infinity. Essentially, if the ratio of functions f(n) and g(n) converges to a positive constant as n goes to infinity, then f (n) = (g(n)).

(2.1) Let f and g be two functions that lim f (n)

n→∞ g(n)

exists and is equal to some number c > 0. Then f (n) = (g(n)).

Proof. We will use the fact that the limit exists and is positive to show that f (n) = O(g(n)) and f (n) = (g(n)), as required by the definition of (·).

Since

lim f(n)=c>0, n→∞ g(n)

it follows from the definition of a limit that there is some n0 beyond which the ratio is always between 1c and 2c. Thus, f(n) ≤ 2cg(n) for all n ≥ n0, which

2

implies that f(n) = O(g(n)); and f(n) ≥ 1cg(n) for all n ≥ n0, which implies 2

that f (n) = (g(n)).

Properties of Asymptotic Growth Rates

Having seen the definitions of O, , and , it is useful to explore some of their basic properties.

Transitivity A first property is transitivity: if a function f is asymptotically upper-bounded by a function g, and if g in turn is asymptotically upper- bounded by a function h, then f is asymptotically upper-bounded by h. A similar property holds for lower bounds. We write this more precisely as follows.

(2.2)

(a) If f =O(g) and g =O(h), then f =O(h). (b) If f =(g) and g =(h), then f =(h).

Proof. We’ll prove part (a) of this claim; the proof of part (b) is very similar.

For (a), we’re given that for some constants c and n0, we have f (n) ≤ cg(n) for all n ≥ n0. Also, for some (potentially different) constants c′ and n0′ , we have g(n) ≤ c′h(n) for all n ≥ n0′ . So consider any number n that is at least as large as both n0 and n0′ . We have f (n) ≤ cg(n) ≤ cc′h(n), and so f (n) ≤ cc′h(n) for all n ≥ max(n0, n0′ ). This latter inequality is exactly what is required for showing that f = O(h).

Combining parts (a) and (b) of (2.2), we can obtain a similar result for asymptotically tight bounds. Suppose we know that f = (g) and that g = (h). Then since f = O(g) and g = O(h), we know from part (a) that f =O(h); since f =(g) and g =(h), we know from part (b) that f =(h). It follows that f = (h). Thus we have shown

(2.3) If f =(g) and g =(h), then f =(h).

Sums of Functions It is also useful to have results that quantify the effect of adding two functions. First, if we have an asymptotic upper bound that applies to each of two functions f and g, then it applies to their sum.

(2.4) Suppose that f and g are two functions such that for some other function h, we have f =O(h) and g =O(h). Then f +g =O(h).

Proof. We’re given that for some constants c and n0, we have f (n) ≤ ch(n) for all n ≥ n0. Also, for some (potentially different) constants c′ and n0′ , we have g(n) ≤ c′h(n) for all n ≥ n0′ . So consider any number n that is at least as large as both n0 and n0′ . We have f (n) + g(n) ≤ ch(n) + c′h(n). Thus f(n) + g(n) ≤ (c + c′)h(n) for all n ≥ max(n0, n0′ ), which is exactly what is required for showing that f + g = O(h).

There is a generalization of this to sums of a fixed constant number of functions k, where k may be larger than two. The result can be stated precisely as follows; we omit the proof, since it is essentially the same as the proof of (2.4), adapted to sums consisting of k terms rather than just two.

(2.5) Let k be a fixed constant, and let f1, f2, . . . , fk and h be functions such that fi = O(h) for all i. Then f1 + f2 + . . . + fk = O(h).

There is also a consequence of (2.4) that covers the following kind of situation. It frequently happens that we’re analyzing an algorithm with two high-level parts, and it is easy to show that one of the two parts is slower than the other. We’d like to be able to say that the running time of the whole algorithm is asymptotically comparable to the running time of the slow part. Since the overall running time is a sum of two functions (the running times of

2.2 Asymptotic Order of Growth

39

40

Chapter 2 Basics of Algorithm Analysis

the two parts), results on asymptotic bounds for sums of functions are directly relevant.

(2.6) Suppose that f and g are two functions (taking nonnegative values) such that g = O(f ). Then f + g = (f ). In other words, f is an asymptotically tight bound for the combined function f + g.

Proof. Clearly f + g = (f), since for all n ≥ 0, we have f(n) + g(n) ≥ f(n). So to complete the proof, we need to show that f + g = O(f ).

But this is a direct consequence of (2.4): we’re given the fact that g = O(f ), and also f = O(f) holds for any function, so by (2.4) we have f + g = O(f).

This result also extends to the sum of any fixed, constant number of functions: the most rapidly growing among the functions is an asymptotically tight bound for the sum.

Asymptotic Bounds for Some Common Functions

There are a number of functions that come up repeatedly in the analysis of algorithms, and it is useful to consider the asymptotic properties of some of the most basic of these: polynomials, logarithms, and exponentials.

Polynomials Recall that a polynomial is a function that can be written in theformf(n)=a0+a1n+a2n2+…+adnd forsomeintegerconstantd>0, where the final coefficient ad is nonzero. This value d is called the degree of the polynomial. For example, the functions of the form pn2 + qn + r (with p ̸= 0) that we considered earlier are polynomials of degree 2.

A basic fact about polynomials is that their asymptotic rate of growth is determined by their “high-order term”—the one that determines the degree. We state this more formally in the following claim. Since we are concerned here only with functions that take nonnegative values, we will restrict our attention to polynomials for which the high-order term has a positive coefficient ad > 0.

(2.7) Let f be a polynomial of degree d, in which the coefficient ad is positive. Then f = O(nd).

Proof. Wewritef=a0+a1n+a2n2+…+adnd,wheread>0.Theupper bound is a direct application of (2.5). First, notice that coefficients aj for j < d may be negative, but in any case we have ajnj ≤ |aj|nd for all n ≥ 1. Thus each term in the polynomial is O(nd). Since f is a sum of a constant number of functions, each of which is O(nd), it follows from (2.5) that f is O(nd).
One can also show that under the conditions of (2.7), we have f = (nd), and hence it follows that in fact f = (nd).
This is a good point at which to discuss the relationship between these types of asymptotic bounds and the notion of polynomial time, which we arrived at in the previous section as a way to formalize the more elusive concept of efficiency. Using O(·) notation, it’s easy to formally define polynomial time: a polynomial-time algorithm is one whose running time T(n) is O(nd) for some constant d, where d is independent of the input size.
So algorithms with running-time bounds like O(n2) and O(n3) are polynomial-time algorithms. But it’s important to realize that an algorithm can be polynomial time even if its running time is not written as n raised to some integer power. To begin with, a number of algorithms have running times of the form O(nx) for some number x that is not an integer. For example, in Chapter 5 we will see an algorithm whose running time is O(n1.59); we will also see exponents less than 1, as in bounds like O(√n) = O(n1/2).
To take another common kind of example, we will see many algorithms whose running times have the form O(n log n). Such algorithms are also polynomial time: as we will see next, log n ≤ n for all n ≥ 1, and hence n log n ≤ n2 for all n ≥ 1. In other words, if an algorithm has running time O(n log n), then it also has running time O(n2), and so it is a polynomial-time algorithm.
Logarithms Recall that logb n is the number x such that bx = n. One way to get an approximate sense of how fast logb n grows is to note that, if we round it down to the nearest integer, it is one less than the number of digits in the base-b representation of the number n. (Thus, for example, 1 + log2 n, rounded down, is the number of bits needed to represent n.)
So logarithms are very slowly growing functions. In particular, for every base b, the function logb n is asymptotically bounded by every function of the form nx, even for (noninteger) values of x arbitrary close to 0.
(2.8) For every b>1and every x>0, we have logb n=O(nx).

One can directly translate between logarithms of different bases using the

following fundamental identity:

loga n = logb n .

logb a

This equation explains why you’ll often notice people writing bounds like

O(log n) without indicating the base of the logarithm. This is not sloppy

usage: the identity above says that log n = 1 · log n, so the point is that a logb a b

loga n = (logb n), and the base of the logarithm is not important when writing bounds using asymptotic notation.

2.2 Asymptotic Order of Growth

41

42

Chapter 2 Basics of Algorithm Analysis

Exponentials Exponential functions are functions of the form f (n) = r n for some constant base r. Here we will be concerned with the case in which r > 1, which results in a very fast-growing function.

In particular, where polynomials raise n to a fixed exponent, exponentials raise a fixed number to n as a power; this leads to much faster rates of growth. One way to summarize the relationship between polynomials and exponentials is as follows.

(2.9) Foreveryr>1andeveryd>0,wehavend=O(rn).

In particular, every exponential grows faster than every polynomial. And as we saw in Table 2.1, when you plug in actual values of n, the differences in growth rates are really quite impressive.

Just as people write O(log n) without specifying the base, you’ll also see people write “The running time of this algorithm is exponential,” without specifying which exponential function they have in mind. Unlike the liberal use of log n, which is justified by ignoring constant factors, this generic use of the term “exponential” is somewhat sloppy. In particular, for different bases r > s > 1, it is never the case that rn = (sn). Indeed, this would require that for some constant c > 0, we would have rn ≤ csn for all sufficiently large n. But rearranging this inequality would give (r/s)n ≤ c for all sufficiently large n. Since r > s, the expression (r/s)n is tending to infinity with n, and so it cannot possibly remain bounded by a fixed constant c.

So asymptotically speaking, exponential functions are all different. Still, it’s usually clear what people intend when they inexactly write “The running time of this algorithm is exponential”—they typically mean that the running time grows at least as fast as some exponential function, and all exponentials grow so fast that we can effectively dismiss this algorithm without working out further details of the exact running time. This is not entirely fair. Occasionally there’s more going on with an exponential algorithm than first appears, as we’ll see, for example, in Chapter 10; but as we argued in the first section of this chapter, it’s a reasonable rule of thumb.

Taken together, then, logarithms, polynomials, and exponentials serve as useful landmarks in the range of possible functions that you encounter when analyzing running times. Logarithms grow more slowly than polynomials, and polynomials grow more slowly than exponentials.

2.3 Implementing the Stable Matching Algorithm Using Lists and Arrays

We’ve now seen a general approach for expressing bounds on the running time of an algorithm. In order to asymptotically analyze the running time of

2.3 Implementing the Stable Matching Algorithm Using Lists and Arrays

43

an algorithm expressed in a high-level fashion—as we expressed the Gale- Shapley Stable Matching algorithm in Chapter 1, for example—one doesn’t have to actually program, compile, and execute it, but one does have to think about how the data will be represented and manipulated in an implementation of the algorithm, so as to bound the number of computational steps it takes.

The implementation of basic algorithms using data structures is something that you probably have had some experience with. In this book, data structures will be covered in the context of implementing specific algorithms, and so we will encounter different data structures based on the needs of the algorithms we are developing. To get this process started, we consider an implementation of the Gale-Shapley Stable Matching algorithm; we showed earlier that the algorithm terminates in at most n2 iterations, and our implementation here provides a corresponding worst-case running time of O(n2), counting actual computational steps rather than simply the total number of iterations. To get such a bound for the Stable Matching algorithm, we will only need to use two of the simplest data structures: lists and arrays. Thus, our implementation also provides a good chance to review the use of these basic data structures as well.

In the Stable Matching Problem, each man and each woman has a ranking of all members of the opposite gender. The very first question we need to discuss is how such a ranking will be represented. Further, the algorithm maintains a matching and will need to know at each step which men and women are free, and who is matched with whom. In order to implement the algorithm, we need to decide which data structures we will use for all these things.

An important issue to note here is that the choice of data structure is up to the algorithm designer; for each algorithm we will choose data structures that make it efficient and easy to implement. In some cases, this may involve preprocessing the input to convert it from its given input representation into a data structure that is more appropriate for the problem being solved.

Arrays and Lists

To start our discussion we will focus on a single list, such as the list of women in order of preference by a single man. Maybe the simplest way to keep a list of n elements is to use an array A of length n, and have A[i] be the ith element of the list. Such an array is simple to implement in essentially all standard programming languages, and it has the following properties.

. We can answer a query of the form “What is the ith element on the list?” in O(1) time, by a direct access to the value A[i].

. If we want to determine whether a particular element e belongs to the list (i.e., whether it is equal to A[i] for some i), we need to check the

44

Chapter 2 Basics of Algorithm Analysis

elements one by one in O(n) time, assuming we don’t know anything about the order in which the elements appear in A.

. If the array elements are sorted in some clear way (either numerically or alphabetically), then we can determine whether an element e belongs to the list in O(log n) time using binary search; we will not need to use binary search for any part of our stable matching implementation, but we will have more to say about it in the next section.

An array is less good for dynamically maintaining a list of elements that changes over time, such as the list of free men in the Stable Matching algorithm; since men go from being free to engaged, and potentially back again, a list of free men needs to grow and shrink during the execution of the algorithm. It is generally cumbersome to frequently add or delete elements to a list that is maintained as an array.

An alternate, and often preferable, way to maintain such a dynamic set of elements is via a linked list. In a linked list, the elements are sequenced together by having each element point to the next in the list. Thus, for each element v on the list, we need to maintain a pointer to the next element; we set this pointer to null if i is the last element. We also have a pointer First that points to the first element. By starting at First and repeatedly following pointers to the next element until we reach null, we can thus traverse the entire contents of the list in time proportional to its length.

A generic way to implement such a linked list, when the set of possible elements may not be fixed in advance, is to allocate a record e for each element that we want to include in the list. Such a record would contain a field e.val that contains the value of the element, and a field e.Next that contains a pointer to the next element in the list. We can create a doubly linked list, which is traversable in both directions, by also having a field e.Prev that contains a pointer to the previous element in the list. (e.Prev = null if e is the first element.) We also include a pointer Last, analogous to First, that points to the last element in the list. A schematic illustration of part of such a list is shown in the first line of Figure 2.1.

A doubly linked list can be modified as follows.

. Deletion. To delete the element e from a doubly linked list, we can just “splice it out” by having the previous element, referenced by e.Prev, and the next element, referenced by e.Next, point directly to each other. The deletion operation is illustrated in Figure 2.1.

. Insertion. To insert element e between elements d and f in a list, we “splice it in” by updating d.Next and f .Prev to point to e, and the Next and Prev pointers of e to point to d and f , respectively. This operation is

2.3 Implementing the Stable Matching Algorithm Using Lists and Arrays

45

Before deleting e:

After deleting e:

Element e

Element e

val

val

val

val

val

val

Figure 2.1 A schematic representation of a doubly linked list, showing the deletion of an element e.

essentially the reverse of deletion, and indeed one can see this operation at work by reading Figure 2.1 from bottom to top.

Inserting or deleting e at the beginning of the list involves updating the First pointer, rather than updating the record of the element before e.

While lists are good for maintaining a dynamically changing set, they also have disadvantages. Unlike arrays, we cannot find the ith element of the list in O(1) time: to find the ith element, we have to follow the Next pointers starting from the beginning of the list, which takes a total of O(i) time.

Given the relative advantages and disadvantages of arrays and lists, it may happen that we receive the input to a problem in one of the two formats and want to convert it into the other. As discussed earlier, such preprocessing is often useful; and in this case, it is easy to convert between the array and list representations in O(n) time. This allows us to freely choose the data structure that suits the algorithm better and not be constrained by the way the information is given as input.

Implementing the Stable Matching Algorithm

Next we will use arrays and linked lists to implement the Stable Matching algo- rithm from Chapter 1. We have already shown that the algorithm terminates in at most n2 iterations, and this provides a type of upper bound on the running time. However, if we actually want to implement the G-S algorithm so that it runs in time proportional to n2, we need to be able to implement each iteration in constant time. We discuss how to do this now.

For simplicity, assume that the set of men and women are both {1, . . . , n}. To ensure this, we can order the men and women (say, alphabetically), and associate number i with the ith man mi or ith women wi in this order. This

46

Chapter 2 Basics of Algorithm Analysis

assumption (or notation) allows us to define an array indexed by all men or all women. We need to have a preference list for each man and for each woman. To do this we will have two arrays, one for women’s preference lists and one for the men’s preference lists; we will use ManPref[m, i] to denote the ith woman on man m’s preference list, and similarly WomanPref[w, i] to be the ith man on the preference list of woman w. Note that the amount of space needed to give the preferences for all 2n individuals is O(n2), as each person has a list of length n.

We need to consider each step of the algorithm and understand what data structure allows us to implement it efficiently. Essentially, we need to be able to do each of four things in constant time.

1. We need to be able to identify a free man.

2. We need, for a man m, to be able to identify the highest-ranked woman to whom he has not yet proposed.

3. For a woman w, we need to decide if w is currently engaged, and if she is, we need to identify her current partner.

4. Forawomanwandtwomenmandm′,weneedtobeabletodecide, again in constant time, which of m or m′ is preferred by w.

First, consider selecting a free man. We will do this by maintaining the set of free men as a linked list. When we need to select a free man, we take the first man m on this list. We delete m from the list if he becomes engaged, and possibly insert a different man m′, if some other man m′ becomes free. In this case, m′ can be inserted at the front of the list, again in constant time.

Next, consider a man m. We need to identify the highest-ranked woman to whom he has not yet proposed. To do this we will need to maintain an extra array Next that indicates for each man m the position of the next woman he will propose to on his list. We initialize Next[m] = 1 for all men m. If a man m needs to propose to a woman, he’ll propose to w = ManPref[m,Next[m]], and once he proposes to w, we increment the value of Next[m] by one, regardless of whether or not w accepts the proposal.

Now assume man m proposes to woman w; we need to be able to identify the man m′ that w is engaged to (if there is such a man). We can do this by maintaining an array Current of length n, where Current[w] is the woman w’s current partner m′. We set Current[w] to a special null symbol when we need to indicate that woman w is not currently engaged; at the start of the algorithm, Current[w] is initialized to this null symbol for all women w.

To sum up, the data structures we have set up thus far can implement the operations (1)–(3) in O(1) time each.

Maybe the trickiest question is how to maintain women’s preferences to keep step (4) efficient. Consider a step of the algorithm, when man m proposes to a woman w. Assume w is already engaged, and her current partner is m′ =Current[w]. We would like to decide in O(1) time if woman w prefers m or m′. Keeping the women’s preferences in an array WomanPref, analogous to the one we used for men, does not work, as we would need to walk through w’s list one by one, taking O(n) time to find m and m′ on the list. While O(n) is still polynomial, we can do a lot better if we build an auxiliary data structure at the beginning.

At the start of the algorithm, we create an n × n array Ranking, where Ranking[w, m] contains the rank of man m in the sorted order of w’s prefer- ences. By a single pass through w’s preference list, we can create this array in linear time for each woman, for a total initial time investment proportional to n2. Then, to decide which of m or m′ is preferred by w, we simply compare the values Ranking[w, m] and Ranking[w, m′].

This allows us to execute step (4) in constant time, and hence we have everything we need to obtain the desired running time.

(2.10) The data structures described above allow us to implement the G-S algorithm in O(n2) time.

2.4 A Survey of Common Running Times

When trying to analyze a new algorithm, it helps to have a rough sense of the “landscape” of different running times. Indeed, there are styles of analysis that recur frequently, and so when one sees running-time bounds like O(n), O(n log n), and O(n2) appearing over and over, it’s often for one of a very small number of distinct reasons. Learning to recognize these common styles of analysis is a long-term goal. To get things under way, we offer the following survey of common running-time bounds and some of the typical approaches that lead to them.

Earlier we discussed the notion that most problems have a natural “search space”—the set of all possible solutions—and we noted that a unifying theme in algorithm design is the search for algorithms whose performance is more efficient than a brute-force enumeration of this search space. In approaching a new problem, then, it often helps to think about two kinds of bounds: one on the running time you hope to achieve, and the other on the size of the problem’s natural search space (and hence on the running time of a brute-force algorithm for the problem). The discussion of running times in this section will begin in many cases with an analysis of the brute-force algorithm, since it is a useful

2.4 A Survey of Common Running Times

47

48

Chapter 2 Basics of Algorithm Analysis

way to get one’s bearings with respect to a problem; the task of improving on such algorithms will be our goal in most of the book.

Linear Time

An algorithm that runs in O(n), or linear, time has a very natural property: its running time is at most a constant factor times the size of the input. One basic way to get an algorithm with this running time is to process the input in a single pass, spending a constant amount of time on each item of input encountered. Other algorithms achieve a linear time bound for more subtle reasons. To illustrate some of the ideas here, we consider two simple linear- time algorithms as examples.

Computing the Maximum Computing the maximum of n numbers, for ex- ample, can be performed in the basic “one-pass” style. Suppose the numbers are provided as input in either a list or an array. We process the numbers a1, a2, . . . , an in order, keeping a running estimate of the maximum as we go. Each time we encounter a number ai, we check whether ai is larger than our current estimate, and if so we update the estimate to ai.

max = a1

For i=2 to n

If ai > max then set max=ai

Endif Endfor

In this way, we do constant work per element, for a total running time of O(n).

Sometimes the constraints of an application force this kind of one-pass algorithm on you—for example, an algorithm running on a high-speed switch on the Internet may see a stream of packets flying past it, and it can try computing anything it wants to as this stream passes by, but it can only perform a constant amount of computational work on each packet, and it can’t save the stream so as to make subsequent scans through it. Two different subareas of algorithms, online algorithms and data stream algorithms, have developed to study this model of computation.

Merging Two Sorted Lists Often, an algorithm has a running time of O(n), but the reason is more complex. We now describe an algorithm for merging two sorted lists that stretches the one-pass style of design just a little, but still has a linear running time.

Suppose we are given two lists of n numbers each, a1, a2, . . . , an and b1, b2, . . . , bn, and each is already arranged in ascending order. We’d like to

merge these into a single list c1, c2, . . . , c2n that is also arranged in ascending order. For example, merging the lists 2, 3, 11, 19 and 4, 9, 16, 25 results in the output 2, 3, 4, 9, 11, 16, 19, 25.

To do this, we could just throw the two lists together, ignore the fact that they’re separately arranged in ascending order, and run a sorting algorithm. But this clearly seems wasteful; we’d like to make use of the existing order in the input. One way to think about designing a better algorithm is to imagine performing the merging of the two lists by hand: suppose you’re given two piles of numbered cards, each arranged in ascending order, and you’d like to produce a single ordered pile containing all the cards. If you look at the top card on each stack, you know that the smaller of these two should go first on the output pile; so you could remove this card, place it on the output, and now iterate on what’s left.

In other words, we have the following algorithm.

To merge sorted lists A = a1,…,an and B=b1,…,bn: Maintain a Current pointer into each list, initialized to

point to the front elements

While both lists are nonempty:

Let ai and bj be the elements pointed to by the Current pointer Append the smaller of these two to the output list

Advance the Current pointer in the list from which the

smaller element was selected

EndWhile

Once one list is empty, append the remainder of the other list

to the output

See Figure 2.2 for a picture of this process.

Append the smaller of ai and bj to the output.

2.4 A Survey of Common Running Times

49

//////

ai

A B

Merged result

///

bj

Figure 2.2 To merge sorted lists A and B, we repeatedly extract the smaller item from the front of the two lists and append it to the output.

50

Chapter 2 Basics of Algorithm Analysis

Now, to show a linear-time bound, one is tempted to describe an argument like what worked for the maximum-finding algorithm: “We do constant work per element, for a total running time of O(n).” But it is actually not true that we do only constant work per element. Suppose that n is an even number, and consider the lists A=1,3,5,…,2n−1and B=n,n+2,n+4,…,3n−2. The number b1 at the front of list B will sit at the front of the list for n/2 iterations while elements from A are repeatedly being selected, and hence it will be involved in (n) comparisons. Now, it is true that each element can be involved in at most O(n) comparisons (at worst, it is compared with each element in the other list), and if we sum this over all elements we get a running-time bound of O(n2). This is a correct bound, but we can show something much stronger.

The better way to argue is to bound the number of iterations of the While loop by an “accounting” scheme. Suppose we charge the cost of each iteration to the element that is selected and added to the output list. An element can be charged only once, since at the moment it is first charged, it is added to the output and never seen again by the algorithm. But there are only 2n elements total, and the cost of each iteration is accounted for by a charge to some element, so there can be at most 2n iterations. Each iteration involves a constant amount of work, so the total running time is O(n), as desired.

While this merging algorithm iterated through its input lists in order, the “interleaved” way in which it processed the lists necessitated a slightly subtle running-time analysis. In Chapter 3 we will see linear-time algorithms for graphs that have an even more complex flow of control: they spend a constant amount of time on each node and edge in the underlying graph, but the order in which they process the nodes and edges depends on the structure of the graph.

O(n log n) Time

O(n log n) is also a very common running time, and in Chapter 5 we will see one of the main reasons for its prevalence: it is the running time of any algorithm that splits its input into two equal-sized pieces, solves each piece recursively, and then combines the two solutions in linear time.

Sorting is perhaps the most well-known example of a problem that can be solved this way. Specifically, the Mergesort algorithm divides the set of input numbers into two equal-sized pieces, sorts each half recursively, and then merges the two sorted halves into a single sorted output list. We have just seen that the merging can be done in linear time; and Chapter 5 will discuss how to analyze the recursion so as to get a bound of O(n log n) on the overall running time.

One also frequently encounters O(n log n) as a running time simply be- cause there are many algorithms whose most expensive step is to sort the input. For example, suppose we are given a set of n time-stamps x1, x2, . . . , xn on which copies of a file arrived at a server, and we’d like to find the largest interval of time between the first and last of these time-stamps during which no copy of the file arrived. A simple solution to this problem is to first sort the time-stamps x1, x2, . . . , xn and then process them in sorted order, determining the sizes of the gaps between each number and its successor in ascending order. The largest of these gaps is the desired subinterval. Note that this algo- rithm requires O(n log n) time to sort the numbers, and then it spends constant work on each number in ascending order. In other words, the remainder of the algorithm after sorting follows the basic recipe for linear time that we discussed earlier.

Quadratic Time

Here’s a basic problem: suppose you are given n points in the plane, each specified by (x, y) coordinates, and you’d like to find the pair of points that are closest together. The natural brute-force algorithm for this problem would enumerate all pairs of points, compute the distance between each pair, and then choose the pair for which this distance is smallest.

What is the running time of this algorithm? The number of pairs of points is n = n(n−1) , and since this quantity is bounded by 1 n2, it is O(n2). More

ways of choosing the first member of the pair (at most n) by the number

of ways of choosing the second member of the pair (also at most n). The

distance between points (x , y ) and (x , y ) can be computed by the formula

(xi − xj)2 + (yi − yj)2 in constant time, so the overall running time is O(n2). This example illustrates a very common way in which a running time of O(n2) arises: performing a search over all pairs of input items and spending constant time per pair.

Quadratic time also arises naturally from a pair of nested loops: An algo- rithm consists of a loop with O(n) iterations, and each iteration of the loop launches an internal loop that takes O(n) time. Multiplying these two factors of n together gives the running time.

The brute-force algorithm for finding the closest pair of points can be written in an equivalent way with two nested loops:

For each input point (xi,yi)

For each other input point (xj, yj)

ii jj

2.4 A Survey of Common Running Times

51

222

crudely, the number of pairs is O(n2) because we multiply the number of

Compute distance d = (xi − xj)2 + (yi − yj)2

52

Chapter 2 Basics of Algorithm Analysis

If d is less than the current minimum, update minimum to d Endfor

Endfor

Note how the “inner” loop, over (xj,yj), has O(n) iterations, each taking constant time; and the “outer” loop, over (xi,yi), has O(n) iterations, each invoking the inner loop once.

It’s important to notice that the algorithm we’ve been discussing for the Closest-Pair Problem really is just the brute-force approach: the natural search space for this problem has size O(n2), and we’re simply enumerating it. At first, one feels there is a certain inevitability about this quadratic algorithm— we have to measure all the distances, don’t we?—but in fact this is an illusion. In Chapter 5 we describe a very clever algorithm that finds the closest pair of points in the plane in only O(n log n) time, and in Chapter 13 we show how randomization can be used to reduce the running time to O(n).

Cubic Time

More elaborate sets of nested loops often lead to algorithms that run in O(n3) time. Consider, for example, the following problem. We are given sets S1,S2,…,Sn, each of which is a subset of {1,2,…,n}, and we would like to know whether some pair of these sets is disjoint—in other words, has no elements in common.

What is the running time needed to solve this problem? Let’s suppose that each set Si is represented in such a way that the elements of Si can be listed in constant time per element, and we can also check in constant time whether a given number p belongs to Si. The following is a direct way to approach the problem.

For pair of sets Si and Sj

Determine whether Si and Sj have an element in common

Endfor

This is a concrete algorithm, but to reason about its running time it helps to open it up (at least conceptually) into three nested loops.

For each set Si

For each other set Sj

For each element p of Si

Determine whether p also belongs to Sj

Endfor

If no element of Si belongs to Sj then

Report that Si and Sj are disjoint Endif

Endfor

Endfor

Each of the sets has maximum size O(n), so the innermost loop takes time O(n). Looping over the sets Sj involves O(n) iterations around this innermost loop; and looping over the sets Si involves O(n) iterations around this. Multi- plying these three factors of n together, we get the running time of O(n3).

For this problem, there are algorithms that improve on O(n3) running time, but they are quite complicated. Furthermore, it is not clear whether the improved algorithms for this problem are practical on inputs of reasonable size.

O(nk) Time

In the same way that we obtained a running time of O(n2) by performing brute- force search over all pairs formed from a set of n items, we obtain a running time of O(nk) for any constant k when we search over all subsets of size k.

Consider, for example, the problem of finding independent sets in a graph, which we discussed in Chapter 1. Recall that a set of nodes is independent if no two are joined by an edge. Suppose, in particular, that for some fixed constant k, we would like to know if a given n-node input graph G has an independent set of size k. The natural brute-force algorithm for this problem would enumerate all subsets of k nodes, and for each subset S it would check whether there is an edge joining any two members of S. That is,

For each subset S of k nodes

Check whether S constitutes an independent set If S is an independent set then

Stop and declare success

Endif

Endfor

If no k-node independent set was found then

Declare failure

Endif

To understand the running time of this algorithm, we need to consider two quantities. First, the total number of k-element subsets in an n-element set is

n n(n − 1)(n − 2) . . . (n − k + 1) nk k = k(k−1)(k−2)…(2)(1) ≤ k!.

2.4 A Survey of Common Running Times

53

54

Chapter 2 Basics of Algorithm Analysis

Since we are treating k as a constant, this quantity is O(nk). Thus, the outer loop in the algorithm above will run for O(nk) iterations as it tries all k-node subsets of the n nodes of the graph.

Inside this loop, we need to test whether a given set S of k nodes constitutes

an independent set. The definition of an independent set tells us that we need

to check, for each pair of nodes, whether there is an edge joining them. Hence

this is a search over pairs, like we saw earlier in the discussion of quadratic

time; it requires looking at k, that is, O(k2), pairs and spending constant time 2

on each.

Thus the total running time is O(k2nk). Since we are treating k as a constant here, and since constants can be dropped in O(·) notation, we can write this running time as O(nk).

Independent Set is a principal example of a problem believed to be compu- tationally hard, and in particular it is believed that no algorithm to find k-node independent sets in arbitrary graphs can avoid having some dependence on k in the exponent. However, as we will discuss in Chapter 10 in the context of a related problem, even once we’ve conceded that brute-force search over k- element subsets is necessary, there can be different ways of going about this that lead to significant differences in the efficiency of the computation.

Beyond Polynomial Time

The previous example of the Independent Set Problem starts us rapidly down the path toward running times that grow faster than any polynomial. In particular, two kinds of bounds that come up very frequently are 2n and n!, and we now discuss why this is so.

Suppose, for example, that we are given a graph and want to find an independent set of maximum size (rather than testing for the existence of one with a given number of nodes). Again, people don’t know of algorithms that improve significantly on brute-force search, which in this case would look as follows.

For each subset S of nodes

Check whether S constitutes an independent set

If S is a larger independent set than the largest seen so far then

Record the size of S as the current maximum Endif

Endfor

This is very much like the brute-force algorithm for k-node independent sets, except that now we are iterating over all subsets of the graph. The total number

of subsets of an n-element set is 2n, and so the outer loop in this algorithm will run for 2n iterations as it tries all these subsets. Inside the loop, we are checking all pairs from a set S that can be as large as n nodes, so each iteration of the loop takes at most O(n2) time. Multiplying these two together, we get a running time of O(n22n).

Thus see that 2n arises naturally as a running time for a search algorithm that must consider all subsets. In the case of Independent Set, something at least nearly this inefficient appears to be necessary; but it’s important to keep in mind that 2n is the size of the search space for many problems, and for many of them we will be able to find highly efficient polynomial- time algorithms. For example, a brute-force search algorithm for the Interval Scheduling Problem that we saw in Chapter 1 would look very similar to the algorithm above: try all subsets of intervals, and find the largest subset that has no overlaps. But in the case of the Interval Scheduling Problem, as opposed to the Independent Set Problem, we will see (in Chapter 4) how to find an optimal solution in O(n log n) time. This is a recurring kind of dichotomy in the study of algorithms: two algorithms can have very similar-looking search spaces, but in one case you’re able to bypass the brute-force search algorithm, and in the other you aren’t.

The function n! grows even more rapidly than 2n, so it’s even more menacing as a bound on the performance of an algorithm. Search spaces of size n! tend to arise for one of two reasons. First, n! is the number of ways to match up n items with n other items—for example, it is the number of possible perfect matchings of n men with n women in an instance of the Stable Matching Problem. To see this, note that there are n choices for how we can match up the first man; having eliminated this option, there are n − 1 choices for how we can match up the second man; having eliminated these two options, there are n − 2 choices for how we can match up the third man; and so forth. Multiplying all these choices out, we get n(n−1)(n−2)…(2)(1)=n!

Despite this enormous set of possible solutions, we were able to solve the Stable Matching Problem in O(n2) iterations of the proposal algorithm. In Chapter 7, we will see a similar phenomenon for the Bipartite Matching Problem we discussed earlier; if there are n nodes on each side of the given bipartite graph, there can be up to n! ways of pairing them up. However, by a fairly subtle search algorithm, we will be able to find the largest bipartite matching in O(n3) time.

The function n! also arises in problems where the search space consists of all ways to arrange n items in order. A basic problem in this genre is the Traveling Salesman Problem: given a set of n cities, with distances between all pairs, what is the shortest tour that visits all cities? We assume that the salesman starts and ends at the first city, so the crux of the problem is the

2.4 A Survey of Common Running Times

55

56

Chapter 2 Basics of Algorithm Analysis

implicit search over all orders of the remaining n − 1 cities, leading to a search space of size (n − 1)!. In Chapter 8, we will see that Traveling Salesman is another problem that, like Independent Set, belongs to the class of NP- complete problems and is believed to have no efficient solution.

Sublinear Time

Finally, there are cases where one encounters running times that are asymp- totically smaller than linear. Since it takes linear time just to read the input, these situations tend to arise in a model of computation where the input can be “queried” indirectly rather than read completely, and the goal is to minimize the amount of querying that must be done.

Perhaps the best-known example of this is the binary search algorithm. Given a sorted array A of n numbers, we’d like to determine whether a given number p belongs to the array. We could do this by reading the entire array, but we’d like to do it much more efficiently, taking advantage of the fact that the array is sorted, by carefully probing particular entries. In particular, we probe the middle entry of A and get its value—say it is q—and we compare q to p. If q=p, we’re done. If q>p, then in order for p to belong to the array A, it must lie in the lower half of A; so we ignore the upper half of A from now on and recursively apply this search in the lower half. Finally, if q < p, then we apply the analogous reasoning and recursively search in the upper half of A.
The point is that in each step, there’s a region of A where p might possibly
be; and we’re shrinking the size of this region by a factor of two with every
probe. So how large is the “active” region of A after k probes? It starts at size
n, so after k probes it has size at most (1)kn. 2
Given this, how long will it take for the size of the active region to be
reduced to a constant? We need k to be large enough so that (1)k =O(1/n), 2
and to do this we can choose k=log2 n. Thus, when k=log2 n, the size of the active region has been reduced to a constant, at which point the recursion bottoms out and we can search the remainder of the array directly in constant time.
So the running time of binary search is O(log n), because of this successive shrinking of the search region. In general, O(log n) arises as a time bound whenever we’re dealing with an algorithm that does a constant amount of work in order to throw away a constant fraction of the input. The crucial fact is that O(log n) such iterations suffice to shrink the input down to constant size, at which point the problem can generally be solved directly.
2.5 A More Complex Data Structure: Priority Queues
57
2.5 A More Complex Data Structure: Priority Queues
Our primary goal in this book was expressed at the outset of the chapter: we seek algorithms that improve qualitatively on brute-force search, and in general we use polynomial-time solvability as the concrete formulation of this. Typically, achieving a polynomial-time solution to a nontrivial problem is not something that depends on fine-grained implementation details; rather, the difference between exponential and polynomial is based on overcoming higher-level obstacles. Once one has an efficient algorithm to solve a problem, however, it is often possible to achieve further improvements in running time by being careful with the implementation details, and sometimes by using more complex data structures.
Some complex data structures are essentially tailored for use in a single kind of algorithm, while others are more generally applicable. In this section, we describe one of the most broadly useful sophisticated data structures, the priority queue. Priority queues will be useful when we describe how to implement some of the graph algorithms developed later in the book. For our purposes here, it is a useful illustration of the analysis of a data structure that, unlike lists and arrays, must perform some nontrivial processing each time it is invoked.
The Problem
In the implementation of the Stable Matching algorithm in Section 2.3, we discussed the need to maintain a dynamically changing set S (such as the set of all free men in that case). In such situations, we want to be able to add elements to and delete elements from the set S, and we want to be able to select an element from S when the algorithm calls for it. A priority queue is designed for applications in which elements have a priority value, or key, and each time we need to select an element from S, we want to take the one with highest priority.
A priority queue is a data structure that maintains a set of elements S, where each element v ∈ S has an associated value key(v) that denotes the priority of element v; smaller keys represent higher priorities. Priority queues support the addition and deletion of elements from the set, and also the selection of the element with smallest key. Our implementation of priority queues will also support some additional operations that we summarize at the end of the section.
A motivating application for priority queues, and one that is useful to keep in mind when considering their general function, is the problem of managing
58
Chapter 2 Basics of Algorithm Analysis
real-time events such as the scheduling of processes on a computer. Each process has a priority, or urgency, but processes do not arrive in order of their priorities. Rather, we have a current set of active processes, and we want to be able to extract the one with the currently highest priority and run it. We can maintain the set of processes in a priority queue, with the key of a process representing its priority value. Scheduling the highest-priority process corresponds to selecting the element with minimum key from the priority queue; concurrent with this, we will also be inserting new processes as they arrive, according to their priority values.
How efficiently do we hope to be able to execute the operations in a priority queue? We will show how to implement a priority queue containing at most n elements at any time so that elements can be added and deleted, and the element with minimum key selected, in O(log n) time per operation.
Before discussing the implementation, let us point out a very basic appli- cation of priority queues that highlights why O(log n) time per operation is essentially the “right” bound to aim for.
(2.11) A sequence of O(n) priority queue operations can be used to sort a set of n numbers.
Proof. Set up a priority queue H, and insert each number into H with its value as a key. Then extract the smallest number one by one until all numbers have been extracted; this way, the numbers will come out of the priority queue in sorted order.
Thus, with a priority queue that can perform insertion and the extraction of minima in O(log n) per operation, we can sort n numbers in O(n log n) time. It is known that, in a comparison-based model of computation (when each operation accesses the input only by comparing a pair of numbers), the time needed to sort must be at least proportional to n log n, so (2.11) highlights a sense in which O(log n) time per operation is the best we can hope for. We should note that the situation is a bit more complicated than this: implementations of priority queues more sophisticated than the one we present here can improve the running time needed for certain operations, and add extra functionality. But (2.11) shows that any sequence of priority queue operations that results in the sorting of n numbers must take time at least proportional to n log n in total.
A Data Structure for Implementing a Priority Queue
We will use a data structure called a heap to implement a priority queue. Before we discuss the structure of heaps, we should consider what happens with some simpler, more natural approaches to implementing the functions
2.5 A More Complex Data Structure: Priority Queues
59
of a priority queue. We could just have the elements in a list, and separately have a pointer labeled Min to the one with minimum key. This makes adding new elements easy, but extraction of the minimum hard. Specifically, finding the minimum is quick—we just consult the Min pointer—but after removing this minimum element, we need to update the Min pointer to be ready for the next operation, and this would require a scan of all elements in O(n) time to find the new minimum.
This complication suggests that we should perhaps maintain the elements in the sorted order of the keys. This makes it easy to extract the element with smallest key, but now how do we add a new element to our set? Should we have the elements in an array, or a linked list? Suppose we want to add s with key value key(s). If the set S is maintained as a sorted array, we can use binary search to find the array position where s should be inserted in O(log n) time, but to insert s in the array, we would have to move all later elements one position to the right. This would take O(n) time. On the other hand, if we maintain the set as a sorted doubly linked list, we could insert it in O(1) time into any position, but the doubly linked list would not support binary search, and hence we may need up to O(n) time to find the position where s should be inserted.
The Definition of a Heap So in all these simple approaches, at least one of the operations can take up to O(n) time—much more than the O(log n) per operation that we’re hoping for. This is where heaps come in. The heap data structure combines the benefits of a sorted array and list for purposes of this application. Conceptually, we think of a heap as a balanced binary tree as shown on the left of Figure 2.3. The tree will have a root, and each node can have up to two children, a left and a right child. The keys in such a binary tree are said to be in heap order if the key of any element is at least as large as the key of the element at its parent node in the tree. In other words,
Heap order: For every element v, at a node i, the element w at i’s parent satisfies key(w) ≤ key(v).
In Figure 2.3 the numbers in the nodes are the keys of the corresponding elements.
Before we discuss how to work with a heap, we need to consider what data structure should be used to represent it. We can use pointers: each node at the heap could keep the element it stores, its key, and three pointers pointing to the two children and the parent of the heap node. We can avoid using pointers, however, if a bound N is known in advance on the total number of elements that will ever be in the heap at any one time. Such heaps can be maintained in an array H indexed by i=1,...,N. We will think of the heap nodes as corresponding to the positions in this array. H [1] is the root, and for any node
60
Chapter 2 Basics of Algorithm Analysis
1
25
Each node’s key is at least as large as its parent’s.
1
2
5
10
3
7
11
15
17
20
9
15
8
16
X
10 3 7 11
15 17 20 9 15 8 16
Figure 2.3 Values in a heap shown as a binary tree on the left, and represented as an array on the right. The arrows show the children for the top three nodes in the tree.
at position i, the children are the nodes at positions leftChild(i) = 2i and rightChild(i) = 2i + 1. So the two children of the root are at positions 2 and 3, and the parent of a node at position i is at position parent(i) = ⌊i/2⌋. If the heap has n < N elements at some time, we will use the first n positions of the array to store the n heap elements, and use length(H) to denote the number of elements in H. This representation keeps the heap balanced at all times. See the right-hand side of Figure 2.3 for the array representation of the heap on the left-hand side.
Implementing the Heap Operations
The heap element with smallest key is at the root, so it takes O(1) time to identify the minimal element. How do we add or delete heap elements? First consider adding a new heap element v, and assume that our heap H has n < N elements so far. Now it will have n + 1 elements. To start with, we can add the new element v to the final position i = n + 1, by setting H[i]= v. Unfortunately, this does not maintain the heap property, as the key of element v may be smaller than the key of its parent. So we now have something that is almost a heap, except for a small “damaged” part where v was pasted on at the end.
We will use the procedure Heapify-up to fix our heap. Let j = parent(i) = ⌊i/2⌋ be the parent of the node i, and assume H[j]=w. If key[v]

let j=parent(i)=⌊i/2⌋

If key[H[i]]

So by the induction hypothesis, applying Heapify-up(j) recursively will produce a heap as required. The process follows the tree-path from position i to the root, so it takes O(log i) time.

To insert a new element in a heap, we first add it as the last element. If the new element has a very large key value, then the array is a heap. Otherwise, it is almost a heap with the key value of the new element too small. We use Heapify-up to fix the heap property.

Now consider deleting an element. Many applications of priority queues don’t require the deletion of arbitrary elements, but only the extraction of the minimum. In a heap, this corresponds to identifying the key at the root (which will be the minimum) and then deleting it; we will refer to this oper- ation as ExtractMin(H). Here we will implement a more general operation Delete(H,i), which will delete the element in position i. Assume the heap currently has n elements. After deleting the element H[i], the heap will have only n − 1 elements; and not only is the heap-order property violated, there is actually a “hole” at position i, since H[i] is now empty. So as a first step, to patch the hole in H, we move the element w in position n to position i. After doing this, H at least has the property that its n − 1 elements are in the first n − 1 positions, as required, but we may well still not have the heap-order property.

However, the only place in the heap where the order might be violated is position i, as the key of element w may be either too small or too big for the position i. If the key is too small (that is, the violation of the heap property is between node i and its parent), then we can use Heapify-up(i) to reestablish the heap order. On the other hand, if key[w] is too big, the heap property may be violated between i and one or both of its children. In this case, we will use a procedure called Heapify-down, closely analogous to Heapify-up, that

2.5 A More Complex Data Structure: Priority Queues

63

The Heapify-down process

4 is moving element w down, 4

toward the leaves.

7 21w 7 7

10 16 7 11 10 16 21w11

15 17 20 17 15 8 16 15 17 20 17 15 8 16

Figure 2.5 The Heapify-down process:. Key 21 (at position 3) is too big (on the left). After swapping keys 21 and 7, the heap violation moves one step closer to the bottom of the tree (on the right).

swaps the element at position i with one of its children and proceeds down the tree recursively. Figure 2.5 shows the first steps of this process.

Heapify-down(H,i): Let n = length(H) If 2i>n then

Terminate with H unchanged Else if 2i

The algorithm repeatedly swaps the element originally at position i down, following a tree-path, so in O(log n) iterations the process results in a heap.

To use the process to remove an element v = H[i]from the heap, we replace H[i]with the last element in the array, H[n]=w. If the resulting array is not a heap, it is almost a heap with the key value of H[i]either too small or too big. We use Heapify-down or Heapify-down to fix the heap property in O(log n) time.

Implementing Priority Queues with Heaps

The heap data structure with the Heapify-down and Heapify-up operations can efficiently implement a priority queue that is constrained to hold at most N elements at any point in time. Here we summarize the operations we will use.

. StartHeap(N) returns an empty heap H that is set up to store at most N elements. This operation takes O(N) time, as it involves initializing the array that will hold the heap.

. Insert(H, v) inserts the item v into heap H. If the heap currently has n elements, this takes O(log n) time.

. FindMin(H) identifies the minimum element in the heap H but does not remove it. This takes O(1) time.

. Delete(H , i) deletes the element in heap position i. This is implemented in O(log n) time for heaps that have n elements.

. ExtractMin(H) identifies and deletes an element with minimum key value from a heap. This is a combination of the preceding two operations, and so it takes O(log n) time.

There is a second class of operations in which we want to operate on elements by name, rather than by their position in the heap. For example, in a number of graph algorithms that use heaps, the heap elements are nodes of the graph with key values that are computed during the algorithm. At various points in these algorithms, we want to operate on a particular node, regardless of where it happens to be in the heap.

To be able to access given elements of the priority queue efficiently, we simply maintain an additional array Position that stores the current position of each element (each node) in the heap. We can now implement the following further operations.

. To delete the element v, we apply Delete(H,Position[v]). Maintaining this array does not increase the overall running time, and so we can delete an element v from a heap with n nodes in O(log n) time.

. An additional operation that is used by some algorithms is ChangeKey (H, v, α), which changes the key value of element v to key(v) = α. To implement this operation in O(log n) time, we first need to be able to identify the position of element v in the array, which we do by using the array Position. Once we have identified the position of element v, we change the key and then apply Heapify-up or Heapify-down as appropriate.

Solved Exercises

Solved Exercise 1

Take the following list of functions and arrange them in ascending order of growth rate. That is, if function g(n) immediately follows function f(n) in your list, then it should be the case that f(n) is O(g(n)).

f1(n) = 10n f2(n) = n1/3

f3(n) = nn

f4(n) = log2 n √

f5(n) = 2 log2 n

We can deal with functions f1, f2, and f4 very easily, since they belong to the basic families of exponentials, polynomials, and logarithms. In particular, by (2.8), we have f4(n)=O(f2(n)); and by (2.9), we have f2(n) = O(f1(n)).

Solved Exercises

65

Solution

66

Chapter 2 Basics of Algorithm Analysis

Now, the function f3 isn’t so hard to deal with. It starts out smaller than 10n, but once n ≥ 10, then clearly 10n ≤ nn. This is exactly what we need for the definition of O(·) notation: for all n ≥ 10, we have 10n ≤ cnn, where in this case c = 1, and so 10n = O(nn).

Finally, we come to function f5, which is admittedly kind of strange- looking. A useful rule of thumb in such situations is to try taking logarithms

to see whether this makes things clearer. In this case, log2 f5(n) = log2 n =

(log2 n)1/2. What do the logarithms of the other functions look like? log f4(n) =

log log n, while log f (n) = 1 log n. All of these can be viewed as functions 22232

of log2 n, and so using the notation z = log2 n, we can write log f2(n) = 1z

3

log f4(n) = log2 z

log f5(n) = z1/2

Now it’s easier to see what’s going on. First, for z ≥ 16, we have log2 z ≤

z1/2. But the condition z ≥ 16 is the same as n ≥ 216 = 65, 536; thus once

n ≥ 216 we have log f4(n) ≤ log f5(n), and so f4(n) ≤ f5(n). Thus we can write

f4(n) = O(f5(n)). Similarly we have z1/2 ≤ 1z once z ≥ 9—in other words, 3

52√52

have discovered that 2 log2 n is a function whose growth rate lies somewhere between that of logarithms and polynomials.

once n ≥ 29 = 512. For n above this bound we have log f5(n) ≤ log f2(n) and hence f (n) ≤ f (n), and so we can write f (n) = O(f (n)). Essentially, we

Since we have sandwiched f5 between f4 and f2, this finishes the task of putting the functions in order.

Solved Exercise 2

Let f and g be two functions that take nonnegative values, and suppose that f = O(g). Show that g = (f ).

Solution This exercise is a way to formalize the intuition that O(·) and (·) are in a sense opposites. It is, in fact, not difficult to prove; it is just a matter of unwinding the definitions.

We’re given that, for some constants c and n0, we have f (n) ≤ cg(n) for all n ≥ n0. Dividing both sides by c, we can conclude that g(n) ≥ 1f(n) for

c

all n ≥ n0. But this is exactly what is required to show that g = (f ): we have

established that g(n) is at least a constant multiple of f (n) (where the constant

is 1), for all sufficiently large n (at least n0). c

Exercises

1. Suppose you have algorithms with the five running times listed below. (Assume these are the exact running times.) How much slower do each of these algorithms get when you (a) double the input size, or (b) increase the input size by one?

(a) n2

(b) n3

(c) 100n2

(d) nlogn

(e) 2n

2. Suppose you have algorithms with the six running times listed below. (Assume these are the exact number of operations performed as a func- tion of the input size n.) Suppose you have a computer that can perform 1010 operations per second, and you need to compute a result in at most an hour of computation. For each of the algorithms, what is the largest input size n for which you would be able to get the result within an hour?

(a) n2

(b) n3

(c) 100n2

(d) nlogn

(e) 2n

(f) 22n

3. Takethefollowinglistoffunctionsandarrangetheminascendingorder of growth rate. That is, if function g(n) immediately follows function f (n) in your list, then it should be the case that f(n) is O(g(n)).

f1(n) =

f2(n) = f3(n) = f4(n) = f5(n) = f6(n) =

n2.5

√

2n

n + 10

10n 100n

n2 log n

Exercises

67

4. Takethefollowinglistoffunctionsandarrangetheminascendingorder of growth rate. That is, if function g(n) immediately follows function f (n) in your list, then it should be the case that f(n) is O(g(n)).

68

Chapter 2

Basics of Algorithm Analysis

√ g1(n) = 2 log n

g2(n) = 2n

g4(n) = n4/3 g3(n) = n(log n)3 g5(n) = nlog n g6(n) = 22n

g7(n) = 2n2

5. Assume you have functions f and g such that f(n) is O(g(n)). For each of the following statements, decide whether you think it is true or false and give a proof or counterexample.

(a) log2 f (n) is O(log2 g(n)).

(b) 2f (n) is O(2g(n)).

(c) f (n)2 is O(g(n)2).

6. Considerthefollowingbasicproblem.You’regivenanarrayAconsisting of n integers A[1], A[2], . . . , A[n]. You’d like to output a two-dimensional n-by-n array B in which B[i, j] (for i < j) contains the sum of array entries A[i] through A[j]—that is, the sum A[i] + A[i + 1] + . . . + A[j]. (The value of array entry B[i, j] is left unspecified whenever i ≥ j, so it doesn’t matter what is output for these values.)
Here’s a simple algorithm to solve this problem.
For i=1, 2,...,n
For j=i+1, i+2,...,n
Add up array entries A[i] through A[j]
Store the result in B[i, j] Endfor
Endfor
(a) For some function f that you should choose, give a bound of the form O(f(n)) on the running time of this algorithm on an input of size n (i.e., a bound on the number of operations performed by the algorithm).
(b) For this same function f , show that the running time of the algorithm on an input of size n is also (f(n)). (This shows an asymptotically tight bound of (f (n)) on the running time.)
(c) Although the algorithm you analyzed in parts (a) and (b) is the most natural way to solve the problem—after all, it just iterates through
the relevant entries of the array B, filling in a value for each—it contains some highly unnecessary sources of inefficiency. Give a different algorithm to solve this problem, with an asymptotically better running time. In other words, you should design an algorithm with running time O(g(n)), where limn→∞ g(n)/f (n) = 0.
7. There’s a class of folk songs and holiday songs in which each verse consists of the previous verse, with one extra line added on. “The Twelve Days of Christmas” has this property; for example, when you get to the fifth verse, you sing about the five golden rings and then, reprising the lines from the fourth verse, also cover the four calling birds, the three French hens, the two turtle doves, and of course the partridge in the pear tree. The Aramaic song “Had gadya” from the Passover Haggadah works like this as well, as do many other songs.
These songs tend to last a long time, despite having relatively short scripts. In particular, you can convey the words plus instructions for one of these songs by specifying just the new line that is added in each verse, without having to write out all the previous lines each time. (So the phrase “five golden rings” only has to be written once, even though it will appear in verses five and onward.)
There’s something asymptotic that can be analyzed here. Suppose, for concreteness, that each line has a length that is bounded by a constant c, and suppose that the song, when sung out loud, runs for n words total. Show how to encode such a song using a script that has length f (n), for a function f (n) that grows as slowly as possible.
8. You’re doing some stress-testing on various models of glass jars to determine the height from which they can be dropped and still not break. The setup for this experiment, on a particular type of jar, is as follows. You have a ladder with n rungs, and you want to find the highest rung from which you can drop a copy of the jar and not have it break. We call this the highest safe rung.
It might be natural to try binary search: drop a jar from the middle rung, see if it breaks, and then recursively try from rung n/4 or 3n/4 depending on the outcome. But this has the drawback that you could break a lot of jars in finding the answer.
If your primary goal were to conserve jars, on the other hand, you could try the following strategy. Start by dropping a jar from the first rung, then the second rung, and so forth, climbing one higher each time until the jar breaks. In this way, you only need a single jar—at the moment
Exercises
69
70
Chapter 2 Basics of Algorithm Analysis
it breaks, you have the correct answer—but you may have to drop it n times (rather than log n as in the binary search solution).
So here is the trade-off: it seems you can perform fewer drops if you’re willing to break more jars. To understand better how this trade- off works at a quantitative level, let’s consider how to run this experiment given a fixed “budget” of k ≥ 1 jars. In other words, you have to determine the correct answer—the highest safe rung—and can use at most k jars in doing so.
(a) Suppose you are given a budget of k = 2 jars. Describe a strategy for finding the highest safe rung that requires you to drop a jar at most f (n) times, for some function f (n) that grows slower than linearly. (In other words, it should be the case that limn→∞ f (n)/n = 0.)
(b) Now suppose you have a budget of k > 2 jars, for some given k. Describe a strategy for finding the highest safe rung using at most k jars. If fk(n) denotes the number of times you need to drop a jar according to your strategy, then the functions f1, f2, f3, . . . should have the property that each grows asymptotically slower than the previous one: limn→∞ fk(n)/fk−1(n) = 0 for each k.

Notes and Further Reading

Polynomial-time solvability emerged as a formal notion of efficiency by a gradual process, motivated by the work of a number of researchers includ- ing Cobham, Rabin, Edmonds, Hartmanis, and Stearns. The survey by Sipser (1992) provides both a historical and technical perspective on these develop- ments. Similarly, the use of asymptotic order of growth notation to bound the running time of algorithms—as opposed to working out exact formulas with leading coefficients and lower-order terms—is a modeling decision that was quite non-obvious at the time it was introduced; Tarjan’s Turing Award lecture (1987) offers an interesting perspective on the early thinking of researchers including Hopcroft, Tarjan, and others on this issue. Further discussion of asymptotic notation and the growth of basic functions can be found in Knuth (1997a).

The implementation of priority queues using heaps, and the application to sorting, is generally credited to Williams (1964) and Floyd (1964). The priority queue is an example of a nontrivial data structure with many applications; in later chapters we will discuss other data structures as they become useful for the implementation of particular algorithms. We will consider the Union-Find data structure in Chapter 4 for implementing an algorithm to find minimum-

cost spanning trees, and we will discuss randomized hashing in Chapter 13. A number of other data structures are discussed in the book by Tarjan (1983). The LEDA library (Library of Efficient Datatypes and Algorithms) of Mehlhorn and Na ̈her (1999) offers an extensive library of data structures useful in combinatorial and geometric applications.

Notes on the Exercises Exercise 8 is based on a problem we learned from Sam Toueg.

Notes and Further Reading

71

This page intentionally left blank

Chapter 3 Graphs

Our focus in this book is on problems with a discrete flavor. Just as continuous mathematics is concerned with certain basic structures such as real numbers, vectors, and matrices, discrete mathematics has developed basic combinatorial structures that lie at the heart of the subject. One of the most fundamental and expressive of these is the graph.

The more one works with graphs, the more one tends to see them ev- erywhere. Thus, we begin by introducing the basic definitions surrounding graphs, and list a spectrum of different algorithmic settings where graphs arise naturally. We then discuss some basic algorithmic primitives for graphs, be- ginning with the problem of connectivity and developing some fundamental graph search techniques.

3.1 Basic Definitions and Applications

Recall from Chapter 1 that a graph G is simply a way of encoding pairwise relationships among a set of objects: it consists of a collection V of nodes and a collection E of edges, each of which “joins” two of the nodes. We thus represent an edge e ∈ E as a two-element subset of V: e = {u, v} for some u,v∈V, where we call u and v the ends of e.

Edges in a graph indicate a symmetric relationship between their ends. Often we want to encode asymmetric relationships, and for this we use the closely related notion of a directed graph. A directed graph G′ consists of a set of nodes V and a set of directed edges E′. Each e′ ∈ E′ is an ordered pair (u, v); in other words, the roles of u and v are not interchangeable, and we call u the tail of the edge and v the head. We will also say that edge e′ leaves node u and enters node v.

74

Chapter 3 Graphs

When we want to emphasize that the graph we are considering is not directed, we will call it an undirected graph; by default, however, the term “graph” will mean an undirected graph. It is also worth mentioning two warnings in our use of graph terminology. First, although an edge e in an undirected graph should properly be written as a set of nodes {u, v}, one will more often see it written (even in this book) in the notation used for ordered pairs: e = (u, v). Second, a node in a graph is also frequently called a vertex; in this context, the two words have exactly the same meaning.

Examples of Graphs Graphs are very simple to define: we just take a collec- tion of things and join some of them by edges. But at this level of abstraction, it’s hard to appreciate the typical kinds of situations in which they arise. Thus, we propose the following list of specific contexts in which graphs serve as important models. The list covers a lot of ground, and it’s not important to remember everything on it; rather, it will provide us with a lot of useful ex- amples against which to check the basic definitions and algorithmic problems that we’ll be encountering later in the chapter. Also, in going through the list, it’s useful to digest the meaning of the nodes and the meaning of the edges in the context of the application. In some cases the nodes and edges both corre- spond to physical objects in the real world, in others the nodes are real objects while the edges are virtual, and in still others both nodes and edges are pure abstractions.

1. Transportation networks. The map of routes served by an airline carrier naturally forms a graph: the nodes are airports, and there is an edge from u to v if there is a nonstop flight that departs from u and arrives at v. Described this way, the graph is directed; but in practice when there is an edge (u, v), there is almost always an edge (v, u), so we would not lose much by treating the airline route map as an undirected graph with edges joining pairs of airports that have nonstop flights each way. Looking at such a graph (you can generally find them depicted in the backs of in- flight airline magazines), we’d quickly notice a few things: there are often a small number of hubs with a very large number of incident edges; and it’s possible to get between any two nodes in the graph via a very small number of intermediate stops.

Other transportation networks can be modeled in a similar way. For example, we could take a rail network and have a node for each terminal, and an edge joining u and v if there’s a section of railway track that goes between them without stopping at any intermediate terminal. The standard depiction of the subway map in a major city is a drawing of such a graph.

2. Communication networks. A collection of computers connected via a communication network can be naturally modeled as a graph in a few

different ways. First, we could have a node for each computer and an edge joining u and v if there is a direct physical link connecting them. Alternatively, for studying the large-scale structure of the Internet, people often define a node to be the set of all machines controlled by a single Internet service provider, with an edge joining u and v if there is a direct peering relationship between them—roughly, an agreement to exchange data under the standard BGP protocol that governs global Internet routing. Note that this latter network is more “virtual” than the former, since the links indicate a formal agreement in addition to a physical connection.

In studying wireless networks, one typically defines a graph where the nodes are computing devices situated at locations in physical space, and there is an edge from u to v if v is close enough to u to receive a signal from it. Note that it’s often useful to view such a graph as directed, since it may be the case that v can hear u’s signal but u cannot hear v’s signal (if, for example, u has a stronger transmitter). These graphs are also interesting from a geometric perspective, since they roughly correspond to putting down points in the plane and then joining pairs that are close together.

3. Informationnetworks.TheWorldWideWebcanbenaturallyviewedasa directed graph, in which nodes correspond to Web pages and there is an edge from u to v if u has a hyperlink to v. The directedness of the graph is crucial here; many pages, for example, link to popular news sites, but these sites clearly do not reciprocate all these links. The structure of all these hyperlinks can be used by algorithms to try inferring the most important pages on the Web, a technique employed by most current search engines.

The hypertextual structure of the Web is anticipated by a number of information networks that predate the Internet by many decades. These include the network of cross-references among articles in an encyclopedia or other reference work, and the network of bibliographic citations among scientific papers.

4. Social networks. Given any collection of people who interact (the em- ployees of a company, the students in a high school, or the residents of a small town), we can define a network whose nodes are people, with an edge joining u and v if they are friends with one another. We could have the edges mean a number of different things instead of friendship: the undirected edge (u, v) could mean that u and v have had a roman- tic relationship or a financial relationship; the directed edge (u, v) could mean that u seeks advice from v, or that u lists v in his or her e-mail address book. One can also imagine bipartite social networks based on a

3.1 Basic Definitions and Applications

75

76

Chapter 3 Graphs

notion of affiliation: given a set X of people and a set Y of organizations, we could define an edge between u ∈ X and v ∈ Y if person u belongs to organization v.

Networks such as this are used extensively by sociologists to study the dynamics of interaction among people. They can be used to identify the most “influential” people in a company or organization, to model trust relationships in a financial or political setting, and to track the spread of fads, rumors, jokes, diseases, and e-mail viruses.

5. Dependencynetworks.Itisnaturaltodefinedirectedgraphsthatcapture the interdependencies among a collection of objects. For example, given the list of courses offered by a college or university, we could have a node for each course and an edge from u to v if u is a prerequisite for v. Given a list of functions or modules in a large software system, we could have a node for each function and an edge from u to v if u invokes v by a function call. Or given a set of species in an ecosystem, we could define a graph—a food web—in which the nodes are the different species and there is an edge from u to v if u consumes v.

This is far from a complete list, too far to even begin tabulating its omissions. It is meant simply to suggest some examples that are useful to keep in mind when we start thinking about graphs in an algorithmic context.

Paths and Connectivity One of the fundamental operations in a graph is that of traversing a sequence of nodes connected by edges. In the examples just listed, such a traversal could correspond to a user browsing Web pages by following hyperlinks; a rumor passing by word of mouth from you to someone halfway around the world; or an airline passenger traveling from San Francisco to Rome on a sequence of flights.

With this notion in mind, we define a path in an undirected graph G=(V,E) to be a sequence P of nodes v1,v2,…,vk−1,vk with the property that each consecutive pair vi , vi+1 is joined by an edge in G. P is often called a path from v1 to vk, or a v1-vk path. For example, the nodes 4, 2, 1, 7, 8 form a path in Figure 3.1. A path is called simple if all its vertices are distinct from oneanother.Acycleisapathv1,v2,…,vk−1,vk inwhichk>2,thefirstk−1 nodes are all distinct, and v1 = vk—in other words, the sequence of nodes “cycles back” to where it began. All of these definitions carry over naturally to directed graphs, with the following change: in a directed path or cycle, each pair of consecutive nodes has the property that (vi , vi+1) is an edge. In other words, the sequence of nodes in the path or cycle must respect the directionality of edges.

We say that an undirected graph is connected if, for every pair of nodes u and v, there is a path from u to v. Choosing how to define connectivity of a

3.1 Basic Definitions and Applications

77

3

42

8 7

1

257

6

1 5

934689

Figure 3.1 Two drawings of the same tree. On the right, the tree is rooted at node 1.

directed graph is a bit more subtle, since it’s possible for u to have a path to v while v has no path to u. We say that a directed graph is strongly connected if, for every two nodes u and v, there is a path from u to v and a path from v to u.

In addition to simply knowing about the existence of a path between some pair of nodes u and v, we may also want to know whether there is a short path. Thus we define the distance between two nodes u and v to be the minimum number of edges in a u-v path. (We can designate some symbol like ∞ to denote the distance between nodes that are not connected by a path.) The term distance here comes from imagining G as representing a communication or transportation network; if we want to get from u to v, we may well want a route with as few “hops” as possible.

Trees We say that an undirected graph is a tree if it is connected and does not contain a cycle. For example, the two graphs pictured in Figure 3.1 are trees. In a strong sense, trees are the simplest kind of connected graph: deleting any edge from a tree will disconnect it.

For thinking about the structure of a tree T, it is useful to root it at a particular node r. Physically, this is the operation of grabbing T at the node r and letting the rest of it hang downward under the force of gravity, like a mobile. More precisely, we “orient” each edge of T away from r; for each other node v, we declare the parent of v to be the node u that directly precedes v on its path from r; we declare w to be a child of v if v is the parent of w. More generally, we say that w is a descendant of v (or v is an ancestor of w) if v lies onthepathfromtheroottow;andwesaythatanodexisaleaf ifithasno descendants. Thus, for example, the two pictures in Figure 3.1 correspond to the same tree T —the same pairs of nodes are joined by edges—but the drawing on the right represents the result of rooting T at node 1.

78

Chapter 3 Graphs

Rooted trees are fundamental objects in computer science, because they encode the notion of a hierarchy. For example, we can imagine the rooted tree in Figure 3.1 as corresponding to the organizational structure of a tiny nine- person company; employees 3 and 4 report to employee 2; employees 2, 5, and 7 report to employee 1; and so on. Many Web sites are organized according to a tree-like structure, to facilitate navigation. A typical computer science department’s Web site will have an entry page as the root; the People page is a child of this entry page (as is the Courses page); pages entitled Faculty and Students are children of the People page; individual professors’ home pages are children of the Faculty page; and so on.

For our purposes here, rooting a tree T can make certain questions about T conceptually easy to answer. For example, given a tree T on n nodes, how many edges does it have? Each node other than the root has a single edge leading “upward” to its parent; and conversely, each edge leads upward from precisely one non-root node. Thus we have very easily proved the following fact.

(3.1) Every n-node tree has exactly n − 1 edges.

In fact, the following stronger statement is true, although we do not prove

it here.

(3.2) Let G be an undirected graph on n nodes. Any two of the following statements implies the third.

(i) G is connected.

(ii) G does not contain a cycle.

(iii) Ghasn−1edges.

We now turn to the role of trees in the fundamental algorithmic idea of

graph traversal.

3.2 Graph Connectivity and Graph Traversal

Having built up some fundamental notions regarding graphs, we turn to a very basic algorithmic question: node-to-node connectivity. Suppose we are given a graph G = (V , E) and two particular nodes s and t. We’d like to find an efficient algorithm that answers the question: Is there a path from s to t in G? We will call this the problem of determining s-t connectivity.

For very small graphs, this question can often be answered easily by visual inspection. But for large graphs, it can take some work to search for a path. Indeed, the s-t Connectivity Problem could also be called the Maze-Solving Problem. If we imagine G as a maze with a room corresponding to each node, and a hallway corresponding to each edge that joins nodes (rooms) together,

3.2 Graph Connectivity and Graph Traversal

79

1 7 9 11

23

4 5 8 10 12

6 13

Figure 3.2 In this graph, node 1 has paths to nodes 2 through 8, but not to nodes 9 through 13.

then the problem is to start in a room s and find your way to another designated room t. How efficient an algorithm can we design for this task?

In this section, we describe two natural algorithms for this problem at a high level: breadth-first search (BFS) and depth-first search (DFS). In the next section we discuss how to implement each of these efficiently, building on a data structure for representing a graph as the input to an algorithm.

Breadth-First Search

Perhaps the simplest algorithm for determining s-t connectivity is breadth-first search (BFS), in which we explore outward from s in all possible directions, adding nodes one “layer” at a time. Thus we start with s and include all nodes that are joined by an edge to s—this is the first layer of the search. We then include all additional nodes that are joined by an edge to any node in the first layer—this is the second layer. We continue in this way until no new nodes are encountered.

In the example of Figure 3.2, starting with node 1 as s, the first layer of the search would consist of nodes 2 and 3, the second layer would consist of nodes 4, 5, 7, and 8, and the third layer would consist just of node 6. At this point the search would stop, since there are no further nodes that could be added (and in particular, note that nodes 9 through 13 are never reached by the search).

As this example reinforces, there is a natural physical interpretation to the algorithm. Essentially, we start at s and “flood” the graph with an expanding wave that grows to visit all nodes that it can reach. The layer containing a node represents the point in time at which the node is reached.

We can define the layers L1, L2, L3, . . . constructed by the BFS algorithm more precisely as follows.

80

Chapter 3 Graphs

. Layer L1 consists of all nodes that are neighbors of s. (For notational reasons, we will sometimes use layer L0 to denote the set consisting just of s.)

. Assuming that we have defined layers L1, . . . , Lj, then layer Lj+1 consists of all nodes that do not belong to an earlier layer and that have an edge to a node in layer Lj.

Recalling our definition of the distance between two nodes as the minimum number of edges on a path joining them, we see that layer L1 is the set of all nodes at distance 1 from s, and more generally layer Lj is the set of all nodes at distance exactly j from s. A node fails to appear in any of the layers if and only if there is no path to it. Thus, BFS is not only determining the nodes that s can reach, it is also computing shortest paths to them. We sum this up in the following fact.

(3.3) For each j ≥ 1, layer Lj produced by BFS consists of all nodes at distance exactly j from s. There is a path from s to t if and only if t appears in some layer.

A further property of breadth-first search is that it produces, in a very natural way, a tree T rooted at s on the set of nodes reachable from s. Specifically, for each such node v (other than s), consider the moment when v is first “discovered” by the BFS algorithm; this happens when some node u in layer Lj is being examined, and we find that it has an edge to the previously unseen node v. At this moment, we add the edge (u,v) to the tree T—u becomes the parent of v, representing the fact that u is “responsible” for completing the path to v. We call the tree T that is produced in this way a breadth-first search tree.

Figure 3.3 depicts the construction of a BFS tree rooted at node 1 for the graph in Figure 3.2. The solid edges are the edges of T; the dotted edges are edges of G that do not belong to T. The execution of BFS that produces this tree can be described as follows.

(a) Starting from node 1, layer L1 consists of the nodes {2, 3}.

(b) LayerL2isthengrownbyconsideringthenodesinlayerL1inorder(say, first 2, then 3). Thus we discover nodes 4 and 5 as soon as we look at 2, so 2 becomes their parent. When we consider node 2, we also discover an edge to 3, but this isn’t added to the BFS tree, since we already know about node 3.

We first discover nodes 7 and 8 when we look at node 3. On the other hand, the edge from 3 to 5 is another edge of G that does not end up in

3.2 Graph Connectivity and Graph Traversal

81

111

232323

45784578

6

(a) (b) (c)

Figure 3.3 The construction of a breadth-first search tree T for the graph in Figure 3.2, with (a), (b), and (c) depicting the successive layers that are added. The solid edges are the edges of T; the dotted edges are in the connected component of G containing node 1, but do not belong to T.

the BFS tree, because by the time we look at this edge out of node 3, we already know about node 5.

(c) We then consider the nodes in layer L2 in order, but the only new node discovered when we look through L2 is node 6, which is added to layer L3. Note that the edges (4, 5) and (7, 8) don’t get added to the BFS tree, because they don’t result in the discovery of new nodes.

(d) Nonewnodesarediscoveredwhennode6isexamined,sonothingisput in layer L4, and the algorithm terminates. The full BFS tree is depicted in Figure 3.3(c).

We notice that as we ran BFS on this graph, the nontree edges all either connected nodes in the same layer, or connected nodes in adjacent layers. We now prove that this is a property of BFS trees in general.

(3.4) Let T be a breadth-first search tree, let x and y be nodes in T belonging to layers Li and Lj respectively, and let (x, y) be an edge of G. Then i and j differ by at most 1.

Proof. Suppose by way of contradiction that i and j differed by more than 1; in particular, suppose i < j − 1. Now consider the point in the BFS algorithm when the edges incident to x were being examined. Since x belongs to layer Li, the only nodes discovered from x belong to layers Li+1 and earlier; hence, if y is a neighbor of x, then it should have been discovered by this point at the latest and hence should belong to layer Li+1 or earlier.
82
Chapter 3
Graphs
Current component containing s
s
It is safe to add v. uv
Figure 3.4 When growing the connected component containing s, we look for nodes like v that have not yet been visited.
Exploring a Connected Component
The set of nodes discovered by the BFS algorithm is precisely those reachable from the starting node s. We will refer to this set R as the connected component of G containing s; and once we know the connected component containing s, we can simply check whether t belongs to it so as to answer the question of s-t connectivity.
Now, if one thinks about it, it’s clear that BFS is just one possible way to produce this component. At a more general level, we can build the component R by “exploring” G in any order, starting from s. To start off, we define R = {s}. Then at any point in time, if we find an edge (u, v) where u ∈ R and v ̸∈ R, we can add v to R. Indeed, if there is a path P from s to u, then there is a path from s to v obtained by first following P and then following the edge (u, v). Figure 3.4 illustrates this basic step in growing the component R.
Suppose we continue growing the set R until there are no more edges leading out of R; in other words, we run the following algorithm.
R will consist of nodes to which s has a path Initially R={s}
While there is an edge (u,v) where u∈R and v̸∈R
Add v to R Endwhile
Here is the key property of this algorithm.
(3.5) The set R produced at the end of the algorithm is precisely the connected component of G containing s.
3.2 Graph Connectivity and Graph Traversal
83
Proof. We have already argued that for any node v ∈ R, there is a path from s to v.
Now, consider a node w ̸∈ R, and suppose by way of contradiction, that there is an s-w path P in G. Since s ∈ R but w ̸∈ R, there must be a first node v on P that does not belong to R; and this node v is not equal to s. Thus there is a node u immediately preceding v on P, so (u, v) is an edge. Moreover, since v is the first node on P that does not belong to R, we must have u ∈ R. It follows that (u, v) is an edge where u ∈ R and v ̸∈ R; this contradicts the stopping rule for the algorithm.
For any node t in the component R, observe that it is easy to recover the actual path from s to t along the lines of the argument above: we simply record, for each node v, the edge (u, v) that was considered in the iteration in which v was added to R. Then, by tracing these edges backward from t, we proceed through a sequence of nodes that were added in earlier and earlier iterations, eventually reaching s; this defines an s-t path.
To conclude, we notice that the general algorithm we have defined to grow R is underspecified, so how do we decide which edge to consider next? The BFS algorithm arises, in particular, as a particular way of ordering the nodes we visit—in successive layers, based on their distance from s. But there are other natural ways to grow the component, several of which lead to efficient algorithms for the connectivity problem while producing search patterns with different structures. We now go on to discuss a different one of these algorithms, depth-first search, and develop some of its basic properties.
Depth-First Search
Another natural method to find the nodes reachable from s is the approach you might take if the graph G were truly a maze of interconnected rooms and you were walking around in it. You’d start from s and try the first edge leading out of it, to a node v. You’d then follow the first edge leading out of v, and continue in this way until you reached a “dead end”—a node for which you had already explored all its neighbors. You’d then backtrack until you got to a node with an unexplored neighbor, and resume from there. We call this algorithm depth- first search (DFS), since it explores G by going as deeply as possible and only retreating when necessary.
DFS is also a particular implementation of the generic component-growing algorithm that we introduced earlier. It is most easily described in recursive form: we can invoke DFS from any starting point but maintain global knowl- edge of which nodes have already been explored.
84
Chapter 3 Graphs
DFS(u):
Mark u as "Explored" and add u to R For each edge (u, v) incident to u
If v is not marked "Explored" then Recursively invoke DFS(v)
Endif Endfor
To apply this to s-t connectivity, we simply declare all nodes initially to be not explored, and invoke DFS(s).
There are some fundamental similarities and some fundamental differ- ences between DFS and BFS. The similarities are based on the fact that they both build the connected component containing s, and we will see in the next section that they achieve qualitatively similar levels of efficiency.
While DFS ultimately visits exactly the same set of nodes as BFS, it typically does so in a very different order; it probes its way down long paths, potentially getting very far from s, before backing up to try nearer unexplored nodes. We can see a reflection of this difference in the fact that, like BFS, the DFS algorithm yields a natural rooted tree T on the component containing s, but the tree will generally have a very different structure. We make s the root of the tree T, and make u the parent of v when u is responsible for the discovery of v. That is, whenever DFS(v) is invoked directly during the call to DFS(u), we add the edge (u, v) to T. The resulting tree is called a depth-first search tree of the component R.
Figure 3.5 depicts the construction of a DFS tree rooted at node 1 for the graph in Figure 3.2. The solid edges are the edges of T; the dotted edges are edges of G that do not belong to T. The execution of DFS begins by building a path on nodes 1, 2, 3, 5, 4. The execution reaches a dead end at 4, since there are no new nodes to find, and so it “backs up” to 5, finds node 6, backs up again to 3, and finds nodes 7 and 8. At this point there are no new nodes to find in the connected component, so all the pending recursive DFS calls terminate, one by one, and the execution comes to an end. The full DFS tree is depicted in Figure 3.5(g).
This example suggests the characteristic way in which DFS trees look different from BFS trees. Rather than having root-to-leaf paths that are as short as possible, they tend to be quite narrow and deep. However, as in the case of BFS, we can say something quite strong about the way in which nontree edges of G must be arranged relative to the edges of a DFS tree T: as in the figure, nontree edges can only connect ancestors of T to descendants.
1111
2222
(a)
333
55
4
(b) (c) (d)
111 222 333 55757
46 46 468
(e) (f) (g)
3.2 Graph Connectivity and Graph Traversal
85
Figure 3.5 The construction of a depth-first search tree T for the graph in Figure 3.2, with (a) through (g) depicting the nodes as they are discovered in sequence. The solid edges are the edges of T; the dotted edges are edges of G that do not belong to T.
To establish this, we first observe the following property of the DFS algorithm and the tree that it produces.
(3.6) For a given recursive call DFS(u), all nodes that are marked “Explored” between the invocation and end of this recursive call are descendants of u in T.
Using (3.6), we prove
(3.7) Let T be a depth-first search tree, let x and y be nodes in T, and let (x, y) be an edge of G that is not an edge of T. Then one of x or y is an ancestor of the other.
86
Chapter 3 Graphs
Proof. Supposethat(x,y)isanedgeofGthatisnotanedgeofT,andsuppose without loss of generality that x is reached first by the DFS algorithm. When the edge (x, y) is examined during the execution of DFS(x), it is not added to T because y is marked “Explored.” Since y was not marked “Explored” when DFS(x) was first invoked, it is a node that was discovered between the invocation and end of the recursive call DFS(x). It follows from (3.6) that y is a descendant of x.
The Set of All Connected Components
So far we have been talking about the connected component containing a particular node s. But there is a connected component associated with each node in the graph. What is the relationship between these components?
In fact, this relationship is highly structured and is expressed in the following claim.
(3.8) For any two nodes s and t in a graph, their connected components are either identical or disjoint.
This is a statement that is very clear intuitively, if one looks at a graph like the example in Figure 3.2. The graph is divided into multiple pieces with no edges between them; the largest piece is the connected component of nodes 1 through 8, the medium piece is the connected component of nodes 11, 12, and 13, and the smallest piece is the connected component of nodes 9 and 10. To prove the statement in general, we just need to show how to define these “pieces” precisely for an arbitrary graph.
Proof. Consider any two nodes s and t in a graph G with the property that there is a path between s and t. We claim that the connected components containing s and t are the same set. Indeed, for any node v in the component of s, the node v must also be reachable from t by a path: we can just walk from t to s, and then on from s to v. The same reasoning works with the roles of s and t reversed, and so a node is in the component of one if and only if it is in the component of the other.
On the other hand, if there is no path between s and t, then there cannot be a node v that is in the connected component of each. For if there were such a node v, then we could walk from s to v and then on to t, constructing a path between s and t. Thus, if there is no path between s and t, then their connected components are disjoint.
This proof suggests a natural algorithm for producing all the connected components of a graph, by growing them one component at a time. We start with an arbitrary node s, and we use BFS (or DFS) to generate its connected
3.3 Implementing Graph Traversal Using Queues and Stacks
87
component. We then find a node v (if any) that was not visited by the search from s, and iterate, using BFS starting from v, to generate its connected component—which, by (3.8), will be disjoint from the component of s. We continue in this way until all nodes have been visited.
3.3 Implementing Graph Traversal Using Queues and Stacks
So far we have been discussing basic algorithmic primitives for working with graphs without mentioning any implementation details. Here we discuss how to use lists and arrays to represent graphs, and we discuss the trade-offs between the different representations. Then we use these data structures to implement the graph traversal algorithms breadth-first search (BFS) and depth- first search (DFS) efficiently. We will see that BFS and DFS differ essentially only in that one uses a queue and the other uses a stack, two simple data structures that we will describe later in this section.
Representing Graphs
There are two basic ways to represent graphs: by an adjacency matrix and by an adjacency list representation. Throughout the book we will use the adjacency list representation. We start, however, by reviewing both of these representations and discussing the trade-offs between them.
A graph G = (V , E) has two natural input parameters, the number of nodes
|V|, and the number of edges |E|. We will use n = |V| and m = |E| to denote
these, respectively. Running times will be given in terms of both of these two
parameters. As usual, we will aim for polynomial running times, and lower-
degree polynomials are better. However, with two parameters in the running
time, the comparison is not always so clear. Is O(m2) or O(n3) a better running
time? This depends on what the relation is between n and m. With at most
one edge between any pair of nodes, the number of edges m can be at most
n ≤ n2. On the other hand, in many applications the graphs of interest are 2
connected, and by (3.1), connected graphs must have at least m ≥ n − 1 edges. But these comparisons do not always tell us which of two running times (such as m2 and n3) are better, so we will tend to keep the running times in terms of both of these parameters. In this section we aim to implement the basic graph search algorithms in time O(m + n). We will refer to this as linear time, since it takes O(m + n) time simply to read the input. Note that when we work with connected graphs, a running time of O(m + n) is the same as O(m), since m ≥ n − 1.
Consider a graph G = (V , E) with n nodes, and assume the set of nodes is V = {1, . . . , n}. The simplest way to represent a graph is by an adjacency
88
Chapter 3 Graphs
matrix, which is an n × n matrix A where A[u, v] is equal to 1 if the graph contains the edge (u, v) and 0 otherwise. If the graph is undirected, the matrix A is symmetric, with A[u, v]= A[v, u] for all nodes u, v ∈ V. The adjacency matrix representation allows us to check in O(1) time if a given edge (u, v) is present in the graph. However, the representation has two basic disadvantages.
. The representation takes (n2) space. When the graph has many fewer edges than n2, more compact representations are possible.
. Many graph algorithms need to examine all edges incident to a given node v. In the adjacency matrix representation, doing this involves considering all other nodes w, and checking the matrix entry A[v, w] to see whether the edge (v, w) is present—and this takes (n) time. In the worst case, v may have (n) incident edges, in which case checking all these edges will take (n) time regardless of the representation. But many graphs in practice have significantly fewer edges incident to most nodes, and so it would be good to be able to find all these incident edges more efficiently.
The representation of graphs used throughout the book is the adjacency list, which works better for sparse graphs—that is, those with many fewer than n2 edges. In the adjacency list representation there is a record for each node v, containing a list of the nodes to which v has edges. To be precise, we have an array Adj, where Adj[v] is a record containing a list of all nodes adjacent to node v. For an undirected graph G = (V, E), each edge e = (v, w) ∈ E occurs on two adjacency lists: node w appears on the list for node v, and node v appears on the list for node w.
Let’s compare the adjacency matrix and adjacency list representations. First consider the space required by the representation. An adjacency matrix requires O(n2) space, since it uses an n × n matrix. In contrast, we claim that the adjacency list representation requires only O(m + n) space. Here is why. First, we need an array of pointers of length n to set up the lists in Adj, and then we need space for all the lists. Now, the lengths of these lists may differ from node to node, but we argued in the previous paragraph that overall, each edge e = (v, w) appears in exactly two of the lists: the one for v and the one for w. Thus the total length of all lists is 2m = O(m).
Another (essentially equivalent) way to justify this bound is as follows.
We define the degree nv of a node v to be the number of incident edges it has.
The length of the list at Adj[v] is list is nv, so the total length over all nodes is
O v∈V nv . Now, the sum of the degrees in a graph is a quantity that often comes up in the analysis of graph algorithms, so it is useful to work out what this sum is.
(3.9) n=2m. v∈V v
3.3 Implementing Graph Traversal Using Queues and Stacks
89
Proof. Each edge e = (v, w) contributes exactly twice to this sum: once in the quantity nv and once in the quantity nw. Since the sum is the total of the contributions of each edge, it is 2m.
We sum up the comparison between adjacency matrices and adjacency lists as follows.
(3.10) The adjacency matrix representation of a graph requires O(n2) space, while the adjacency list representation requires only O(m + n) space.
Since we have already argued that m ≤ n2, the bound O(m + n) is never worse than O(n2); and it is much better when the underlying graph is sparse, with m much smaller than n2.
Now we consider the ease of accessing the information stored in these two different representations. Recall that in an adjacency matrix we can check in O(1) time if a particular edge (u, v) is present in the graph. In the adjacency list representation, this can take time proportional to the degree O(nv): we have to follow the pointers on u’s adjacency list to see if edge v occurs on the list. On the other hand, if the algorithm is currently looking at a node u, it can read the list of neighbors in constant time per neighbor.
In view of this, the adjacency list is a natural representation for exploring graphs. If the algorithm is currently looking at a node u, it can read this list of neighbors in constant time per neighbor; move to a neighbor v once it encounters it on this list in constant time; and then be ready to read the list associated with node v. The list representation thus corresponds to a physical notion of “exploring” the graph, in which you learn the neighbors of a node u once you arrive at u, and can read them off in constant time per neighbor.
Queues and Stacks
Many algorithms have an inner step in which they need to process a set of elements, such the set of all edges adjacent to a node in a graph, the set of visited nodes in BFS and DFS, or the set of all free men in the Stable Matching algorithm. For this purpose, it is natural to maintain the set of elements to be considered in a linked list, as we have done for maintaining the set of free men in the Stable Matching algorithm.
One important issue that arises is the order in which to consider the elements in such a list. In the Stable Matching algorithm, the order in which we considered the free men did not affect the outcome, although this required a fairly subtle proof to verify. In many other algorithms, such as DFS and BFS, the order in which elements are considered is crucial.
90
Chapter 3 Graphs
Two of the simplest and most natural options are to maintain a set of elements as either a queue or a stack. A queue is a set from which we extract elements in first-in, first-out (FIFO) order: we select elements in the same order in which they were added. A stack is a set from which we extract elements in last-in, first-out (LIFO) order: each time we select an element, we choose the one that was added most recently. Both queues and stacks can be easily implemented via a doubly linked list. In both cases, we always select the first element on our list; the difference is in where we insert a new element. In a queue a new element is added to the end of the list as the last element, while in a stack a new element is placed in the first position on the list. Recall that a doubly linked list has explicit First and Last pointers to the beginning and end, respectively, so each of these insertions can be done in constant time.
Next we will discuss how to implement the search algorithms of the previous section in linear time. We will see that BFS can be thought of as using a queue to select which node to consider next, while DFS is effectively using a stack.
Implementing Breadth-First Search
The adjacency list data structure is ideal for implementing breadth-first search. The algorithm examines the edges leaving a given node one by one. When we are scanning the edges leaving u and come to an edge (u, v), we need to know whether or not node v has been previously discovered by the search. To make this simple, we maintain an array Discovered of length n and set Discovered[v]=true as soon as our search first sees v. The algorithm, as described in the previous section, constructs layers of nodes L1, L2, . . . , where Li is the set of nodes at distance i from the source s. To maintain the nodes in a layer Li, we have a list L[i]for each i=0,1,2,....
BFS(s):
Set Discovered[s] = true and Discovered[v] = false for all other v Initialize L[0] to consist of the single element s
Set the layer counter i = 0
Set the current BFS tree T = ∅
While L[i] is not empty
Initialize an empty list L[i + 1] For each node u ∈ L[i]
Consider each edge (u, v) incident to u If Discovered[v] = false then
Set Discovered[v] = true
Add edge (u, v) to the tree T
3.3 Implementing Graph Traversal Using Queues and Stacks
91
Add v to the list L[i+1] Endif
Endfor
Increment the layer counter i by one Endwhile
In this implementation it does not matter whether we manage each list L[i] as a queue or a stack, since the algorithm is allowed to consider the nodes in a layer Li in any order.
(3.11) The above implementation of the BFS algorithm runs in time O(m + n) (i.e., linear in the input size), if the graph is given by the adjacency list representation.
Proof. As a first step, it is easy to bound the running time of the algorithm by O(n2) (a weaker bound than our claimed O(m + n)). To see this, note that there are at most n lists L[i] that we need to set up, so this takes O(n) time. Now we need to consider the nodes u on these lists. Each node occurs on at most one list, so the For loop runs at most n times over all iterations of the While loop. When we consider a node u, we need to look through all edges (u, v) incident to u. There can be at most n such edges, and we spend O(1) time considering each edge. So the total time spent on one iteration of the For loop is at most O(n). We’ve thus concluded that there are at most n iterations of the For loop, and that each iteration takes at most O(n) time, so the total time is at most O(n2).
To get the improved O(m + n) time bound, we need to observe that the
For loop processing a node u can take less than O(n) time if u has only a
few neighbors. As before, let nu denote the degree of node u, the number of
edges incident to u. Now, the time spent in the For loop considering edges
incident to node u is O(n ), so the total over all nodes is O( n ). Recall u u∈Vu
from (3.9) that u∈V nu = 2m, and so the total time spent considering edges over the whole algorithm is O(m). We need O(n) additional time to set up lists and manage the array Discovered. So the total time spent is O(m + n) as claimed.
We described the algorithm using up to n separate lists L[i] for each layer Li. Instead of all these distinct lists, we can implement the algorithm using a single list L that we maintain as a queue. In this way, the algorithm processes nodes in the order they are first discovered: each time a node is discovered, it is added to the end of the queue, and the algorithm always processes the edges out of the node that is currently first in the queue.
92
Chapter 3 Graphs
If we maintain the discovered nodes in this order, then all nodes in layer Li willappearinthequeueaheadofallnodesinlayerLi+1,fori=0,1,2....Thus, all nodes in layer Li will be considered in a contiguous sequence, followed by all nodes in layer Li+1, and so forth. Hence this implementation in terms of a single queue will produce the same result as the BFS implementation above.
Implementing Depth-First Search
We now consider the depth-first search algorithm. In the previous section we presented DFS as a recursive procedure, which is a natural way to specify it. However, it can also be viewed as almost identical to BFS, with the difference that it maintains the nodes to be processed in a stack, rather than in a queue. Essentially, the recursive structure of DFS can be viewed as pushing nodes onto a stack for later processing, while moving on to more freshly discovered nodes. We now show how to implement DFS by maintaining this stack of nodes to be processed explicitly.
In both BFS and DFS, there is a distinction between the act of discovering a node v—the first time it is seen, when the algorithm finds an edge leading to v—and the act of exploring a node v, when all the incident edges to v are scanned, resulting in the potential discovery of further nodes. The difference between BFS and DFS lies in the way in which discovery and exploration are interleaved.
In BFS, once we started to explore a node u in layer Li, we added all its newly discovered neighbors to the next layer Li+1, and we deferred actually exploring these neighbors until we got to the processing of layer Li+1. In contrast, DFS is more impulsive: when it explores a node u, it scans the neighbors of u until it finds the first not-yet-explored node v (if any), and then it immediately shifts attention to exploring v.
To implement the exploration strategy of DFS, we first add all of the nodes adjacent to u to our list of nodes to be considered, but after doing this we proceed to explore a new neighbor v of u. As we explore v, in turn, we add the neighbors of v to the list we’re maintaining, but we do so in stack order, so that these neighbors will be explored before we return to explore the other neighbors of u. We only come back to other nodes adjacent to u when there are no other nodes left.
In addition, we use an array Explored analogous to the Discovered array we used for BFS. The difference is that we only set Explored[v] to be true when we scan v’s incident edges (when the DFS search is at v), while BFS sets Discovered[v] to true as soon as v is first discovered. The implementation in full looks as follows.
3.3 Implementing Graph Traversal Using Queues and Stacks
93
DFS(s):
Initialize S to be a stack with one element s While S is not empty
Take a node u from S
If Explored[u] = false then
Set Explored[u] = true
For each edge (u, v) incident to u
Add v to the stack S Endfor
Endif
Endwhile
There is one final wrinkle to mention. Depth-first search is underspecified, since the adjacency list of a node being explored can be processed in any order. Note that the above algorithm, because it pushes all adjacent nodes onto the stack before considering any of them, in fact processes each adjacency list in the reverse order relative to the recursive version of DFS in the previous section.
(3.12) The above algorithm implements DFS, in the sense that it visits the nodes in exactly the same order as the recursive DFS procedure in the previous section (except that each adjacency list is processed in reverse order).
If we want the algorithm to also find the DFS tree, we need to have each node u on the stack S maintain the node that “caused” u to get added to the stack. This can be easily done by using an array parent and setting parent[v]=u when we add node v to the stack due to edge (u,v). When we mark a node u ̸= s as Explored, we also can add the edge (u,parent[u]) to the tree T. Note that a node v may be in the stack S multiple times, as it can be adjacent to multiple nodes u that we explore, and each such node adds a copy of v to the stack S. However, we will only use one of these copies to explore node v, the copy that we add last. As a result, it suffices to maintain one value parent[v] for each node v by simply overwriting the value parent[v] every time we add a new copy of v to the stack S.
The main step in the algorithm is to add and delete nodes to and from the stack S, which takes O(1) time. Thus, to bound the running time, we need to bound the number of these operations. To count the number of stack operations, it suffices to count the number of nodes added to S, as each node needs to be added once for every time it can be deleted from S.
How many elements ever get added to S? As before, let nv denote the degree of node v. Node v will be added to the stack S every time one of its nv adjacent nodes is explored, so the total number of nodes added to S is at
94
Chapter 3 Graphs
most n = 2m. This proves the desired O(m + n) bound on the running uv
time of DFS.
(3.13) The above implementation of the DFS algorithm runs in time O(m + n) (i.e., linear in the input size), if the graph is given by the adjacency list representation.
Finding the Set of All Connected Components
In the previous section we talked about how one can use BFS (or DFS) to find all connected components of a graph. We start with an arbitrary node s, and we use BFS (or DFS) to generate its connected component. We then find a node v (if any) that was not visited by the search from s and iterate, using BFS (or DFS) starting from v to generate its connected component—which, by (3.8), will be disjoint from the component of s. We continue in this way until all nodes have been visited.
Although we earlier expressed the running time of BFS and DFS as O(m + n), where m and n are the total number of edges and nodes in the graph, both BFS and DFS in fact spend work only on edges and nodes in the connected component containing the starting node. (They never see any of the other nodes or edges.) Thus the above algorithm, although it may run BFS or DFS a number of times, only spends a constant amount of work on a given edge or node in the iteration when the connected component it belongs to is under consideration. Hence the overall running time of this algorithm is still O(m + n).
3.4 Testing Bipartiteness: An Application of Breadth-First Search
Recall the definition of a bipartite graph: it is one where the node set V can be partitioned into sets X and Y in such a way that every edge has one end in X and the other end in Y. To make the discussion a little smoother, we can imagine that the nodes in the set X are colored red, and the nodes in the set Y are colored blue. With this imagery, we can say a graph is bipartite if it is possible to color its nodes red and blue so that every edge has one red end and one blue end.
The Problem
In the earlier chapters, we saw examples of bipartite graphs. Here we start by asking: What are some natural examples of a nonbipartite graph, one where no such partition of V is possible?
3.4 Testing Bipartiteness: An Application of Breadth-First Search
95
Clearly a triangle is not bipartite, since we can color one node red, another one blue, and then we can’t do anything with the third node. More generally, consider a cycle C of odd length, with nodes numbered 1, 2, 3, . . . , 2k, 2k + 1. If we color node 1 red, then we must color node 2 blue, and then we must color node 3 red, and so on—coloring odd-numbered nodes red and even-numbered nodes blue. But then we must color node 2k + 1 red, and it has an edge to node 1, which is also red. This demonstrates that there’s no way to partition C into red and blue nodes as required. More generally, if a graph G simply contains an odd cycle, then we can apply the same argument; thus we have established the following.
(3.14) If a graph G is bipartite, then it cannot contain an odd cycle.
It is easy to recognize that a graph is bipartite when appropriate sets X and Y (i.e., red and blue nodes) have actually been identified for us; and in many settings where bipartite graphs arise, this is natural. But suppose we encounter a graph G with no annotation provided for us, and we’d like to determine for ourselves whether it is bipartite—that is, whether there exists a partition into red and blue nodes, as required. How difficult is this? We see from (3.14) that an odd cycle is one simple “obstacle” to a graph’s being bipartite. Are there other, more complex obstacles to bipartitness?
Designing the Algorithm
In fact, there is a very simple procedure to test for bipartiteness, and its analysis can be used to show that odd cycles are the only obstacle. First we assume the graph G is connected, since otherwise we can first compute its connected components and analyze each of them separately. Next we pick any node s ∈ V and color it red; there is no loss in doing this, since s must receive some color. It follows that all the neighbors of s must be colored blue, so we do this. It then follows that all the neighbors of these nodes must be colored red, their neighbors must be colored blue, and so on, until the whole graph is colored. At this point, either we have a valid red/blue coloring of G, in which every edge has ends of opposite colors, or there is some edge with ends of the same color. In this latter case, it seems clear that there’s nothing we could have done: G simply is not bipartite. We now want to argue this point precisely and also work out an efficient way to perform the coloring.
The first thing to notice is that the coloring procedure we have just described is essentially identical to the description of BFS: we move outward from s, coloring nodes as soon as we first encounter them. Indeed, another way to describe the coloring algorithm is as follows: we perform BFS, coloring
96
Chapter 3 Graphs
The cycle through x, y, and z has odd length.
s red, all of layer L1 blue, all of layer L2 red, and so on, coloring odd-numbered layers blue and even-numbered layers red.
We can implement this on top of BFS, by simply taking the implementation of BFS and adding an extra array Color over the nodes. Whenever we get to a step in BFS where we are adding a node v to a list L[i+1], we assign Color[v] = red if i + 1 is an even number, and Color[v] = blue if i + 1 is an odd number. At the end of this procedure, we simply scan all the edges and determine whether there is any edge for which both ends received the same color. Thus, the total running time for the coloring algorithm is O(m + n), just as it is for BFS.
Analyzing the Algorithm
We now prove a claim that shows this algorithm correctly determines whether G is bipartite, and it also shows that we can find an odd cycle in G whenever it is not bipartite.
(3.15) Let G be a connected graph, and let L1, L2, . . . be the layers produced by BFS starting at node s. Then exactly one of the following two things must hold.
(i) There is no edge of G joining two nodes of the same layer. In this case G is a bipartite graph in which the nodes in even-numbered layers can be colored red, and the nodes in odd-numbered layers can be colored blue.
(ii) There is an edge of G joining two nodes of the same layer. In this case, G contains an odd-length cycle, and so it cannot be bipartite.
Proof. First consider case (i), where we suppose that there is no edge joining two nodes of the same layer. By (3.4), we know that every edge of G joins nodes either in the same layer or in adjacent layers. Our assumption for case (i) is precisely that the first of these two alternatives never happens, so this means that every edge joins two nodes in adjacent layers. But our coloring procedure gives nodes in adjacent layers the opposite colors, and so every edge has ends with opposite colors. Thus this coloring establishes that G is bipartite.
Now suppose we are in case (ii); why must G contain an odd cycle? We are told that G contains an edge joining two nodes of the same layer. Suppose this is the edge e = (x, y), with x, y ∈ Lj. Also, for notational reasons, recall that L0 (“layer 0”) is the set consisting of just s. Now consider the BFS tree T produced by our algorithm, and let z be the node whose layer number is as large as possible, subject to the condition that z is an ancestor of both x and y in T; for obvious reasons, we can call z the lowest common ancestor of x and y. Suppose z ∈ Li, where i < j. We now have the situation pictured in Figure 3.6. We consider the cycle C defined by following the z-x path in T, then the edge e,
s
Layer Li
Layer Lj x
z
Figure 3.6 If two nodes x and y in the same layer are joined by an edge, then the cycle through x, y, and their lowest common ancestor z has odd length, demonstrating that the graph cannot be bipartite.
y
and then the y-z path in T. The length of this cycle is (j − i) + 1+ (j − i), adding the length of its three parts separately; this is equal to 2(j − i) + 1, which is an odd number.
3.5 Connectivity in Directed Graphs
Thus far, we have been looking at problems on undirected graphs; we now consider the extent to which these ideas carry over to the case of directed graphs.
Recall that in a directed graph, the edge (u, v) has a direction: it goes from u to v. In this way, the relationship between u and v is asymmetric, and this has qualitative effects on the structure of the resulting graph. In Section 3.1, for example, we discussed the World Wide Web as an instance of a large, complex directed graph whose nodes are pages and whose edges are hyperlinks. The act of browsing the Web is based on following a sequence of edges in this directed graph; and the directionality is crucial, since it’s not generally possible to browse “backwards” by following hyperlinks in the reverse direction.
At the same time, a number of basic definitions and algorithms have natural analogues in the directed case. This includes the adjacency list repre- sentation and graph search algorithms such as BFS and DFS. We now discuss these in turn.
Representing Directed Graphs
In order to represent a directed graph for purposes of designing algorithms, we use a version of the adjacency list representation that we employed for undirected graphs. Now, instead of each node having a single list of neighbors, each node has two lists associated with it: one list consists of nodes to which it has edges, and a second list consists of nodes from which it has edges. Thus an algorithm that is currently looking at a node u can read off the nodes reachable by going one step forward on a directed edge, as well as the nodes that would be reachable if one went one step in the reverse direction on an edge from u.
The Graph Search Algorithms
Breadth-first search and depth-first search are almost the same in directed graphs as they are in undirected graphs. We will focus here on BFS. We start at a node s, define a first layer of nodes to consist of all those to which s has an edge, define a second layer to consist of all additional nodes to which these first-layer nodes have an edge, and so forth. In this way, we discover nodes layer by layer as they are reached in this outward search from s, and the nodes in layer j are precisely those for which the shortest path from s has exactly j edges. As in the undirected case, this algorithm performs at most constant work for each node and edge, resulting in a running time of O(m + n).
3.5 Connectivity in Directed Graphs
97
98
Chapter 3 Graphs
It is important to understand what this directed version of BFS is comput- ing. In directed graphs, it is possible for a node s to have a path to a node t even though t has no path to s; and what directed BFS is computing is the set of all nodes t with the property that s has a path to t. Such nodes may or may not have paths back to s.
There is a natural analogue of depth-first search as well, which also runs in linear time and computes the same set of nodes. It is again a recursive procedure that tries to explore as deeply as possible, in this case only following edges according to their inherent direction. Thus, when DFS is at a node u, it recursively launches a depth-first search, in order, for each node to which u has an edge.
Suppose that, for a given node s, we wanted the set of nodes with paths to s, rather than the set of nodes to which s has paths. An easy way to do this would be to define a new directed graph, Grev, that we obtain from G simply by reversing the direction of every edge. We could then run BFS or DFS in Grev; anodehasapathfromsinGrev ifandonlyifithasapathtosinG.
Strong Connectivity
Recall that a directed graph is strongly connected if, for every two nodes u and v, there is a path from u to v and a path from v to u. It’s worth also formulating some terminology for the property at the heart of this definition; let’s say that two nodes u and v in a directed graph are mutually reachable if there is a path from u to v and also a path from v to u. (So a graph is strongly connected if every pair of nodes is mutually reachable.)
Mutual reachability has a number of nice properties, many of them stem- ming from the following simple fact.
(3.16) If u and v are mutually reachable, and v and w are mutually reachable, then u and w are mutually reachable.
Proof. To construct a path from u to w, we first go from u to v (along the path guaranteed by the mutual reachability of u and v), and then on from v to w (along the path guaranteed by the mutual reachability of v and w). To construct a path from w to u, we just reverse this reasoning: we first go from w to v (along the path guaranteed by the mutual reachability of v and w), and then on from v to u (along the path guaranteed by the mutual reachability of u and v).
There is a simple linear-time algorithm to test if a directed graph is strongly connected, implicitly based on (3.16). We pick any node s and run BFS in G starting from s. We then also run BFS starting from s in Grev. Now, if one of these two searches fails to reach every node, then clearly G is not strongly connected. But suppose we find that s has a path to every node, and that
3.6 Directed Acyclic Graphs and Topological Ordering
99
every node has a path to s. Then s and v are mutually reachable for every v, and so it follows that every two nodes u and v are mutually reachable: s and u are mutually reachable, and s and v are mutually reachable, so by (3.16) we also have that u and v are mutually reachable.
By analogy with connected components in an undirected graph, we can define the strong component containing a node s in a directed graph to be the set of all v such that s and v are mutually reachable. If one thinks about it, the algorithm in the previous paragraph is really computing the strong component containing s: we run BFS starting from s both in G and in Grev; the set of nodes reached by both searches is the set of nodes with paths to and from s, and hence this set is the strong component containing s.
There are further similarities between the notion of connected components in undirected graphs and strong components in directed graphs. Recall that connected components naturally partitioned the graph, since any two were either identical or disjoint. Strong components have this property as well, and for essentially the same reason, based on (3.16).
(3.17) For any two nodes s and t in a directed graph, their strong components are either identical or disjoint.
Proof. Consider any two nodes s and t that are mutually reachable; we claim that the strong components containing s and t are identical. Indeed, for any node v, if s and v are mutually reachable, then by (3.16), t and v are mutually reachable as well. Similarly, if t and v are mutually reachable, then again by (3.16), s and v are mutually reachable.
On the other hand, if s and t are not mutually reachable, then there cannot be a node v that is in the strong component of each. For if there were such a node v, then s and v would be mutually reachable, and v and t would be mutually reachable, so from (3.16) it would follow that s and t were mutually reachable.
In fact, although we will not discuss the details of this here, with more work it is possible to compute the strong components for all nodes in a total time of O(m + n).
3.6 Directed Acyclic Graphs and Topological Ordering
If an undirected graph has no cycles, then it has an extremely simple structure: each of its connected components is a tree. But it is possible for a directed graph to have no (directed) cycles and still have a very rich structure. For example, such graphs can have a large number of edges: if we start with the node
100
Chapter 3
Graphs
v6
v2
v7
v5
(b)
v3
v1
In a topological ordering, all edges point from left to right.
v4 v1 v2 v3 v4 v5 v6 v7
(c)
(a)
Figure 3.7 (a) A directed acyclic graph. (b) The same DAG with a topological ordering, specified by the labels on each node. (c) A different drawing of the same DAG, arranged so as to emphasize the topological ordering.
set {1, 2, . . . , n} and include an edge (i, j) whenever i < j, then the resulting directed graph has n edges but no cycles.
2
If a directed graph has no cycles, we call it—naturally enough—a directed
acyclic graph, or a DAG for short. (The term DAG is typically pronounced as a word, not spelled out as an acronym.) In Figure 3.7(a) we see an example of a DAG, although it may take some checking to convince oneself that it really has no directed cycles.
The Problem
DAGs are a very common structure in computer science, because many kinds of dependency networks of the type we discussed in Section 3.1 are acyclic. Thus DAGs can be used to encode precedence relations or dependencies in a natural way. Suppose we have a set of tasks labeled {1, 2, . . . , n} that need to be performed, and there are dependencies among them stipulating, for certain pairs i and j, that i must be performed before j. For example, the tasks may be courses, with prerequisite requirements stating that certain courses must be taken before others. Or the tasks may correspond to a pipeline of computing jobs, with assertions that the output of job i is used in determining the input to job j, and hence job i must be done before job j.
We can represent such an interdependent set of tasks by introducing a node for each task, and a directed edge (i, j) whenever i must be done before j. If the precedence relation is to be at all meaningful, the resulting graph G must be a DAG. Indeed, if it contained a cycle C, there would be no way to do any of the tasks in C: since each task in C cannot begin until some other one completes, no task in C could ever be done, since none could be done first.
3.6 Directed Acyclic Graphs and Topological Ordering
101
Let’s continue a little further with this picture of DAGs as precedence relations. Given a set of tasks with dependencies, it would be natural to seek a valid order in which the tasks could be performed, so that all dependencies are respected. Specifically, for a directed graph G, we say that a topological ordering of G is an ordering of its nodes as v1, v2, . . . , vn so that for every edge (vi, vj), we have i < j. In other words, all edges point “forward” in the ordering. A topological ordering on tasks provides an order in which they can be safely performed; when we come to the task vj, all the tasks that are required to precede it have already been done. In Figure 3.7(b) we’ve labeled the nodes of the DAG from part (a) with a topological ordering; note that each edge indeed goes from a lower-indexed node to a higher-indexed node.
In fact, we can view a topological ordering of G as providing an immediate “proof” that G has no cycles, via the following.
(3.18) If G has a topological ordering, then G is a DAG.
Proof. Suppose, by way of contradiction, that G has a topological ordering v1, v2, . . . , vn, and also has a cycle C. Let vi be the lowest-indexed node on C, and let vj be the node on C just before vi—thus (vj , vi) is an edge. But by our choice of i, we have j > i, which contradicts the assumption that v1, v2, . . . , vn was a topological ordering.

The proof of acyclicity that a topological ordering provides can be very useful, even visually. In Figure 3.7(c), we have drawn the same graph as in (a) and (b), but with the nodes laid out in the topological ordering. It is immediately clear that the graph in (c) is a DAG since each edge goes from left to right.

Computing a Topological Ordering The main question we consider here is the converse of (3.18): Does every DAG have a topological ordering, and if so, how do we find one efficiently? A method to do this for every DAG would be very useful: it would show that for any precedence relation on a set of tasks without cycles, there is an efficiently computable order in which to perform the tasks.

Designing and Analyzing the Algorithm

In fact, the converse of (3.18) does hold, and we establish this via an efficient algorithm to compute a topological ordering. The key to this lies in finding a way to get started: which node do we put at the beginning of the topological ordering? Such a node v1 would need to have no incoming edges, since any such incoming edge would violate the defining property of the topological

102

Chapter 3 Graphs

ordering, that all edges point forward. Thus, we need to prove the following fact.

(3.19) In every DAG G, there is a node v with no incoming edges.

Proof. LetGbeadirectedgraphinwhicheverynodehasatleastoneincoming edge. We show how to find a cycle in G; this will prove the claim. We pick any node v, and begin following edges backward from v: since v has at least one incoming edge (u, v), we can walk backward to u; then, since u has at least one incoming edge (x, u), we can walk backward to x; and so on. We can continue this process indefinitely, since every node we encounter has an incoming edge. But after n + 1 steps, we will have visited some node w twice. If we let C denote the sequence of nodes encountered between successive visits to w, then clearly C forms a cycle.

In fact, the existence of such a node v is all we need to produce a topological ordering of G by induction. Specifically, let us claim by induction that every DAG has a topological ordering. This is clearly true for DAGs on one or two nodes. Now suppose it is true for DAGs with up to some number of nodes n. Then, given a DAG G on n + 1 nodes, we find a node v with no incoming edges, as guaranteed by (3.19). We place v first in the topological ordering; this is safe, since all edges out of v will point forward. Now G−{v} is a DAG, since deleting v cannot create any cycles that weren’t there previously. Also, G−{v} has n nodes, so we can apply the induction hypothesis to obtain a topological ordering of G−{v}. We append the nodes of G−{v} in this order after v; this is an ordering of G in which all edges point forward, and hence it is a topological ordering.

Thus we have proved the desired converse of (3.18).

(3.20) If G is a DAG, then G has a topological ordering.

The inductive proof contains the following algorithm to compute a topo-

logical ordering of G.

To compute a topological ordering of G:

Find a node v with no incoming edges and order it first Delete v from G

Recursively compute a topological ordering of G−{v}

and append this order after v

In Figure 3.8 we show the sequence of node deletions that occurs when this algorithm is applied to the graph in Figure 3.7. The shaded nodes in each iteration are those with no incoming edges; the crucial point, which is what

3.6 Directed Acyclic Graphs and Topological Ordering

103

v2 v3 v2 v3 v3

v6 v5 v4 v6 v5 v4 v6 v5 v4

v7v1v7 v7

(a) (b) (c)

v6 v5 v4 v6 v5 v6

v7 v7 v7 (d) (e) (f)

Figure 3.8 Starting from the graph in Figure 3.7, nodes are deleted one by one so as to be added to a topological ordering. The shaded nodes are those with no incoming edges; note that there is always at least one such edge at every stage of the algorithm’s execution.

(3.19) guarantees, is that when we apply this algorithm to a DAG, there will always be at least one such node available to delete.

To bound the running time of this algorithm, we note that identifying a node v with no incoming edges, and deleting it from G, can be done in O(n) time. Since the algorithm runs for n iterations, the total running time is O(n2).

This is not a bad running time; and if G is very dense, containing (n2) edges, then it is linear in the size of the input. But we may well want something better when the number of edges m is much less than n2. In such a case, a running time of O(m + n) could be a significant improvement over (n2).

In fact, we can achieve a running time of O(m + n) using the same high- level algorithm—iteratively deleting nodes with no incoming edges. We simply have to be more efficient in finding these nodes, and we do this as follows.

We declare a node to be “active” if it has not yet been deleted by the algorithm, and we explicitly maintain two things:

(a) for each node w, the number of incoming edges that w has from active nodes; and

(b) thesetSofallactivenodesinGthathavenoincomingedgesfromother active nodes.

104

Chapter 3 Graphs

b

ae

cd

Figure 3.9 How many topo- logical orderings does this graph have?

At the start, all nodes are active, so we can initialize (a) and (b) with a single pass through the nodes and edges. Then, each iteration consists of selecting a node v from the set S and deleting it. After deleting v, we go through all nodes w to which v had an edge, and subtract one from the number of active incoming edges that we are maintaining for w. If this causes the number of active incoming edges to w to drop to zero, then we add w to the set S. Proceeding in this way, we keep track of nodes that are eligible for deletion at all times, while spending constant work per edge over the course of the whole algorithm.

Solved Exercises

Solved Exercise 1

Consider the directed acyclic graph G in Figure 3.9. How many topological orderings does it have?

Solution Recall that a topological ordering of G is an ordering of the nodes as v1, v2, . . . , vn so that all edges point “forward”: for every edge (vi, vj), we have i < j.
So one way to answer this question would be to write down all 5 · 4 · 3 · 2 · 1 = 120 possible orderings and check whether each is a topological ordering. But this would take a while.
Instead, we think about this as follows. As we saw in the text (or reasoning directly from the definition), the first node in a topological ordering must be one that has no edge coming into it. Analogously, the last node must be one that has no edge leaving it. Thus, in every topological ordering of G, the node a must come first and the node e must come last.
Now we have to figure how the nodes b, c, and d can be arranged in the middle of the ordering. The edge (c, d) enforces the requirement that c must come before d; but b can be placed anywhere relative to these two: before both, between c and d, or after both. This exhausts all the possibilities, and so we conclude that there are three possible topological orderings:
a,b,c,d,e a,c,b,d,e a,c,d,b,e
Solved Exercise 2
Some friends of yours are working on techniques for coordinating groups of mobile robots. Each robot has a radio transmitter that it uses to communicate
with a base station, and your friends find that if the robots get too close to one another, then there are problems with interference among the transmitters. So a natural problem arises: how to plan the motion of the robots in such a way that each robot gets to its intended destination, but in the process the robots don’t come close enough together to cause interference problems.
We can model this problem abstractly as follows. Suppose that we have an undirected graph G = (V , E), representing the floor plan of a building, and there are two robots initially located at nodes a and b in the graph. The robot at node a wants to travel to node c along a path in G, and the robot at node b wants to travel to node d. This is accomplished by means of a schedule: at each time step, the schedule specifies that one of the robots moves across a single edge, from one node to a neighboring node; at the end of the schedule, the robot from node a should be sitting on c, and the robot from b should be sitting on d.
A schedule is interference-free if there is no point at which the two robots occupy nodes that are at a distance ≤ r from one another in the graph, for a given parameter r. We’ll assume that the two starting nodes a and b are at a distance greater than r, and so are the two ending nodes c and d.
Give a polynomial-time algorithm that decides whether there exists an interference-free schedule by which each robot can get to its destination.
Solution This is a problem of the following general flavor. We have a set of possible configurations for the robots, where we define a configuration to be a choice of location for each one. We are trying to get from a given starting configuration (a, b) to a given ending configuration (c, d), subject to constraints on how we can move between configurations (we can only change one robot’s location to a neighboring node), and also subject to constraints on which configurations are “legal.”
This problem can be tricky to think about if we view things at the level of the underlying graph G: for a given configuration of the robots—that is, the current location of each one—it’s not clear what rule we should be using to decide how to move one of the robots next. So instead we apply an idea that can be very useful for situations in which we’re trying to perform this type of search. We observe that our problem looks a lot like a path-finding problem, not in the original graph G but in the space of all possible configurations.
Let us define the following (larger) graph H. The node set of H is the set of all possible configurations of the robots; that is, H consists of all possible pairs of nodes in G. We join two nodes of H by an edge if they represent configurations that could be consecutive in a schedule; that is, (u, v) and (u′,v′)willbejoinedbyanedgeinH ifoneofthepairsu,u′ orv,v′ areequal, and the other pair corresponds to an edge in G.
Solved Exercises
105
106
Chapter 3 Graphs
We can already observe that paths in H from (a, b) to (c, d) correspond to schedules for the robots: such a path consists precisely of a sequence of configurations in which, at each step, one robot crosses a single edge in G. However, we have not yet encoded the notion that the schedule should be interference-free.
To do this, we simply delete from H all nodes that correspond to configura- tions in which there would be interference. Thus we define H′ to be the graph obtained from H by deleting all nodes (u, v) for which the distance between u and v in G is at most r.
The full algorithm is then as follows. We construct the graph H′, and then run the connectivity algorithm from the text to determine whether there is a path from (a, b) to (c, d). The correctness of the algorithm follows from the fact that paths in H′ correspond to schedules, and the nodes in H′ correspond precisely to the configurations in which there is no interference.
Finally, we need to consider the running time. Let n denote the number of nodes in G, and m denote the number of edges in G. We’ll analyze the running time by doing three things: (1) bounding the size of H′ (which will in general be larger than G), (2) bounding the time it takes to construct H′, and (3) bounding the time it takes to search for a path from (a, b) to (c, d) in H.
1. First, then, let’s consider the size of H′. H′ has at most n2 nodes, since its nodes correspond to pairs of nodes in G. Now, how many edges does H′ have? A node (u, v) will have edges to (u′, v) for each neighbor u′ of u in G, and to (u,v′) for each neighbor v′ of v in G. A simple upper bound says that there can be at most n choices for (u′, v), and at most n choices for (u, v′), so there are at most 2n edges incident to each node of H′. Summing over the (at most) n2 nodes of H′, we have O(n3) edges.
(We can actually give a better bound of O(mn) on the number of edges in H′, by using the bound (3.9) we proved in Section 3.3 on the sum of the degrees in a graph. We’ll leave this as a further exercise.)
2. Now we bound the time needed to construct H′. We first build H by enumerating all pairs of nodes in G in time O(n2), and constructing edges using the definition above in time O(n) per node, for a total of O(n3). Now we need to figure out which nodes to delete from H so as to produce H′. We can do this as follows. For each node u in G, we run a breadth- first search from u and identify all nodes v within distance r of u. We list all these pairs (u, v) and delete them from H. Each breadth-first search in G takes time O(m + n), and we’re doing one from each node, so the total time for this part is O(mn + n2).
3. Now we have H′, and so we just need to decide whether there is a path from (a, b) to (c, d). This can be done using the connectivity algorithm from the text in time that is linear in the number of nodes and edges of H′. Since H′ has O(n2) nodes and O(n3) edges, this final step takes polynomial time as well.
Exercises
1. Consider the directed acyclic graph G in Figure 3.10. How many topolog- ical orderings does it have?
2. Give an algorithm to detect whether a given undirected graph contains a cycle. If the graph contains a cycle, then your algorithm should output one. (It should not output all cycles in the graph, just one of them.) The running time of your algorithm should be O(m + n) for a graph with n nodes and m edges.
3. ThealgorithmdescribedinSection3.6forcomputingatopologicalorder- ing of a DAG repeatedly finds a node with no incoming edges and deletes it. This will eventually produce a topological ordering, provided that the input graph really is a DAG.
But suppose that we’re given an arbitrary graph that may or may not be a DAG. Extend the topological ordering algorithm so that, given an input directed graph G, it outputs one of two things: (a) a topological ordering, thus establishing that G is a DAG; or (b) a cycle in G, thus establishing that G is not a DAG. The running time of your algorithm should be O(m + n) for a directed graph with n nodes and m edges.
4. InspiredbytheexampleofthatgreatCornellian,VladimirNabokov,some of your friends have become amateur lepidopterists (they study butter- flies). Often when they return from a trip with specimens of butterflies, it is very difficult for them to tell how many distinct species they’ve caught—thanks to the fact that many species look very similar to one another.
One day they return with n butterflies, and they believe that each belongs to one of two different species, which we’ll call A and B for purposes of this discussion. They’d like to divide the n specimens into two groups—those that belong to A and those that belong to B—but it’s very hard for them to directly label any one specimen. So they decide to adopt the following approach.
bc
af
de
Figure 3.10 How many topo- logical orderings does this graph have?
Exercises
107
108
Chapter 3 Graphs
For each pair of specimens i and j, they study them carefully side by side. If they’re confident enough in their judgment, then they label the pair (i, j) either “same” (meaning they believe them both to come from the same species) or “different” (meaning they believe them to come from different species). They also have the option of rendering no judgment on a given pair, in which case we’ll call the pair ambiguous.
So now they have the collection of n specimens, as well as a collection of m judgments (either “same” or “different”) for the pairs that were not declared to be ambiguous. They’d like to know if this data is consistent with the idea that each butterfly is from one of species A or B. So more concretely, we’ll declare the m judgments to be consistent if it is possible to label each specimen either A or B in such a way that for each pair (i, j) labeled “same,” it is the case that i and j have the same label; and for each pair (i, j) labeled “different,” it is the case that i and j have different labels. They’re in the middle of tediously working out whether their judgments are consistent, when one of them realizes that you probably have an algorithm that would answer this question right away.
Give an algorithm with running time O(m + n) that determines whether the m judgments are consistent.
5. Abinarytreeisarootedtreeinwhicheachnodehasatmosttwochildren. Show by induction that in any binary tree the number of nodes with two children is exactly one less than the number of leaves.
6. WehaveaconnectedgraphG=(V,E),andaspecificvertexu∈V.Suppose we compute a depth-first search tree rooted at u, and obtain a tree T that includes all nodes of G. Suppose we then compute a breadth-first search tree rooted at u, and obtain the same tree T. Prove that G = T. (In other words, if T is both a depth-first search tree and a breadth-first search tree rooted at u, then G cannot contain any edges that do not belong to T.)
7. Some friends of yours work on wireless networks, and they’re currently studying the properties of a network of n mobile devices. As the devices move around (actually, as their human owners move around), they define a graph at any point in time as follows: there is a node representing each of the n devices, and there is an edge between device i and device j if the physical locations of i and j are no more than 500 meters apart. (If so, we say that i and j are “in range” of each other.)
They’d like it to be the case that the network of devices is connected at all times, and so they’ve constrained the motion of the devices to satisfy
the following property: at all times, each device i is within 500 meters of at least n/2 of the other devices. (We’ll assume n is an even number.) What they’d like to know is: Does this property by itself guarantee that the network will remain connected?
Here’s a concrete way to formulate the question as a claim about graphs.
Claim: Let G be a graph on n nodes, where n is an even number. If every node of G has degree at least n/2, then G is connected.
Decide whether you think the claim is true or false, and give a proof of either the claim or its negation.
8. A number of stories in the press about the structure of the Internet and the Web have focused on some version of the following question: How far apart are typical nodes in these networks? If you read these stories carefully, you find that many of them are confused about the difference between the diameter of a network and the average distance in a network; they often jump back and forth between these concepts as though they’re the same thing.
As in the text, we say that the distance between two nodes u and v in a graph G = (V , E) is the minimum number of edges in a path joining them; we’ll denote this by dist(u, v). We say that the diameter of G is the maximum distance between any pair of nodes; and we’ll denote this quantity by diam(G).
Let’s define a related quantity, which we’ll call the average pairwise
distance in G (denoted apd(G)). We define apd(G) to be the average, over
all n sets of two distinct nodes u and v, of the distance between u and v. 2
That is,
⎡ ⎤ apd(G)=⎣ dist(u,v)⎦/ n .
{u , v}⊆V
2
Here’s a simple example to convince yourself that there are graphs G for which diam(G) ̸= apd(G). Let G be a graph with three nodes u, v, w, and with the two edges {u, v} and {v, w}. Then
while
diam(G) = dist(u, w) = 2,
apd(G) = [dist(u, v) + dist(u, w) + dist(v, w)]/3 = 4/3.
Exercises
109
110
Chapter 3 Graphs
Of course, these two numbers aren’t all that far apart in the case of this three-node graph, and so it’s natural to ask whether there’s always a close relation between them. Here’s a claim that tries to make this precise.
Claim: There exists a positive natural number c so that for all connected graphs G, it is the case that
diam(G) ≤ c. apd(G)
Decide whether you think the claim is true or false, and give a proof of either the claim or its negation.
9. There’s a natural intuition that two nodes that are far apart in a com- munication network—separated by many hops—have a more tenuous connection than two nodes that are close together. There are a number of algorithmic results that are based to some extent on different ways of making this notion precise. Here’s one that involves the susceptibility of paths to the deletion of nodes.
Suppose that an n-node undirected graph G = (V , E) contains two nodes s and t such that the distance between s and t is strictly greater than n/2. Show that there must exist some node v, not equal to either s or t, such that deleting v from G destroys all s-t paths. (In other words, the graph obtained from G by deleting v contains no path from s to t.) Give an algorithm with running time O(m + n) to find such a node v.
10. Anumberofartmuseumsaroundthecountryhavebeenfeaturingwork by an artist named Mark Lombardi (1951–2000), consisting of a set of intricately rendered graphs. Building on a great deal of research, these graphs encode the relationships among people involved in major political scandals over the past several decades: the nodes correspond to partici- pants, and each edge indicates some type of relationship between a pair of participants. And so, if you peer closely enough at the drawings, you can trace out ominous-looking paths from a high-ranking U.S. govern- ment official, to a former business partner, to a bank in Switzerland, to a shadowy arms dealer.
Such pictures form striking examples of social networks, which, as we discussed in Section 3.1, have nodes representing people and organi- zations, and edges representing relationships of various kinds. And the short paths that abound in these networks have attracted considerable attention recently, as people ponder what they mean. In the case of Mark Lombardi’s graphs, they hint at the short set of steps that can carry you from the reputable to the disreputable.
Of course, a single, spurious short path between nodes v and w in such a network may be more coincidental than anything else; a large number of short paths between v and w can be much more convincing. So in addition to the problem of computing a single shortest v-w path in a graph G, social networks researchers have looked at the problem of determining the number of shortest v-w paths.
This turns out to be a problem that can be solved efficiently. Suppose we are given an undirected graph G = (V , E), and we identify two nodes v and w in G. Give an algorithm that computes the number of shortest v-w paths in G. (The algorithm should not list all the paths; just the number suffices.) The running time of your algorithm should be O(m + n) for a graph with n nodes and m edges.
11. You’rehelpingsomesecurityanalystsmonitoracollectionofnetworked computers, tracking the spread of an online virus. There are n computers in the system, labeled C1, C2, . . . , Cn, and as input you’re given a collection of trace data indicating the times at which pairs of computers commu- nicated. Thus the data is a sequence of ordered triples (Ci , Cj , tk ); such a triple indicates that Ci and Cj exchanged bits at time tk. There are m triples total.
We’ll assume that the triples are presented to you in sorted order of time. For purposes of simplicity, we’ll assume that each pair of computers communicates at most once during the interval you’re observing.
The security analysts you’re working with would like to be able to answer questions of the following form: If the virus was inserted into computer Ca at time x, could it possibly have infected computer Cb by time y? The mechanics of infection are simple: if an infected computer Ci communicates with an uninfected computer Cj at time tk (in other words, if one of the triples (Ci , Cj , tk) or (Cj , Ci , tk) appears in the trace data), then computer Cj becomes infected as well, starting at time tk. Infection can thus spread from one machine to another across a sequence of communications, provided that no step in this sequence involves a move backward in time. Thus, for example, if Ci is infected by time tk, and the trace data contains triples (Ci, Cj, tk) and (Cj, Cq, tr), where tk ≤ tr, then Cq will become infected via Cj. (Note that it is okay for tk to be equal to tr; this would mean that Cj had open connections to both Ci and Cq at the same time, and so a virus could move from Ci to Cq.)
For example, suppose n = 4, the trace data consists of the triples (C1,C2,4), (C2,C4,8), (C3,C4,8), (C1,C4,12),
Exercises
111
112
Chapter 3 Graphs
12.
and the virus was inserted into computer C1 at time 2. Then C3 would be infected at time 8 by a sequence of three steps: first C2 becomes infected at time 4, then C4 gets the virus from C2 at time 8, and then C3 gets the virus from C4 at time 8. On the other hand, if the trace data were
(C2,C3,8), (C1,C4,12), (C1,C2,14),
and again the virus was inserted into computer C1 at time 2, then C3 would not become infected during the period of observation: although C2 becomes infected at time 14, we see that C3 only communicates with C2 before C2 was infected. There is no sequence of communications moving forward in time by which the virus could get from C1 to C3 in this second example.
Design an algorithm that answers questions of this type: given a collection of trace data, the algorithm should decide whether a virus introduced at computer Ca at time x could have infected computer Cb by time y. The algorithm should run in time O(m + n).
You’re helping a group of ethnographers analyze some oral history data they’ve collected by interviewing members of a village to learn about the lives of people who’ve lived there over the past two hundred years.
From these interviews, they’ve learned about a set of n people (all of them now deceased), whom we’ll denote P1, P2, . . . , Pn. They’ve also collected facts about when these people lived relative to one another. Each fact has one of the following two forms:
. For some i and j, person Pi died before person Pj was born; or
. for some i and j, the life spans of Pi and Pj overlapped at least partially.
Naturally, they’re not sure that all these facts are correct; memories are not so good, and a lot of this was passed down by word of mouth. So what they’d like you to determine is whether the data they’ve collected is at least internally consistent, in the sense that there could have existed a set of people for which all the facts they’ve learned simultaneously hold.
Give an efficient algorithm to do this: either it should produce pro- posed dates of birth and death for each of the n people so that all the facts hold true, or it should report (correctly) that no such dates can exist—that is, the facts collected by the ethnographers are not internally consistent.
Notes and Further Reading
The theory of graphs is a large topic, encompassing both algorithmic and non- algorithmic issues. It is generally considered to have begun with a paper by
Euler (1736), grown through interest in graph representations of maps and chemical compounds in the nineteenth century, and emerged as a systematic area of study in the twentieth century, first as a branch of mathematics and later also through its applications to computer science. The books by Berge (1976), Bollobas (1998), and Diestel (2000) provide substantial further coverage of graph theory. Recently, extensive data has become available for studying large networks that arise in the physical, biological, and social sciences, and there has been interest in understanding properties of networks that span all these different domains. The books by Barabasi (2002) and Watts (2002) discuss this emerging area of research, with presentations aimed at a general audience.
The basic graph traversal techniques covered in this chapter have numer- ous applications. We will see a number of these in subsequent chapters, and we refer the reader to the book by Tarjan (1983) for further results.
Notes on the Exercises Exercise 12 is based on a result of Martin Golumbic and Ron Shamir.
Notes and Further Reading
113
This page intentionally left blank
Chapter 4 Greedy Algorithms
In Wall Street, that iconic movie of the 1980s, Michael Douglas gets up in front of a room full of stockholders and proclaims, “Greed . . . is good. Greed is right. Greed works.” In this chapter, we’ll be taking a much more understated perspective as we investigate the pros and cons of short-sighted greed in the design of algorithms. Indeed, our aim is to approach a number of different computational problems with a recurring set of questions: Is greed good? Does greed work?
It is hard, if not impossible, to define precisely what is meant by a greedy algorithm. An algorithm is greedy if it builds up a solution in small steps, choosing a decision at each step myopically to optimize some underlying criterion. One can often design many different greedy algorithms for the same problem, each one locally, incrementally optimizing some different measure on its way to a solution.
When a greedy algorithm succeeds in solving a nontrivial problem opti- mally, it typically implies something interesting and useful about the structure of the problem itself; there is a local decision rule that one can use to con- struct optimal solutions. And as we’ll see later, in Chapter 11, the same is true of problems in which a greedy algorithm can produce a solution that is guar- anteed to be close to optimal, even if it does not achieve the precise optimum. These are the kinds of issues we’ll be dealing with in this chapter. It’s easy to invent greedy algorithms for almost any problem; finding cases in which they work well, and proving that they work well, is the interesting challenge.
The first two sections of this chapter will develop two basic methods for proving that a greedy algorithm produces an optimal solution to a problem. One can view the first approach as establishing that the greedy algorithm stays ahead. By this we mean that if one measures the greedy algorithm’s progress
116
Chapter 4 Greedy Algorithms
in a step-by-step fashion, one sees that it does better than any other algorithm at each step; it then follows that it produces an optimal solution. The second approach is known as an exchange argument, and it is more general: one considers any possible solution to the problem and gradually transforms it into the solution found by the greedy algorithm without hurting its quality. Again, it will follow that the greedy algorithm must have found a solution that is at least as good as any other solution.
Following our introduction of these two styles of analysis, we focus on several of the most well-known applications of greedy algorithms: shortest paths in a graph, the Minimum Spanning Tree Problem, and the construc- tion of Huffman codes for performing data compression. They each provide nice examples of our analysis techniques. We also explore an interesting re- lationship between minimum spanning trees and the long-studied problem of clustering. Finally, we consider a more complex application, the Minimum- Cost Arborescence Problem, which further extends our notion of what a greedy algorithm is.
4.1 Interval Scheduling: The Greedy Algorithm Stays Ahead
Let’s recall the Interval Scheduling Problem, which was the first of the five representative problems we considered in Chapter 1. We have a set of requests {1, 2, . . . , n}; the ith request corresponds to an interval of time starting at s(i) and finishing at f(i). (Note that we are slightly changing the notation from Section 1.2, where we used si rather than s(i) and fi rather than f(i). This change of notation will make things easier to talk about in the proofs.) We’ll say that a subset of the requests is compatible if no two of them overlap in time, and our goal is to accept as large a compatible subset as possible. Compatible sets of maximum size will be called optimal.
Designing a Greedy Algorithm
Using the Interval Scheduling Problem, we can make our discussion of greedy algorithms much more concrete. The basic idea in a greedy algorithm for interval scheduling is to use a simple rule to select a first request i1. Once a request i1 is accepted, we reject all requests that are not compatible with i1. We then select the next request i2 to be accepted, and again reject all requests that are not compatible with i2. We continue in this fashion until we run out of requests. The challenge in designing a good greedy algorithm is in deciding which simple rule to use for the selection—and there are many natural rules for this problem that do not give good solutions.
Let’s try to think of some of the most natural rules and see how they work.
4.1 Interval Scheduling: The Greedy Algorithm Stays Ahead
117
. The most obvious rule might be to always select the available request that starts earliest—that is, the one with minimal start time s(i). This way our resource starts being used as quickly as possible.
This method does not yield an optimal solution. If the earliest request i is for a very long interval, then by accepting request i we may have to reject a lot of requests for shorter time intervals. Since our goal is to satisfy as many requests as possible, we will end up with a suboptimal solution. In a really bad case—say, when the finish time f(i) is the maximum among all requests—the accepted request i keeps our resource occupied for the whole time. In this case our greedy method would accept a single request, while the optimal solution could accept many. Such a situation is depicted in Figure 4.1(a).
. This might suggest that we should start out by accepting the request that requires the smallest interval of time—namely, the request for which f (i) − s(i) is as small as possible. As it turns out, this is a somewhat better rule than the previous one, but it still can produce a suboptimal schedule. For example, in Figure 4.1(b), accepting the short interval in the middle would prevent us from accepting the other two, which form an optimal solution.
(a)
(b)
(c)
Figure 4.1 Some instances of the Interval Scheduling Problem on which natural greedy algorithms fail to find the optimal solution. In (a), it does not work to select the interval that starts earliest; in (b), it does not work to select the shortest interval; and in (c), it does not work to select the interval with the fewest conflicts.
118
Chapter 4 Greedy Algorithms
. In the previous greedy rule, our problem was that the second request competes with both the first and the third—that is, accepting this request made us reject two other requests. We could design a greedy algorithm that is based on this idea: for each request, we count the number of other requests that are not compatible, and accept the request that has the fewest number of noncompatible requests. (In other words, we select the interval with the fewest “conflicts.”) This greedy choice would lead to the optimum solution in the previous example. In fact, it is quite a bit harder to design a bad example for this rule; but it can be done, and we’ve drawn an example in Figure 4.1(c). The unique optimal solution in this example is to accept the four requests in the top row. The greedy method suggested here accepts the middle request in the second row and thereby ensures a solution of size no greater than three.
A greedy rule that does lead to the optimal solution is based on a fourth idea: we should accept first the request that finishes first, that is, the request i for which f (i) is as small as possible. This is also quite a natural idea: we ensure that our resource becomes free as soon as possible while still satisfying one request. In this way we can maximize the time left to satisfy other requests.
Let us state the algorithm a bit more formally. We will use R to denote the set of requests that we have neither accepted nor rejected yet, and use A to denote the set of accepted requests. For an example of how the algorithm runs, see Figure 4.2.
Initially let R be the set of all requests, and let A be empty While R is not yet empty
Choose a request i ∈ R that has the smallest finishing time
Add request i to A
Delete all requests from R that are not compatible with request i
EndWhile
Return the set A as the set of accepted requests
Analyzing the Algorithm
While this greedy method is quite natural, it is certainly not obvious that it returns an optimal set of intervals. Indeed, it would only be sensible to reserve judgment on its optimality: the ideas that led to the previous nonoptimal versions of the greedy method also seemed promising at first.
As a start, we can immediately declare that the intervals in the set A returned by the algorithm are all compatible.
(4.1) A is a compatible set of requests.
4.1 Interval Scheduling: The Greedy Algorithm Stays Ahead
119
68 1359
247
8 1359
47
8 1359
7
8 1359
8
135
Figure 4.2 Sample run of the Interval Scheduling Algorithm. At each step the selected intervals are darker lines, and the intervals deleted at the corresponding step are indicated with dashed lines.
What we need to show is that this solution is optimal. So, for purposes of comparison, let O be an optimal set of intervals. Ideally one might want to show that A = O, but this is too much to ask: there may be many optimal solutions, and at best A is equal to a single one of them. So instead we will simply show that |A| = |O|, that is, that A contains the same number of intervals as O and hence is also an optimal solution.
The idea underlying the proof, as we suggested initially, will be to find a sense in which our greedy algorithm “stays ahead” of this solution O. We will compare the partial solutions that the greedy algorithm constructs to initial segments of the solution O, and show that the greedy algorithm is doing better in a step-by-step fashion.
We introduce some notation to help with this proof. Let i1, . . . , ik be the set of requests in A in the order they were added to A. Note that |A| = k. Similarly, let the set of requests in O be denoted by j1,...,jm. Our goal is to prove that k = m. Assume that the requests in O are also ordered in the natural left-to- right order of the corresponding intervals, that is, in the order of the start and finish points. Note that the requests in O are compatible, which implies that the start points have the same order as the finish points.
Intervals numbered in order
Selecting interval 1
Selecting interval 3
Selecting interval 5
Selecting interval 8
120
Chapter 4
Greedy Algorithms
ir–1
jr–1
Can the greedy algorithm’s rth interval really finish later?
ir ? jr
Figure 4.3 The inductive step in the proof that the greedy algorithm stays ahead.
Our intuition for the greedy method came from wanting our resource to become free again as soon as possible after satisfying the first request. And indeed, our greedy rule guarantees that f (i1) ≤ f (j1). This is the sense in which we want to show that our greedy rule “stays ahead”—that each of its intervals finishes at least as soon as the corresponding interval in the set O. Thus we now prove that for each r ≥ 1, the rth accepted request in the algorithm’s schedule finishes no later than the rth request in the optimal schedule.
(4.2) For all indices r ≤ k we have f(ir) ≤ f(jr).
Proof. We will prove this statement by induction. For r = 1 the statement is clearly true: the algorithm starts by selecting the request i1 with minimum finish time.
Now let r > 1. We will assume as our induction hypothesis that the statement is true for r − 1, and we will try to prove it for r. As shown in Figure 4.3, the induction hypothesis lets us assume that f(ir−1)≤f(jr−1). In order for the algorithm’s rth interval not to finish earlier as well, it would need to “fall behind” as shown. But there’s a simple reason why this could not happen: rather than choose a later-finishing interval, the greedy algorithm always has the option (at worst) of choosing jr and thus fulfilling the induction step.

We can make this argument precise as follows. We know (since O consists of compatible intervals) that f (jr −1) ≤ s(jr ). Combining this with the induction hypothesisf(ir−1)≤f(jr−1),wegetf(ir−1)≤s(jr).Thustheintervaljr isinthe set R of available intervals at the time when the greedy algorithm selects ir. The greedy algorithm selects the available interval with smallest finish time; since interval jr is one of these available intervals, we have f (ir ) ≤ f (jr ). This completes the induction step.

Thus we have formalized the sense in which the greedy algorithm is remaining ahead of O: for each r, the rth interval it selects finishes at least as soon as the rth interval in O. We now see why this implies the optimality of the greedy algorithm’s set A.

4.1 Interval Scheduling: The Greedy Algorithm Stays Ahead

121

(4.3) The greedy algorithm returns an optimal set A.

Proof. We will prove the statement by contradiction. If A is not optimal, then an optimal set O must have more requests, that is, we must have m > k. Applying (4.2) with r = k, we get that f(ik) ≤ f(jk). Since m > k, there is a request jk+1 in O. This request starts after request jk ends, and hence after ik ends. So after deleting all requests that are not compatible with requests i1, . . . , ik, the set of possible requests R still contains jk+1. But the greedy algorithm stops with request ik, and it is only supposed to stop when R is empty—a contradiction.

Implementation and Running Time We can make our algorithm run in time O(n log n) as follows. We begin by sorting the n requests in order of finishing time and labeling them in this order; that is, we will assume that f (i) ≤ f (j) when i < j. This takes time O(n log n). In an additional O(n) time, we construct an array S[1 . . . n] with the property that S[i] contains the value s(i).
We now select requests by processing the intervals in order of increasing f (i). We always select the first interval; we then iterate through the intervals in order until reaching the first interval j for which s(j) ≥ f (1); we then select this one as well. More generally, if the most recent interval we’ve selected ends at time f , we continue iterating through subsequent intervals until we reach the first j for which s(j) ≥ f . In this way, we implement the greedy algorithm analyzed above in one pass through the intervals, spending constant time per interval. Thus this part of the algorithm takes time O(n).
Extensions
The Interval Scheduling Problem we considered here is a quite simple schedul- ing problem. There are many further complications that could arise in practical settings. The following point out issues that we will see later in the book in various forms.
. In defining the problem, we assumed that all requests were known to the scheduling algorithm when it was choosing the compatible subset. It would also be natural, of course, to think about the version of the problem in which the scheduler needs to make decisions about accepting or rejecting certain requests before knowing about the full set of requests. Customers (requestors) may well be impatient, and they may give up and leave if the scheduler waits too long to gather information about all other requests. An active area of research is concerned with such on- line algorithms, which must make decisions as time proceeds, without knowledge of future input.
122
Chapter 4 Greedy Algorithms
. Our goal was to maximize the number of satisfied requests. But we could picture a situation in which each request has a different value to us. For example, each request i could also have a value vi (the amount gained by satisfying request i), and the goal would be to maximize our income: the sum of the values of all satisfied requests. This leads to the Weighted Interval Scheduling Problem, the second of the representative problems we described in Chapter 1.
There are many other variants and combinations that can arise. We now discuss one of these further variants in more detail, since it forms another case in which a greedy algorithm can be used to produce an optimal solution.
A Related Problem: Scheduling All Intervals
The Problem In the Interval Scheduling Problem, there is a single resource and many requests in the form of time intervals, so we must choose which requests to accept and which to reject. A related problem arises if we have many identical resources available and we wish to schedule all the requests using as few resources as possible. Because the goal here is to partition all intervals across multiple resources, we will refer to this as the Interval Partitioning Problem.1
For example, suppose that each request corresponds to a lecture that needs to be scheduled in a classroom for a particular interval of time. We wish to satisfy all these requests, using as few classrooms as possible. The classrooms at our disposal are thus the multiple resources, and the basic constraint is that any two lectures that overlap in time must be scheduled in different classrooms. Equivalently, the interval requests could be jobs that need to be processed for a specific period of time, and the resources are machines capable of handling these jobs. Much later in the book, in Chapter 10, we will see a different application of this problem in which the intervals are routing requests that need to be allocated bandwidth on a fiber-optic cable.
As an illustration of the problem, consider the sample instance in Fig- ure 4.4(a). The requests in this example can all be scheduled using three resources; this is indicated in Figure 4.4(b), where the requests are rearranged into three rows, each containing a set of nonoverlapping intervals. In general, one can imagine a solution using k resources as a rearrangement of the requests into k rows of nonoverlapping intervals: the first row contains all the intervals
1 The problem is also referred to as the Interval Coloring Problem; the terminology arises from thinking of the different resources as having distinct colors—all the intervals assigned to a particular resource are given the corresponding color.
4.1 Interval Scheduling: The Greedy Algorithm Stays Ahead
123
ej cdg
bh
a
fi
(a)
cdfj bgi aeh
(b)
Figure 4.4 (a) An instance of the Interval Partitioning Problem with ten intervals (a through j). (b) A solution in which all intervals are scheduled using three resources: each row represents a set of intervals that can all be scheduled on a single resource.
assigned to the first resource, the second row contains all those assigned to the second resource, and so forth.
Now, is there any hope of using just two resources in this sample instance? Clearly the answer is no. We need at least three resources since, for example, intervals a, b, and c all pass over a common point on the time-line, and hence they all need to be scheduled on different resources. In fact, one can make this last argument in general for any instance of Interval Partitioning. Suppose we define the depth of a set of intervals to be the maximum number that pass over any single point on the time-line. Then we claim
(4.4) In any instance of Interval Partitioning, the number of resources needed is at least the depth of the set of intervals.
Proof. Suppose a set of intervals has depth d, and let I1, . . . , Id all pass over a common point on the time-line. Then each of these intervals must be scheduled on a different resource, so the whole instance needs at least d resources.
We now consider two questions, which turn out to be closely related. First, can we design an efficient algorithm that schedules all intervals using the minimum possible number of resources? Second, is there always a schedule using a number of resources that is equal to the depth? In effect, a positive answer to this second question would say that the only obstacles to partitioning intervals are purely local—a set of intervals all piled over the same point. It’s not immediately clear that there couldn’t exist other, “long-range” obstacles that push the number of required resources even higher.
124
Chapter 4 Greedy Algorithms
We now design a simple greedy algorithm that schedules all intervals using a number of resources equal to the depth. This immediately implies the optimality of the algorithm: in view of (4.4), no solution could use a number of resources that is smaller than the depth. The analysis of our algorithm will therefore illustrate another general approach to proving optimality: one finds a simple, “structural” bound asserting that every possible solution must have at least a certain value, and then one shows that the algorithm under consideration always achieves this bound.
Designing the Algorithm Let d be the depth of the set of intervals; we show how to assign a label to each interval, where the labels come from the set of numbers {1, 2, . . . , d}, and the assignment has the property that overlapping intervals are labeled with different numbers. This gives the desired solution, since we can interpret each number as the name of a resource, and the label of each interval as the name of the resource to which it is assigned.
The algorithm we use for this is a simple one-pass greedy strategy that orders intervals by their starting times. We go through the intervals in this order, and try to assign to each interval we encounter a label that hasn’t already been assigned to any previous interval that overlaps it. Specifically, we have the following description.
Sort the intervals by their start times, breaking ties arbitrarily Let I1, I2, . . . , In denote the intervals in this order
For j=1,2,3,...,n
For each interval Ii that precedes Ij in sorted order and overlaps it Exclude the label of Ii from consideration for Ij
Endfor
If there is any label from {1, 2, . . . , d} that has not been excluded then
Assign a nonexcluded label to Ij Else
Leave Ij unlabeled Endif
Endfor
Analyzing the Algorithm We claim the following.
(4.5) If we use the greedy algorithm above, every interval will be assigned a
label, and no two overlapping intervals will receive the same label.
Proof. First let’s argue that no interval ends up unlabeled. Consider one of the intervals Ij, and suppose there are t intervals earlier in the sorted order that overlap it. These t intervals, together with Ij, form a set of t + 1 intervals that all pass over a common point on the time-line (namely, the start time of
4.2 Scheduling to Minimize Lateness: An Exchange Argument
125
Ij), and so t + 1≤ d. Thus t ≤ d − 1. It follows that at least one of the d labels is not excluded by this set of t intervals, and so there is a label that can be assigned to Ij.
Next we claim that no two overlapping intervals are assigned the same label. Indeed, consider any two intervals I and I′ that overlap, and suppose I precedes I′ in the sorted order. Then when I′ is considered by the algorithm, I is in the set of intervals whose labels are excluded from consideration; consequently, the algorithm will not assign to I′ the label that it used for I.
The algorithm and its analysis are very simple. Essentially, if you have d labels at your disposal, then as you sweep through the intervals from left to right, assigning an available label to each interval you encounter, you can never reach a point where all the labels are currently in use.
Since our algorithm is using d labels, we can use (4.4) to conclude that it is, in fact, always using the minimum possible number of labels. We sum this up as follows.
(4.6) The greedy algorithm above schedules every interval on a resource, using a number of resources equal to the depth of the set of intervals. This is the optimal number of resources needed.
4.2 SchedulingtoMinimizeLateness:AnExchange Argument
We now discuss a scheduling problem related to the one with which we began the chapter. Despite the similarities in the problem formulation and in the greedy algorithm to solve it, the proof that this algorithm is optimal will require a more sophisticated kind of analysis.
The Problem
Consider again a situation in which we have a single resource and a set of n requests to use the resource for an interval of time. Assume that the resource is available starting at time s. In contrast to the previous problem, however, each request is now more flexible. Instead of a start time and finish time, the request i has a deadline di, and it requires a contiguous time interval of length ti, but it is willing to be scheduled at any time before the deadline. Each accepted request must be assigned an interval of time of length ti, and different requests must be assigned nonoverlapping intervals.
There are many objective functions we might seek to optimize when faced with this situation, and some are computationally much more difficult than
126
Chapter 4 Greedy Algorithms
Job 1
Job 2
Job 3
Solution:
Deadline 4
Job 3: done at time 1+2+3=6
Deadline 6
Length 1
Length 2
Length 3
Job 1: done at time 1
Deadline 2
Job 2: done at time 1+2=3
Figure 4.5 A sample instance of scheduling to minimize lateness.
others. Here we consider a very natural goal that can be optimized by a greedy algorithm. Suppose that we plan to satisfy each request, but we are allowed to let certain requests run late. Thus, beginning at our overall start time s, we will assign each request i an interval of time of length ti; let us denote this interval by [s(i), f (i)], with f (i) = s(i) + ti. Unlike the previous problem, then, the algorithm must actually determine a start time (and hence a finish time) for each interval.
We say that a request i is late if it misses the deadline, that is, if f (i) > di. The lateness of such a request i is defined to be li = f (i) − di. We will say that li = 0 if request i is not late. The goal in our new optimization problem will be to schedule all requests, using nonoverlapping intervals, so as to minimize the maximum lateness, L = maxi li. This problem arises naturally when scheduling jobs that need to use a single machine, and so we will refer to our requests as jobs.

Figure 4.5 shows a sample instance of this problem, consisting of three jobs: the first has length t1 = 1 and deadline d1 = 2; the second has t2 = 2 and d2 =4; and the third has t3 =3 and d3 =6. It is not hard to check that scheduling the jobs in the order 1, 2, 3 incurs a maximum lateness of 0.

Designing the Algorithm

What would a greedy algorithm for this problem look like? There are several natural greedy approaches in which we look at the data (ti , di) about the jobs and use this to order them according to some simple rule.

. One approach would be to schedule the jobs in order of increasing length ti, so as to get the short jobs out of the way quickly. This immediately

4.2 Scheduling to Minimize Lateness: An Exchange Argument

127

looks too simplistic, since it completely ignores the deadlines of the jobs. And indeed, consider a two-job instance where the first job has t1 = 1 and d1 = 100, while the second job has t2 = 10 and d2 = 10. Then the second job has to be started right away if we want to achieve lateness L = 0, and scheduling the second job first is indeed the optimal solution.

. The previous example suggests that we should be concerned about jobs whose available slack time di − ti is very small—they’re the ones that need to be started with minimal delay. So a more natural greedy algorithm would be to sort jobs in order of increasing slack di − ti.

Unfortunately, this greedy rule fails as well. Consider a two-job instance where the first job has t1 = 1 and d1 = 2, while the second job has t2 = 10 and d2 = 10. Sorting by increasing slack would place the second job first in the schedule, and the first job would incur a lateness of 9. (It finishes at time 11, nine units beyond its deadline.) On the other hand, if we schedule the first job first, then it finishes on time and the second job incurs a lateness of only 1.

There is, however, an equally basic greedy algorithm that always produces an optimal solution. We simply sort the jobs in increasing order of their deadlines di, and schedule them in this order. (This rule is often called Earliest Deadline First.) There is an intuitive basis to this rule: we should make sure that jobs with earlier deadlines get completed earlier. At the same time, it’s a little hard to believe that this algorithm always produces optimal solutions— specifically because it never looks at the lengths of the jobs. Earlier we were skeptical of the approach that sorted by length on the grounds that it threw away half the input data (i.e., the deadlines); but now we’re considering a solution that throws away the other half of the data. Nevertheless, Earliest Deadline First does produce optimal solutions, and we will now prove this.

First we specify some notation that will be useful in talking about the algorithm. By renaming the jobs if necessary, we can assume that the jobs are labeled in the order of their deadlines, that is, we have

d1 ≤ . . . ≤ dn.

We will simply schedule all jobs in this order. Again, let s be the start time for all jobs. Job 1 will start at time s=s(1) and end at time f(1)=s(1)+t1; Job 2 will start at time s(2) = f(1) and end at time f(2) = s(2) + t2; and so forth. We will use f to denote the finishing time of the last scheduled job. We write this algorithm here.

Order the jobs in order of their deadlines

Assume for simplicity of notation that d1 ≤ . . . ≤ dn Initially, f = s

128

Chapter 4 Greedy Algorithms

Consider the jobs i=1,…,n in this order

Assign job i to the time interval from s(i)=f to f(i)=f +ti Let f = f + ti

End

Return the set of scheduled intervals [s(i), f (i)] for i = 1, . . . , n

Analyzing the Algorithm

To reason about the optimality of the algorithm, we first observe that the schedule it produces has no “gaps”—times when the machine is not working yet there are jobs left. The time that passes during a gap will be called idle time: there is work to be done, yet for some reason the machine is sitting idle. Not only does the schedule A produced by our algorithm have no idle time; it is also very easy to see that there is an optimal schedule with this property. We do not write down a proof for this.

(4.7) There is an optimal schedule with no idle time.

Now, how can we prove that our schedule A is optimal, that is, its maximum lateness L is as small as possible? As in previous analyses, we will start by considering an optimal schedule O. Our plan here is to gradually modify O, preserving its optimality at each step, but eventually transforming it into a schedule that is identical to the schedule A found by the greedy algorithm. We refer to this type of analysis as an exchange argument, and we will see that it is a powerful way to think about greedy algorithms in general.

We first try characterizing schedules in the following way. We say that a schedule A′ has an inversion if a job i with deadline di is scheduled before another job j with earlier deadline dj < di. Notice that, by definition, the schedule A produced by our algorithm has no inversions. If there are jobs with identical deadlines then there can be many different schedules with no inversions. However, we can show that all these schedules have the same maximum lateness L.
(4.8) All schedules with no inversions and no idle time have the same maximum lateness.
Proof. If two different schedules have neither inversions nor idle time, then they might not produce exactly the same order of jobs, but they can only differ in the order in which jobs with identical deadlines are scheduled. Consider such a deadline d. In both schedules, the jobs with deadline d are all scheduled consecutively (after all jobs with earlier deadlines and before all jobs with later deadlines). Among the jobs with deadline d, the last one has the greatest lateness, and this lateness does not depend on the order of the jobs.
4.2 Scheduling to Minimize Lateness: An Exchange Argument
129
The main step in showing the optimality of our algorithm is to establish that there is an optimal schedule that has no inversions and no idle time. To do this, we will start with any optimal schedule having no idle time; we will then convert it into a schedule with no inversions without increasing its maximum lateness. Thus the resulting scheduling after this conversion will be optimal as well.
(4.9) There is an optimal schedule that has no inversions and no idle time. Proof. By (4.7), there is an optimal schedule O with no idle time. The proof
will consist of a sequence of statements. The first of these is simple to establish. (a) If O has an inversion, then there is a pair of jobs i and j such that j is
scheduled immediately after i and has dj < di.
Indeed, consider an inversion in which a job a is scheduled sometime before a job b, and da > db. If we advance in the scheduled order of jobs from a to b one at a time, there has to come a point at which the deadline we see decreases for the first time. This corresponds to a pair of consecutive jobs that form an inversion.

Now suppose O has at least one inversion, and by (a), let i and j be a pair of inverted requests that are consecutive in the scheduled order. We will decrease the number of inversions in O by swapping the requests i and j in the schedule O. The pair (i, j) formed an inversion in O, this inversion is eliminated by the swap, and no new inversions are created. Thus we have

(b) After swapping i and j we get a schedule with one less inversion.

The hardest part of this proof is to argue that the inverted schedule is also

optimal.

(c) The new swapped schedule has a maximum lateness no larger than that

of O.

It is clear that if we can prove (c), then we are done. The initial schedule O

can have at most n inversions (if all pairs are inverted), and hence after at n 2

most 2 swaps we get an optimal schedule with no inversions.

So we now conclude by proving (c), showing that by swapping a pair of consecutive, inverted jobs, we do not increase the maximum lateness L of the schedule.

Proof of (c). We invent some notation to describe the schedule O: assume that each request r is scheduled for the time interval [s(r), f (r)] and has lateness lr′ . Let L′ = maxr lr′ denote the maximum lateness of this schedule.

130

Chapter 4 Greedy Algorithms

Before swapping:

dj di

After swapping:

dj di

Only the finishing times of i and j are affected by the swap.

(a)

(b)

Job i

Job j

Job j

Job i

Figure 4.6 The effect of swapping two consecutive, inverted jobs.

Let O denote the swapped schedule; we will use s(r), f(r), lr, and L to denote the corresponding quantities in the swapped schedule.

Now recall our two adjacent, inverted jobs i and j. The situation is roughly as pictured in Figure 4.6. The finishing time of j before the swap is exactly equal to the finishing time of i after the swap. Thus all jobs other than jobs i and j finish at the same time in the two schedules. Moreover, job j will get finished earlier in the new schedule, and hence the swap does not increase the lateness of job j.

Thus the only thing to worry about is job i: its lateness may have been increased, and what if this actually raises the maximum lateness of the whole schedule? After the swap, job i finishes at time f(j), when job j was finished in the schedule O. If job i is late in this new schedule, its lateness is li = f(i) − di = f(j) − di. But the crucial point is that i cannot be more late in the schedule O than j was in the schedule O. Specifically, our assumption di > dj implies that

li =f(j)−di

does not increase the maximum lateness of the schedule.

The optimality of our greedy algorithm now follows immediately.

4.3 Optimal Caching: A More Complex Exchange Argument

131

(4.10) The schedule A produced by the greedy algorithm has optimal maxi- mum lateness L.

Proof. Statement (4.9) proves that an optimal schedule with no inversions exists. Now by (4.8) all schedules with no inversions have the same maximum lateness, and so the schedule obtained by the greedy algorithm is optimal.

Extensions

There are many possible generalizations of this scheduling problem. For ex- ample, we assumed that all jobs were available to start at the common start time s. A natural, but harder, version of this problem would contain requests i that, in addition to the deadline di and the requested time ti, would also have an earliest possible starting time ri. This earliest possible starting time is usu- ally referred to as the release time. Problems with release times arise naturally in scheduling problems where requests can take the form: Can I reserve the room for a two-hour lecture, sometime between 1 P.M. and 5 P.M.? Our proof that the greedy algorithm finds an optimal solution relied crucially on the fact that all jobs were available at the common start time s. (Do you see where?) Unfortunately, as we will see later in the book, in Chapter 8, this more general version of the problem is much more difficult to solve optimally.

4.3 Optimal Caching: A More Complex Exchange Argument

We now consider a problem that involves processing a sequence of requests of a different form, and we develop an algorithm whose analysis requires a more subtle use of the exchange argument. The problem is that of cache maintenance.

The Problem

To motivate caching, consider the following situation. You’re working on a long research paper, and your draconian library will only allow you to have eight books checked out at once. You know that you’ll probably need more than this over the course of working on the paper, but at any point in time, you’d like to have ready access to the eight books that are most relevant at that time. How should you decide which books to check out, and when should you return some in exchange for others, to minimize the number of times you have to exchange a book at the library?

This is precisely the problem that arises when dealing with a memory hierarchy: There is a small amount of data that can be accessed very quickly,

132

Chapter 4 Greedy Algorithms

and a large amount of data that requires more time to access; and you must decide which pieces of data to have close at hand.

Memory hierarchies have been a ubiquitous feature of computers since very early in their history. To begin with, data in the main memory of a processor can be accessed much more quickly than the data on its hard disk; but the disk has much more storage capacity. Thus, it is important to keep the most regularly used pieces of data in main memory, and go to disk as infrequently as possible. The same phenomenon, qualitatively, occurs with on-chip caches in modern processors. These can be accessed in a few cycles, and so data can be retrieved from cache much more quickly than it can be retrieved from main memory. This is another level of hierarchy: small caches have faster access time than main memory, which in turn is smaller and faster to access than disk. And one can see extensions of this hierarchy in many other settings. When one uses a Web browser, the disk often acts as a cache for frequently visited Web pages, since going to disk is still much faster than downloading something over the Internet.

Caching is a general term for the process of storing a small amount of data in a fast memory so as to reduce the amount of time spent interacting with a slow memory. In the previous examples, the on-chip cache reduces the need to fetch data from main memory, the main memory acts as a cache for the disk, and the disk acts as a cache for the Internet. (Much as your desk acts as a cache for the campus library, and the assorted facts you’re able to remember without looking them up constitute a cache for the books on your desk.)

For caching to be as effective as possible, it should generally be the case that when you go to access a piece of data, it is already in the cache. To achieve this, a cache maintenance algorithm determines what to keep in the cache and what to evict from the cache when new data needs to be brought in.

Of course, as the caching problem arises in different settings, it involves various different considerations based on the underlying technology. For our purposes here, though, we take an abstract view of the problem that underlies most of these settings. We consider a set U of n pieces of data stored in main memory. We also have a faster memory, the cache, that can hold k < n pieces of data at any one time. We will assume that the cache initially holds some set of k items. A sequence of data items D=d1,d2,...,dm drawn from U is presented to us—this is the sequence of memory references we must process— and in processing them we must decide at all times which k items to keep in the cache. When item di is presented, we can access it very quickly if it is already in the cache; otherwise, we are required to bring it from main memory into the cache and, if the cache is full, to evict some other piece of data that is currently in the cache to make room for di. This is called a cache miss, and we want to have as few of these as possible.
4.3 Optimal Caching: A More Complex Exchange Argument
133
Thus, on a particular sequence of memory references, a cache main- tenance algorithm determines an eviction schedule—specifying which items should be evicted from the cache at which points in the sequence—and this determines the contents of the cache and the number of misses over time. Let’s consider an example of this process.
. Suppose we have three items {a, b, c}, the cache size is k = 2, and we are presented with the sequence
a,b,c,b,c,a,b.
Suppose that the cache initially contains the items a and b. Then on the third item in the sequence, we could evict a so as to bring in c; and on the sixth item we could evict c so as to bring in a; we thereby incur two cache misses over the whole sequence. After thinking about it, one concludes that any eviction schedule for this sequence must include at least two cache misses.
Under real operating conditions, cache maintenance algorithms must process memory references d1, d2, . . . without knowledge of what’s coming in the future; but for purposes of evaluating the quality of these algorithms, systems researchers very early on sought to understand the nature of the optimal solution to the caching problem. Given a full sequence S of memory references, what is the eviction schedule that incurs as few cache misses as possible?
Designing and Analyzing the Algorithm
In the 1960s, Les Belady showed that the following simple rule will always incur the minimum number of misses:
When di needs to be brought into the cache,
evict the item that is needed the farthest into the future
We will call this the Farthest-in-Future Algorithm. When it is time to evict something, we look at the next time that each item in the cache will be referenced, and choose the one for which this is as late as possible.
This is a very natural algorithm. At the same time, the fact that it is optimal on all sequences is somewhat more subtle than it first appears. Why evict the item that is needed farthest in the future, as opposed, for example, to the one that will be used least frequently in the future? Moreover, consider a sequence like
a,b,c,d,a,d,e,a,d,b,c
134
Chapter 4 Greedy Algorithms
with k = 3 and items {a, b, c} initially in the cache. The Farthest-in-Future rule will produce a schedule S that evicts c on the fourth step and b on the seventh step. But there are other eviction schedules that are just as good. Consider the schedule S′ that evicts b on the fourth step and c on the seventh step, incurring the same number of misses. So in fact it’s easy to find cases where schedules produced by rules other than Farthest-in-Future are also optimal; and given this flexibility, why might a deviation from Farthest-in-Future early on not yield an actual savings farther along in the sequence? For example, on the seventh step in our example, the schedule S′ is actually evicting an item (c) that is needed farther into the future than the item evicted at this point by Farthest-in-Future, since Farthest-in-Future gave up c earlier on.
These are some of the kinds of things one should worry about before concluding that Farthest-in-Future really is optimal. In thinking about the example above, we quickly appreciate that it doesn’t really matter whether b or c is evicted at the fourth step, since the other one should be evicted at the seventh step; so given a schedule where b is evicted first, we can swap the choices of b and c without changing the cost. This reasoning—swapping one decision for another—forms the first outline of an exchange argument that proves the optimality of Farthest-in-Future.
Before delving into this analysis, let’s clear up one important issue. All the cache maintenance algorithms we’ve been considering so far produce schedules that only bring an item d into the cache in a step i if there is a request to d in step i, and d is not already in the cache. Let us call such a schedule reduced—it does the minimal amount of work necessary in a given step. But in general one could imagine an algorithm that produced schedules that are not reduced, by bringing in items in steps when they are not requested. We now show that for every nonreduced schedule, there is an equally good reduced schedule.
Let S be a schedule that may not be reduced. We define a new schedule S—the reduction of S—as follows. In any step i where S brings in an item d that has not been requested, our construction of S “pretends” to do this but actually leaves d in main memory. It only really brings d into the cache in the next step j after this in which d is requested. In this way, the cache miss incurred by S in step j can be charged to the earlier cache operation performed by S in step i, when it brought in d. Hence we have the following fact.
(4.11) S is a reduced schedule that brings in at most as many items as the schedule S.
Note that for any reduced schedule, the number of items that are brought in is exactly the number of misses.
4.3 Optimal Caching: A More Complex Exchange Argument
135
Proving the Optimalthy of Farthest-in-Future We now proceed with the exchange argument showing that Farthest-in-Future is optimal. Consider an arbitrary sequence D of memory references; let SFF denote the schedule produced by Farthest-in-Future, and let S∗ denote a schedule that incurs the minimum possible number of misses. We will now gradually “transform” the schedule S∗ into the schedule SFF, one eviction decision at a time, without increasing the number of misses.
Here is the basic fact we use to perform one step in the transformation.
(4.12) Let S be a reduced schedule that makes the same eviction decisions as SFF through the first j items in the sequence, for a number j. Then there is a reduced schedule S′ that makes the same eviction decisions as SFF through the first j + 1 items, and incurs no more misses than S does.
Proof. Consider the (j + 1)st request, to item d = dj+1. Since S and SFF have agreed up to this point, they have the same cache contents. If d is in the cache for both, then no eviction decision is necessary (both schedules are reduced), and so S in fact agrees with SFF through step j+1, and we can set S′=S. Similarly, if d needs to be brought into the cache, but S and SFF both evict the same item to make room for d, then we can again set S′ = S.
So the interesting case arises when d needs to be brought into the cache, and to do this S evicts item f while SFF evicts item e ̸= f. Here S and SFF do not already agree through step j + 1 since S has e in cache while SFF has f in cache. Hence we must actually do something nontrivial to construct S′.
As a first step, we should have S′ evict e rather than f. Now we need to further ensure that S′ incurs no more misses than S. An easy way to do this would be to have S′ agree with S for the remainder of the sequence; but this is no longer possible, since S and S′ have slightly different caches from this point onward. So instead we’ll have S′ try to get its cache back to the same state as S as quickly as possible, while not incurring unnecessary misses. Once the caches are the same, we can finish the construction of S′ by just having it behave like S.
Specifically, from request j + 2 onward, S′ behaves exactly like S until one of the following things happens for the first time.
(i) There is a request to an item g ̸=e,f that is not in the cache of S, and S evicts e to make room for it. Since S′ and S only differ on e and f , it must be that g is not in the cache of S′ either; so we can have S′ evict f, and now the caches of S and S′ are the same. We can then have S′ behave exactly like S for the rest of the sequence.
(ii) There is a request to f, and S evicts an item e′. If e′ = e, then we’re all set: S′ can simply access f from the cache, and after this step the caches
136
Chapter 4 Greedy Algorithms
of S and S′ will be the same. If e′ ̸= e, then we have S′ evict e′ as well, and bring in e from main memory; this too results in S and S′ having the same caches. However, we must be careful here, since S′ is no longer a reduced schedule: it brought in e when it wasn’t immediately needed. So to finish this part of the construction, we further transform S′ to its reduction S′ using (4.11); this doesn’t increase the number of items brought in by S′, and it still agrees with SFF through step j + 1.
Hence, in both these cases, we have a new reduced schedule S′ that agrees with SFF through the first j + 1 items and incurs no more misses than S does. And crucially—here is where we use the defining property of the Farthest-in- Future Algorithm—one of these two cases will arise before there is a reference to e. This is because in step j + 1, Farthest-in-Future evicted the item (e) that would be needed farthest in the future; so before there could be a request to e, there would have to be a request to f , and then case (ii) above would apply.
Using this result, it is easy to complete the proof of optimality. We begin with an optimal schedule S∗, and use (4.12) to construct a schedule S1 that agrees with SFF through the first step. We continue applying (4.12) inductively for j = 1, 2, 3, . . . , m, producing schedules Sj that agree with SFF through the first j steps. Each schedule incurs no more misses than the previous one; and by definition Sm = SFF , since it agrees with it through the whole sequence. Thus we have
(4.13) SFF incurs no more misses than any other schedule S∗ and hence is optimal.
Extensions: Caching under Real Operating Conditions
As mentioned in the previous subsection, Belady’s optimal algorithm provides a benchmark for caching performance; but in applications, one generally must make eviction decisions on the fly without knowledge of future requests. Experimentally, the best caching algorithms under this requirement seem to be variants of the Least-Recently-Used (LRU) Principle, which proposes evicting the item from the cache that was referenced longest ago.
If one thinks about it, this is just Belady’s Algorithm with the direction of time reversed—longest in the past rather than farthest in the future. It is effective because applications generally exhibit locality of reference: a running program will generally keep accessing the things it has just been accessing. (It is easy to invent pathological exceptions to this principle, but these are relatively rare in practice.) Thus one wants to keep the more recently referenced items in the cache.
Long after the adoption of LRU in practice, Sleator and Tarjan showed that one could actually provide some theoretical analysis of the performance of LRU, bounding the number of misses it incurs relative to Farthest-in-Future. We will discuss this analysis, as well as the analysis of a randomized variant on LRU, when we return to the caching problem in Chapter 13.
4.4 Shortest Paths in a Graph
Some of the basic algorithms for graphs are based on greedy design principles. Here we apply a greedy algorithm to the problem of finding shortest paths, and in the next section we look at the construction of minimum-cost spanning trees.
The Problem
As we’ve seen, graphs are often used to model networks in which one trav- els from one point to another—traversing a sequence of highways through interchanges, or traversing a sequence of communication links through inter- mediate routers. As a result, a basic algorithmic problem is to determine the shortest path between nodes in a graph. We may ask this as a point-to-point question: Given nodes u and v, what is the shortest u-v path? Or we may ask for more information: Given a start node s, what is the shortest path from s to each other node?
The concrete setup of the shortest paths problem is as follows. We are given a directed graph G = (V , E), with a designated start node s. We assume that s has a path to every other node in G. Each edge e has a length le ≥0, indicating the time (or distance, or cost) it takes to traverse e. For a path P, the length of P—denoted l(P)—is the sum of the lengths of all edges in P. Our goal is to determine the shortest path from s to every other node in the graph. We should mention that although the problem is specified for a directed graph, we can handle the case of an undirected graph by simply replacing each undirected edge e = (u, v) of length le by two directed edges (u, v) and (v, u), each of length le.
Designing the Algorithm
In 1959, Edsger Dijkstra proposed a very simple greedy algorithm to solve the single-source shortest-paths problem. We begin by describing an algorithm that just determines the length of the shortest path from s to each other node in the graph; it is then easy to produce the paths as well. The algorithm maintains a set S of vertices u for which we have determined a shortest-path distance d(u) from s; this is the “explored” part of the graph. Initially S = {s}, and d(s) = 0. Now, for each node v∈V−S, we determine the shortest path that can be constructed by traveling along a path through the explored part S to some u ∈ S, followed by the single edge (u, v). That is, we consider the quantity
4.4 Shortest Paths in a Graph
137
138
Chapter 4 Greedy Algorithms
d′(v) = mine=(u,v):u∈S d(u) + le. We choose the node v ∈ V −S for which this quantity is minimized, add v to S, and define d(v) to be the value d′(v).
Dijkstra’s Algorithm (G, l)
Let S be the set of explored nodes
For each u ∈ S, we store a distance d(u) Initially S={s} and d(s)=0
While S̸=V
Select a node v ̸∈ S with at least one edge from S for which d′(v) = mine=(u , v):u∈S d(u) + le is as small as possible
Add v to S and define d(v)=d′(v) EndWhile
It is simple to produce the s-u paths corresponding to the distances found by Dijkstra’s Algorithm. As each node v is added to the set S, we simply record the edge (u, v) on which it achieved the value mine=(u,v):u∈S d(u) + le. The path Pv is implicitly represented by these edges: if (u, v) is the edge we have stored for v, then Pv is just (recursively) the path Pu followed by the single edge (u, v). In other words, to construct Pv, we simply start at v; follow the edge we have stored for v in the reverse direction to u; then follow the edge we have stored for u in the reverse direction to its predecessor; and so on until we reach s. Note that s must be reached, since our backward walk from v visits nodes that were added to S earlier and earlier.
To get a better sense of what the algorithm is doing, consider the snapshot of its execution depicted in Figure 4.7. At the point the picture is drawn, two iterations have been performed: the first added node u, and the second added node v. In the iteration that is about to be performed, the node x will be added because it achieves the smallest value of d′(x); thanks to the edge (u, x), we have d′(x)=d(u)+lux =2. Note that attempting to add y or z to the set S at this point would lead to an incorrect value for their shortest-path distances; ultimately, they will be added because of their edges from x.
Analyzing the Algorithm
We see in this example that Dijkstra’s Algorithm is doing the right thing and avoiding recurring pitfalls: growing the set S by the wrong node can lead to an overestimate of the shortest-path distance to that node. The question becomes: Is it always true that when Dijkstra’s Algorithm adds a node v, we get the true shortest-path distance to v?
We now answer this by proving the correctness of the algorithm, showing that the paths Pu really are shortest paths. Dijkstra’s Algorithm is greedy in
4.4 Shortest Paths in a Graph
139
3
11
y
1
2
u
s4x 22
Set S:
nodes already v explored
3
Figure 4.7 A snapshot of the execution of Dijkstra’s Algorithm. The next node that will be added to the set S is x, due to the path through u.
the sense that we always form the shortest new s-v path we can make from a path in S followed by a single edge. We prove its correctness using a variant of our first style of analysis: we show that it “stays ahead” of all other solutions by establishing, inductively, that each time it selects a path to a node v, that path is shorter than every other possible path to v.
(4.14) Consider the set S at any point in the algorithm’s execution. For each u ∈ S, the path Pu is a shortest s-u path.
Note that this fact immediately establishes the correctness of Dijkstra’s Algorithm, since we can apply it when the algorithm terminates, at which point S includes all nodes.
Proof. We prove this by induction on the size of S. The case |S| = 1 is easy, since then we have S = {s} and d(s) = 0. Suppose the claim holds when |S| = k for some value of k≥1; we now grow S to size k+1by adding the node v. Let (u, v) be the final edge on our s-v path Pv.
By induction hypothesis, Pu is the shortest s-u path for each u ∈ S. Now consider any other s-v path P; we wish to show that it is at least as long as Pv. In order to reach v, this path P must leave the set S somewhere; let y be the first node on P that is not in S, and let x∈S be the node just before y.
The situation is now as depicted in Figure 4.8, and the crux of the proof is very simple: P cannot be shorter than Pv because it is already at least as
z
140
Chapter 4
Greedy Algorithms
xy P
The alternate s–v path P through x and y is already too long by the time it has left the set S.
s
Pu
Set S
uv
Figure 4.8 The shortest path Pv and an alternate s-v path P through the node y.
long as Pv by the time it has left the set S. Indeed, in iteration k + 1, Dijkstra’s Algorithm must have considered adding node y to the set S via the edge (x, y) and rejected this option in favor of adding v. This means that there is no path from s to y through x that is shorter than Pv. But the subpath of P up to y is such a path, and so this subpath is at least as long as Pv. Since edge lengths are nonnegative, the full path P is at least as long as Pv as well.
This is a complete proof; one can also spell out the argument in the previous paragraph using the following inequalities. Let P′ be the subpath of P from s to x. Since x ∈ S, we know by the induction hypothesis that Px is a shortest s-x path (of length d(x)), and so l(P′) ≥ l(Px) = d(x). Thus the subpath of P out to node y has length l(P′)+l(x,y)≥d(x)+l(x,y)≥d′(y), and the full path P is at least as long as this subpath. Finally, since Dijkstra’s Algorithm selected v in this iteration, we know that d′(y) ≥ d′(v) = l(Pv). Combining these inequalities shows that l(P) ≥ l(P′) + l(x, y) ≥ l(Pv).
Here are two observations about Dijkstra’s Algorithm and its analysis. First, the algorithm does not always find shortest paths if some of the edges can have negative lengths. (Do you see where the proof breaks?) Many shortest-path applications involve negative edge lengths, and a more com- plex algorithm—due to Bellman and Ford—is required for this case. We will see this algorithm when we consider the topic of dynamic programming.
The second observation is that Dijkstra’s Algorithm is, in a sense, even simpler than we’ve described here. Dijkstra’s Algorithm is really a “contin- uous” version of the standard breadth-first search algorithm for traversing a graph, and it can be motivated by the following physical intuition. Suppose the edges of G formed a system of pipes filled with water, joined together at the nodes; each edge e has length le and a fixed cross-sectional area. Now suppose an extra droplet of water falls at node s and starts a wave from s. As the wave expands out of node s at a constant speed, the expanding sphere
of wavefront reaches nodes in increasing order of their distance from s. It is easy to believe (and also true) that the path taken by the wavefront to get to any node v is a shortest path. Indeed, it is easy to see that this is exactly the path to v found by Dijkstra’s Algorithm, and that the nodes are discovered by the expanding water in the same order that they are discovered by Dijkstra’s Algorithm.
Implementation and Running Time To conclude our discussion of Dijkstra’s Algorithm, we consider its running time. There are n − 1 iterations of the While loop for a graph with n nodes, as each iteration adds a new node v to S. Selecting the correct node v efficiently is a more subtle issue. One’s first impression is that each iteration would have to consider each node v ̸∈ S, and go through all the edges between S and v to determine the minimum mine=(u,v):u∈S d(u) + le, so that we can select the node v for which this minimum is smallest. For a graph with m edges, computing all these minima can take O(m) time, so this would lead to an implementation that runs in O(mn) time.
We can do considerably better if we use the right data structures. First, we will explicitly maintain the values of the minima d′(v) = mine=(u,v):u∈S d(u) + le for each node v ∈ V − S, rather than recomputing them in each iteration. We can further improve the efficiency by keeping the nodes V − S in a priority queue with d′(v) as their keys. Priority queues were discussed in Chapter 2; they are data structures designed to maintain a set of n elements, each with a key. A priority queue can efficiently insert elements, delete elements, change an element’s key, and extract the element with the minimum key. We will need the third and fourth of the above operations: ChangeKey and ExtractMin.
How do we implement Dijkstra’s Algorithm using a priority queue? We put the nodes V in a priority queue with d′(v) as the key for v ∈ V. To select the node v that should be added to the set S, we need the ExtractMin operation. To see how to update the keys, consider an iteration in which node v is added to S, and let w ̸∈ S be a node that remains in the priority queue. What do we have to do to update the value of d′(w)? If (v, w) is not an edge, then we don’t have to do anything: the set of edges considered in the minimum mine=(u,w):u∈S d(u) + le is exactly the same before and after adding v to S. If e′ = (v, w) ∈ E, on the other hand, then the new value for the key is min(d′(w), d(v) + le′). If d′(w) > d(v) + le′ then we need to use the ChangeKey operation to decrease the key of node w appropriately. This ChangeKey operation can occur at most once per edge, when the tail of the edge e′ is added to S. In summary, we have the following result.

4.4 Shortest Paths in a Graph

141

142

Chapter 4 Greedy Algorithms

(4.15) Using a priority queue, Dijkstra’s Algorithm can be implemented on a graph with n nodes and m edges to run in O(m) time, plus the time for n ExtractMin and m ChangeKey operations.

Using the heap-based priority queue implementation discussed in Chap- ter 2, each priority queue operation can be made to run in O(log n) time. Thus the overall time for the implementation is O(m log n).

4.5 The Minimum Spanning Tree Problem

We now apply an exchange argument in the context of a second fundamental problem on graphs: the Minimum Spanning Tree Problem.

The Problem

Suppose we have a set of locations V = {v1, v2, . . . , vn}, and we want to build a communication network on top of them. The network should be connected— there should be a path between every pair of nodes—but subject to this requirement, we wish to build it as cheaply as possible.

For certain pairs (vi, vj), we may build a direct link between vi and vj for

a certain cost c(vi, vj) > 0. Thus we can represent the set of possible links that

may be built using a graph G = (V , E), with a positive cost ce associated with

each edge e = (vi, vj). The problem is to find a subset of the edges T ⊆ E so

that the graph (V , T ) is connected, and the total cost c is as small as e∈T e

possible. (We will assume that the full graph G is connected; otherwise, no solution is possible.)

Here is a basic observation.

(4.16) Let T be a minimum-cost solution to the network design problem defined above. Then (V , T ) is a tree.

Proof. By definition, (V,T) must be connected; we show that it also will contain no cycles. Indeed, suppose it contained a cycle C, and let e be any edge on C. We claim that (V , T − {e}) is still connected, since any path that previously used the edge e can now go “the long way” around the remainder of the cycle C instead. It follows that (V , T − {e}) is also a valid solution to the problem, and it is cheaper—a contradiction.

If we allow some edges to have 0 cost (that is, we assume only that the costs ce are nonnegative), then a minimum-cost solution to the network design problem may have extra edges—edges that have 0 cost and could optionally be deleted. But even in this case, there is always a minimum-cost solution that is a tree. Starting from any optimal solution, we could keep deleting edges on

4.5 The Minimum Spanning Tree Problem

143

cycles until we had a tree; with nonnegative edges, the cost would not increase during this process.

We will call a subset T ⊆ E a spanning tree of G if (V , T ) is a tree. Statement (4.16) says that the goal of our network design problem can be rephrased as that of finding the cheapest spanning tree of the graph; for this reason, it is generally called the Minimum Spanning Tree Problem. Unless G is a very simple graph, it will have exponentially many different spanning trees, whose structures may look very different from one another. So it is not at all clear how to efficiently find the cheapest tree from among all these options.

Designing Algorithms

As with the previous problems we’ve seen, it is easy to come up with a number of natural greedy algorithms for the problem. But curiously, and fortunately, this is a case where many of the first greedy algorithms one tries turn out to be correct: they each solve the problem optimally. We will review a few of these algorithms now and then discover, via a nice pair of exchange arguments, some of the underlying reasons for this plethora of simple, optimal algorithms.

Here are three greedy algorithms, each of which correctly finds a minimum spanning tree.

. One simple algorithm starts without any edges at all and builds a span- ning tree by successively inserting edges from E in order of increasing cost. As we move through the edges in this order, we insert each edge e as long as it does not create a cycle when added to the edges we’ve already inserted. If, on the other hand, inserting e would result in a cycle, then we simply discard e and continue. This approach is called Kruskal’s Algorithm.

. Another simple greedy algorithm can be designed by analogy with Dijk- stra’s Algorithm for paths, although, in fact, it is even simpler to specify than Dijkstra’s Algorithm. We start with a root node s and try to greedily grow a tree from s outward. At each step, we simply add the node that can be attached as cheaply as possibly to the partial tree we already have.

More concretely, we maintain a set S ⊆ V on which a spanning tree has been constructed so far. Initially, S = {s}. In each iteration, we grow S by one node, adding the node v that minimizes the “attachment cost” mine=(u,v):u∈S ce, and including the edge e = (u, v) that achieves this minimum in the spanning tree. This approach is called Prim’s Algorithm.

. Finally, we can design a greedy algorithm by running sort of a “back- ward” version of Kruskal’s Algorithm. Specifically, we start with the full graph (V , E) and begin deleting edges in order of decreasing cost. As we get to each edge e (starting from the most expensive), we delete it as

144

Chapter 4

Greedy Algorithms

a

r

b

a

r

b

cg f

d

(a)

e

h

cg f

d

(b)

e

h

Figure 4.9 Sample run of the Minimum Spanning Tree Algorithms of (a) Prim and (b) Kruskal, on the same input. The first 4 edges added to the spanning tree are indicated by solid lines; the next edge to be added is a dashed line.

long as doing so would not actually disconnect the graph we currently have. For want of a better name, this approach is generally called the Reverse-Delete Algorithm (as far as we can tell, it’s never been named after a specific person).

For example, Figure 4.9 shows the first four edges added by Prim’s and Kruskal’s Algorithms respectively, on a geometric instance of the Minimum Spanning Tree Problem in which the cost of each edge is proportional to the geometric distance in the plane.

The fact that each of these algorithms is guaranteed to produce an opti- mal solution suggests a certain “robustness” to the Minimum Spanning Tree Problem—there are many ways to get to the answer. Next we explore some of the underlying reasons why so many different algorithms produce minimum- cost spanning trees.

Analyzing the Algorithms

All these algorithms work by repeatedly inserting or deleting edges from a partial solution. So, to analyze them, it would be useful to have in hand some basic facts saying when it is “safe” to include an edge in the minimum spanning tree, and, correspondingly, when it is safe to eliminate an edge on the grounds that it couldn’t possibly be in the minimum spanning tree. For purposes of the analysis, we will make the simplifying assumption that all edge costs are distinct from one another (i.e., no two are equal). This assumption makes it

4.5 The Minimum Spanning Tree Problem

145

easier to express the arguments that follow, and we will show later in this section how this assumption can be easily eliminated.

When Is It Safe to Include an Edge in the Minimum Spanning Tree? The crucial fact about edge insertion is the following statement, which we will refer to as the Cut Property.

(4.17) Assume that all edge costs are distinct. Let S be any subset of nodes that is neither empty nor equal to all of V, and let edge e = (v, w) be the minimum- cost edge with one end in S and the other in V − S. Then every minimum spanning tree contains the edge e.

Proof. LetTbeaspanningtreethatdoesnotcontaine;weneedtoshowthatT does not have the minimum possible cost. We’ll do this using an exchange argument: we’ll identify an edge e′ in T that is more expensive than e, and with the property exchanging e for e′ results in another spanning tree. This resulting spanning tree will then be cheaper than T, as desired.

The crux is therefore to find an edge that can be successfully exchanged with e. Recall that the ends of e are v and w. T is a spanning tree, so there must be a path P in T from v to w. Starting at v, suppose we follow the nodes of P in sequence; there is a first node w′ on P that is in V − S. Let v′ ∈ S be the node just before w′ on P, and let e′ = (v′, w′) be the edge joining them. Thus, e′ isanedgeofT withoneendinSandtheotherinV−S.SeeFigure4.10for the situation at this stage in the proof.

If we exchange e for e′, we get a set of edges T′=T−{e′}∪{e}. We claim that T′ is a spanning tree. Clearly (V,T′) is connected, since (V,T) is connected, and any path in (V, T) that used the edge e′ = (v′, w′) can now be “rerouted” in (V, T′) to follow the portion of P from v′ to v, then the edge e, and then the portion of P from w to w′. To see that (V, T′) is also acyclic, note that the only cycle in (V, T′ ∪ {e′}) is the one composed of e and the path P, and this cycle is not present in (V, T′) due to the deletion of e′.

We noted above that the edge e′ has one end in S and the other in V −S. But e is the cheapest edge with this property, and so ce < ce′. (The inequality is strict since no two edges have the same cost.) Thus the total cost of T′ is less than that of T, as desired.
The proof of (4.17) is a bit more subtle than it may first appear. To appreciate this subtlety, consider the following shorter but incorrect argument for (4.17). Let T be a spanning tree that does not contain e. Since T is a spanning tree, it must contain an edge f with one end in S and the other in V − S. Since e is the cheapest edge with this property, we have ce < cf , and hence T − {f} ∪ {e} is a spanning tree that is cheaper than T.
146
Chapter 4
Greedy Algorithms
S
v e w
e vw
f
e can be swapped for e. h
Figure 4.10 Swapping the edge e for the edge e′ in the spanning tree T, as described in the proof of (4.17).
The problem with this argument is not in the claim that f exists, or that T −{f}∪{e} is cheaper than T. The difficulty is that T −{f}∪{e} may not be a spanning tree, as shown by the example of the edge f in Figure 4.10. The point is that we can’t prove (4.17) by simply picking any edge in T that crosses from S to V − S; some care must be taken to find the right one.
The Optimality of Kruskal’s and Prim’s Algorithms We can now easily prove the optimality of both Kruskal’s Algorithm and Prim’s Algorithm. The point is that both algorithms only include an edge when it is justified by the Cut Property (4.17).
(4.18) Kruskal’s Algorithm produces a minimum spanning tree of G.
Proof. Consider any edge e = (v, w) added by Kruskal’s Algorithm, and let S be the set of all nodes to which v has a path at the moment just before e is added. Clearly v ∈ S, but w ̸∈ S, since adding e does not create a cycle. Moreover, no edge from S to V − S has been encountered yet, since any such edge could have been added without creating a cycle, and hence would have been added by Kruskal’s Algorithm. Thus e is the cheapest edge with one end in S and the other in V − S, and so by (4.17) it belongs to every minimum spanning tree.
4.5 The Minimum Spanning Tree Problem
147
So if we can show that the output (V , T ) of Kruskal’s Algorithm is in fact a spanning tree of G, then we will be done. Clearly (V, T) contains no cycles, since the algorithm is explicitly designed to avoid creating cycles. Further, if (V , T ) were not connected, then there would exist a nonempty subset of nodes S (not equal to all of V) such that there is no edge from S to V −S. But this contradicts the behavior of the algorithm: we know that since G is connected, there is at least one edge between S and V − S, and the algorithm will add the first of these that it encounters.
(4.19) Prim’s Algorithm produces a minimum spanning tree of G.
Proof. For Prim’s Algorithm, it is also very easy to show that it only adds edges belonging to every minimum spanning tree. Indeed, in each iteration of the algorithm, there is a set S ⊆ V on which a partial spanning tree has been constructed, and a node v and edge e are added that minimize the quantity mine=(u,v):u∈S ce. By definition, e is the cheapest edge with one end in S and the other end in V − S, and so by the Cut Property (4.17) it is in every minimum spanning tree.
It is also straightforward to show that Prim’s Algorithm produces a span- ning tree of G, and hence it produces a minimum spanning tree.
When Can We Guarantee an Edge Is Not in the Minimum Spanning Tree? The crucial fact about edge deletion is the following statement, which we will refer to as the Cycle Property.
(4.20) Assume that all edge costs are distinct. Let C be any cycle in G, and let edge e = (v, w) be the most expensive edge belonging to C. Then e does not belong to any minimum spanning tree of G.
Proof. Let T be a spanning tree that contains e; we need to show that T does not have the minimum possible cost. By analogy with the proof of the Cut Property (4.17), we’ll do this with an exchange argument, swapping e for a cheaper edge in such a way that we still have a spanning tree.
So again the question is: How do we find a cheaper edge that can be exchanged in this way with e? Let’s begin by deleting e from T; this partitions the nodes into two components: S, containing node v; and V − S, containing node w. Now, the edge we use in place of e should have one end in S and the other in V − S, so as to stitch the tree back together.
We can find such an edge by following the cycle C. The edges of C other than e form, by definition, a path P with one end at v and the other at w. If we follow P from v to w, we begin in S and end up in V − S, so there is some
148
Chapter 4
Greedy Algorithms
S
e vw
Cycle C e
e can be swapped for e.
Figure 4.11 Swapping the edge e′ for the edge e in the spanning tree T, as described in
the proof of (4.20).
edge e′ on P that crosses from S to V − S. See Figure 4.11 for an illustration of this.
Now consider the set of edges T′ = T − {e} ∪ {e′}. Arguing just as in the proof of the Cut Property (4.17), the graph (V, T′) is connected and has no cycles, so T′ is a spanning tree of G. Moreover, since e is the most expensive edge on the cycle C, and e′ belongs to C, it must be that e′ is cheaper than e, and hence T′ is cheaper than T, as desired.
The Optimality of the Reverse-Delete Algorithm Now that we have the Cycle Property (4.20), it is easy to prove that the Reverse-Delete Algorithm produces a minimum spanning tree. The basic idea is analogous to the optimality proofs for the previous two algorithms: Reverse-Delete only adds an edge when it is justified by (4.20).
(4.21) The Reverse-Delete Algorithm produces a minimum spanning tree of G.
Proof. Consider any edge e = (v, w) removed by Reverse-Delete. At the time that e is removed, it lies on a cycle C; and since it is the first edge encountered by the algorithm in decreasing order of edge costs, it must be the most expensive edge on C. Thus by (4.20), e does not belong to any minimum spanning tree.
So if we show that the output (V , T ) of Reverse-Delete is a spanning tree of G, we will be done. Clearly (V, T) is connected, since the algorithm never removes an edge when this will disconnect the graph. Now, suppose by way of
4.5 The Minimum Spanning Tree Problem
149
contradiction that (V , T ) contains a cycle C . Consider the most expensive edge e on C, which would be the first one encountered by the algorithm. This edge should have been removed, since its removal would not have disconnected the graph, and this contradicts the behavior of Reverse-Delete.
While we will not explore this further here, the combination of the Cut Property (4.17) and the Cycle Property (4.20) implies that something even more general is going on. Any algorithm that builds a spanning tree by repeatedly including edges when justified by the Cut Property and deleting edges when justified by the Cycle Property—in any order at all—will end up with a minimum spanning tree. This principle allows one to design natural greedy algorithms for this problem beyond the three we have considered here, and it provides an explanation for why so many greedy algorithms produce optimal solutions for this problem.
Eliminating the Assumption that All Edge Costs Are Distinct Thus far, we have assumed that all edge costs are distinct, and this assumption has made the analysis cleaner in a number of places. Now, suppose we are given an instance of the Minimum Spanning Tree Problem in which certain edges have the same cost – how can we conclude that the algorithms we have been discussing still provide optimal solutions?
There turns out to be an easy way to do this: we simply take the instance and perturb all edge costs by different, extremely small numbers, so that they all become distinct. Now, any two costs that differed originally will still have the same relative order, since the perturbations are so small; and since all of our algorithms are based on just comparing edge costs, the perturbations effectively serve simply as “tie-breakers” to resolve comparisons among costs that used to be equal.
Moreover, we claim that any minimum spanning tree T for the new, perturbed instance must have also been a minimum spanning tree for the original instance. To see this, we note that if T cost more than some tree T∗ in the original instance, then for small enough perturbations, the change in the cost of T cannot be enough to make it better than T ∗ under the new costs. Thus, if we run any of our minimum spanning tree algorithms, using the perturbed costs for comparing edges, we will produce a minimum spanning tree T that is also optimal for the original instance.
Implementing Prim’s Algorithm
We next discuss how to implement the algorithms we have been considering so as to obtain good running-time bounds. We will see that both Prim’s and Kruskal’s Algorithms can be implemented, with the right choice of data struc- tures, to run in O(m log n) time. We will see how to do this for Prim’s Algorithm
150
Chapter 4 Greedy Algorithms
here, and defer discussing the implementation of Kruskal’s Algorithm to the next section. Obtaining a running time close to this for the Reverse-Delete Algorithm is difficult, so we do not focus on Reverse-Delete in this discussion.
For Prim’s Algorithm, while the proof of correctness was quite different from the proof for Dijkstra’s Algorithm for the Shortest-Path Algorithm, the implementations of Prim and Dijkstra are almost identical. By analogy with Dijkstra’s Algorithm, we need to be able to decide which node v to add next to the growing set S, by maintaining the attachment costs a(v) = mine=(u,v):u∈S ce for each node v ∈ V − S. As before, we keep the nodes in a priority queue with these attachment costs a(v) as the keys; we select a node with an ExtractMin operation, and update the attachment costs using ChangeKey operations. There are n − 1 iterations in which we perform ExtractMin, and we perform ChangeKey at most once for each edge. Thus we have
(4.22) Using a priority queue, Prim’s Algorithm can be implemented on a graph with n nodes and m edges to run in O(m) time, plus the time for n ExtractMin, and m ChangeKey operations.
As with Dijkstra’s Algorithm, if we use a heap-based priority queue we can implement both ExtractMin and ChangeKey in O(log n) time, and so get an overall running time of O(m log n).
Extensions
The minimum spanning tree problem emerged as a particular formulation of a broader network design goal—finding a good way to connect a set of sites by installing edges between them. A minimum spanning tree optimizes a particular goal, achieving connectedness with minimum total edge cost. But there are a range of further goals one might consider as well.
We may, for example, be concerned about point-to-point distances in the spanning tree we build, and be willing to reduce these even if we pay more for the set of edges. This raises new issues, since it is not hard to construct examples where the minimum spanning tree does not minimize point-to-point distances, suggesting some tension between these goals.
Alternately, we may care more about the congestion on the edges. Given traffic that needs to be routed between pairs of nodes, one could seek a spanning tree in which no single edge carries more than a certain amount of this traffic. Here too, it is easy to find cases in which the minimum spanning tree ends up concentrating a lot of traffic on a single edge.
More generally, it is reasonable to ask whether a spanning tree is even the right kind of solution to our network design problem. A tree has the property that destroying any one edge disconnects it, which means that trees are not at
4.6 Implementing Kruskal’s Algorithm: The Union-Find Data Structure
151
all robust against failures. One could instead make resilience an explicit goal, for example seeking the cheapest connected network on the set of sites that remains connected after the deletion of any one edge.
All of these extensions lead to problems that are computationally much harder than the basic Minimum Spanning Tree problem, though due to their importance in practice there has been research on good heuristics for them.
4.6 Implementing Kruskal’s Algorithm: The Union-Find Data Structure
One of the most basic graph problems is to find the set of connected compo- nents. In Chapter 3 we discussed linear-time algorithms using BFS or DFS for finding the connected components of a graph.
In this section, we consider the scenario in which a graph evolves through the addition of edges. That is, the graph has a fixed population of nodes, but it grows over time by having edges appear between certain pairs of nodes. Our goal is to maintain the set of connected components of such a graph throughout this evolution process. When an edge is added to the graph, we don’t want to have to recompute the connected components from scratch. Rather, we will develop a data structure that we call the Union-Find structure, which will store a representation of the components in a way that supports rapid searching and updating.
This is exactly the data structure needed to implement Kruskal’s Algorithm efficiently. As each edge e = (v, w) is considered, we need to efficiently find the identities of the connected components containing v and w. If these components are different, then there is no path from v and w, and hence edge e should be included; but if the components are the same, then there is a v-w path on the edges already included, and so e should be omitted. In the event that e is included, the data structure should also support the efficient merging of the components of v and w into a single new component.
The Problem
The Union-Find data structure allows us to maintain disjoint sets (such as the components of a graph) in the following sense. Given a node u, the operation Find(u) will return the name of the set containing u. This operation can be used to test if two nodes u and v are in the same set, by simply checking if Find(u) = Find(v). The data structure will also implement an operation Union(A, B) to take two sets A and B and merge them to a single set.
These operations can be used to maintain connected components of an evolving graph G = (V , E) as edges are added. The sets will be the connected components of the graph. For a node u, the operation Find(u) will return the
152
Chapter 4 Greedy Algorithms
name of the component containing u. If we add an edge (u, v) to the graph, then we first test if u and v are already in the same connected component (by testing if Find(u) = Find(v)). If they are not, then Union(Find(u),Find(v)) can be used to merge the two components into one. It is important to note that the Union-Find data structure can only be used to maintain components of a graph as we add edges; it is not designed to handle the effects of edge deletion, which may result in a single component being “split” into two.
To summarize, the Union-Find data structure will support three oper- ations.
. MakeUnionFind(S) for a set S will return a Union-Find data structure on set S where all elements are in separate sets. This corresponds, for example, to the connected components of a graph with no edges. Our goal will be to implement MakeUnionFind in time O(n) where n = |S|.
. For an element u ∈ S, the operation Find(u) will return the name of the set containing u. Our goal will be to implement Find(u) in O(log n) time. Some implementations that we discuss will in fact take only O(1) time for this operation.
. For two sets A and B, the operation Union(A, B) will change the data structure by merging the sets A and B into a single set. Our goal will be to implement Union in O(log n) time.
Let’s briefly discuss what we mean by the name of a set—for example, as returned by the Find operation. There is a fair amount of flexibility in defining the names of the sets; they should simply be consistent in the sense that Find(v) and Find(w) should return the same name if v and w belong to the same set, and different names otherwise. In our implementations, we will name each set using one of the elements it contains.
A Simple Data Structure for Union-Find
Maybe the simplest possible way to implement a Union-Find data structure is to maintain an array Component that contains the name of the set currently containing each element. Let S be a set, and assume it has n elements denoted {1, . . . , n}. We will set up an array Component of size n, where Component[s] is the name of the set containing s. To implement MakeUnionFind(S), we set up the array and initialize it to Component[s] = s for all s ∈ S. This implementation makes Find(v) easy: it is a simple lookup and takes only O(1) time. However, Union(A, B) for two sets A and B can take as long as O(n) time, as we have to update the values of Component[s] for all elements in sets A and B.
To improve this bound, we will do a few simple optimizations. First, it is useful to explicitly maintain the list of elements in each set, so we don’t have to look through the whole array to find the elements that need updating. Further,
4.6 Implementing Kruskal’s Algorithm: The Union-Find Data Structure
153
we save some time by choosing the name for the union to be the name of one of the sets, say, set A: this way we only have to update the values Component[s] for s∈B, but not for any s∈A. Of course, if set B is large, this idea by itself doesn’t help very much. Thus we add one further optimization. When set B is big, we may want to keep its name and change Component[s] for all s ∈ A instead. More generally, we can maintain an additional array size of length n, where size[A] is the size of set A, and when a Union(A, B) operation is performed, we use the name of the larger set for the union. This way, fewer elements need to have their Component values updated.
Even with these optimizations, the worst case for a Union operation is still O(n) time; this happens if we take the union of two large sets A and B, each containing a constant fraction of all the elements. However, such bad cases for Union cannot happen very often, as the resulting set A ∪ B is even bigger. How can we make this statement more precise? Instead of bounding the worst-case running time of a single Union operation, we can bound the total (or average) running time of a sequence of k Union operations.
(4.23) Consider the array implementation of the Union-Find data structure for some set S of size n, where unions keep the name of the larger set. The Find operation takes O(1) time, MakeUnionFind(S) takes O(n) time, and any sequence of k Union operations takes at most O(k log k) time.
Proof. The claims about the MakeUnionFind and Find operations are easy to verify. Now consider a sequence of k Union operations. The only part of a Union operation that takes more than O(1) time is updating the array Component. Instead of bounding the time spent on one Union operation, we will bound the total time spent updating Component[v] for an element v throughout the sequence of k operations.
Recall that we start the data structure from a state when all n elements are in their own separate sets. A single Union operation can consider at most two of these original one-element sets, so after any sequence of k Union operations, all but at most 2k elements of S have been completely untouched. Now consider a particular element v. As v’s set is involved in a sequence of Union operations, its size grows. It may be that in some of these Unions, the value of Component[v] is updated, and in others it is not. But our convention is that the union uses the name of the larger set, so in every update to Component[v] the size of the set containing v at least doubles. The size of v’s set starts out at 1, and the maximum possible size it can reach is 2k (since we argued above that all but at most 2k elements are untouched by Union operations). Thus Component[v] gets updated at most log2(2k) times throughout the process. Moreover, at most 2k elements are involved in any Union operations at all, so
154
Chapter 4 Greedy Algorithms
we get a bound of O(k log k) for the time spent updating Component values in a sequence of k Union operations.
While this bound on the average running time for a sequence of k opera- tions is good enough in many applications, including implementing Kruskal’s Algorithm, we will try to do better and reduce the worst-case time required. We’ll do this at the expense of raising the time required for the Find operation to O(log n).
A Better Data Structure for Union-Find
The data structure for this alternate implementation uses pointers. Each node v ∈ S will be contained in a record with an associated pointer to the name of the set that contains v. As before, we will use the elements of the set S as possible set names, naming each set after one of its elements. For the MakeUnionFind(S) operation, we initialize a record for each element v ∈ S with a pointer that points to itself (or is defined as a null pointer), to indicate that v is in its own set.
Consider a Union operation for two sets A and B, and assume that the name we used for set A is a node v ∈ A, while set B is named after node u ∈ B. The idea is to have either u or v be the name of the combined set; assume we select v as the name. To indicate that we took the union of the two sets, and that the name of the union set is v, we simply update u’s pointer to point to v. We do not update the pointers at the other nodes of set B.
As a result, for elements w ∈ B other than u, the name of the set they belong to must be computed by following a sequence of pointers, first leading them to the “old name” u and then via the pointer from u to the “new name” v. See Figure 4.12 for what such a representation looks like. For example, the two sets in Figure 4.12 could be the outcome of the following sequence of Union operations: Union(w, u), Union(s, u), Union(t, v), Union(z, v), Union(i, x), Union(y, j), Union(x, j), and Union(u, v).
This pointer-based data structure implements Union in O(1) time: all we have to do is to update one pointer. But a Find operation is no longer constant time, as we have to follow a sequence of pointers through a history of old names the set had, in order to get to the current name. How long can a Find(u) operation take? The number of steps needed is exactly the number of times the set containing node u had to change its name, that is, the number of times the Component[u] array position would have been updated in our previous array representation. This can be as large as O(n) if we are not careful with choosing set names. To reduce the time required for a Find operation, we will use the same optimization we used before: keep the name of the larger set as the name of the union. The sequence of Unions that produced the data
4.6 Implementing Kruskal’s Algorithm: The Union-Find Data Structure
155
The set {s, u, w} was merged into {t, v, z}.
j
u
v
w
s
t
z
x
y
i
Figure 4.12 A Union-Find data structure using pointers. The data structure has only two sets at the moment, named after nodes v and j. The dashed arrow from u to v is the result of the last Union operation. To answer a Find query, we follow the arrows until we get to a node that has no outgoing arrow. For example, answering the query Find(i) would involve following the arrows i to x, and then x to j.
structure in Figure 4.12 followed this convention. To implement this choice efficiently, we will maintain an additional field with the nodes: the size of the corresponding set.
(4.24) Consider the above pointer-based implementation of the Union-Find data structure for some set S of size n, where unions keep the name of the larger set. A Union operation takes O(1) time, MakeUnionFind(S) takes O(n) time, and a Find operation takes O(log n) time.
Proof. The statements about Union and MakeUnionFind are easy to verify. The time to evaluate Find(v) for a node v is the number of times the set containing node v changes its name during the process. By the convention that the union keeps the name of the larger set, it follows that every time the name of the set containing node v changes, the size of this set at least doubles. Since the set containing v starts at size 1 and is never larger than n, its size can double at most log2 n times, and so there can be at most log2 n name changes.
Further Improvements
Next we will briefly discuss a natural optimization in the pointer-based Union- Find data structure that has the effect of speeding up the Find operations. Strictly speaking, this improvement will not be necessary for our purposes in this book: for all the applications of Union-Find data structures that we con- sider, the O(log n) time per operation is good enough in the sense that further improvement in the time for operations would not translate to improvements
156
Chapter 4 Greedy Algorithms
in the overall running time of the algorithms where we use them. (The Union- Find operations will not be the only computational bottleneck in the running time of these algorithms.)
To motivate the improved version of the data structure, let us first discuss a bad case for the running time of the pointer-based Union-Find data structure. First we build up a structure where one of the Find operations takes about log n time. To do this, we can repeatedly take Unions of equal-sized sets. Assume v is a node for which the Find(v) operation takes about log n time. Now we can issue Find(v) repeatedly, and it takes log n for each such call. Having to follow the same sequence of log n pointers every time for finding the name of the set containing v is quite redundant: after the first request for Find(v), we already “know” the name x of the set containing v, and we also know that all other nodes that we touched during our path from v to the current name also are all contained in the set x. So in the improved implementation, we will compress the path we followed after every Find operation by resetting all pointers along the path to point to the current name of the set. No information is lost by doing this, and it makes subsequent Find operations run more quickly. See Figure 4.13 for a Union-Find data structure and the result of Find(v) using path compression.
Now consider the running time of the operations in the resulting imple- mentation. As before, a Union operation takes O(1) time and MakeUnion- Find(S) takes O(n) time to set up a data structure for a set of size n. How did the time required for a Find(v) operation change? Some Find operations can still take up to log n time; and for some Find operations we actually increase
Everything on the path from v to x now points directly to x.
x
v
v
(a) (b)
Figure 4.13 (a) An instance of a Union-Find data structure; and (b) the result of the operation Find(v) on this structure, using path compression.
x
the time, since after finding the name x of the set containing v, we have to go back through the same path of pointers from v to x, and reset each of these pointers to point to x directly. But this additional work can at most double the time required, and so does not change the fact that a Find takes at most O(log n) time. The real gain from compression is in making subsequent calls to Find cheaper, and this can be made precise by the same type of argument we used in (4.23): bounding the total time for a sequence of n Find operations, rather than the worst-case time for any one of them. Although we do not go into the details here, a sequence of n Find operations employing compression requires an amount of time that is extremely close to linear in n; the actual upper bound is O(nα(n)), where α(n) is an extremely slow-growing function of n called the inverse Ackermann function. (In particular, α(n) ≤ 4 for any value of n that could be encountered in practice.)
Implementing Kruskal’s Algorithm
Now we’ll use the Union-Find data structure to implement Kruskal’s Algo- rithm. First we need to sort the edges by cost. This takes time O(m log m). Since we have at most one edge between any pair of nodes, we have m ≤ n2 and hence this running time is also O(m log n).
After the sorting operation, we use the Union-Find data structure to maintain the connected components of (V,T) as edges are added. As each edge e = (v, w) is considered, we compute Find(u) and Find(v) and test if they are equal to see if v and w belong to different components. We use Union(Find(u),Find(v)) to merge the two components, if the algorithm decides to include edge e in the tree T.
We are doing a total of at most 2m Find and n − 1 Union operations over the course of Kruskal’s Algorithm. We can use either (4.23) for the array-based implementation of Union-Find, or (4.24) for the pointer-based implementation, to conclude that this is a total of O(m log n) time. (While more efficient implementations of the Union-Find data structure are possible, this would not help the running time of Kruskal’s Algorithm, which has an unavoidable O(m log n) term due to the initial sorting of the edges by cost.)
To sum up, we have
(4.25) Kruskal’s Algorithm can be implemented on a graph with n nodes and m edges to run in O(m log n) time.
4.7 Clustering
We motivated the construction of minimum spanning trees through the prob- lem of finding a low-cost network connecting a set of sites. But minimum
4.7 Clustering
157
158
Chapter 4 Greedy Algorithms
spanning trees arise in a range of different settings, several of which appear on the surface to be quite different from one another. An appealing example is the role that minimum spanning trees play in the area of clustering.
The Problem
Clustering arises whenever one has a collection of objects—say, a set of photographs, documents, or microorganisms—that one is trying to classify or organize into coherent groups. Faced with such a situation, it is natural to look first for measures of how similar or dissimilar each pair of objects is. One common approach is to define a distance function on the objects, with the interpretation that objects at a larger distance from one another are less similar to each other. For points in the physical world, distance may actually be related to their physical distance; but in many applications, distance takes on a much more abstract meaning. For example, we could define the distance between two species to be the number of years since they diverged in the course of evolution; we could define the distance between two images in a video stream as the number of corresponding pixels at which their intensity values differ by at least some threshold.
Now, given a distance function on the objects, the clustering problem seeks to divide them into groups so that, intuitively, objects within the same group are “close,” and objects in different groups are “far apart.” Starting from this vague set of goals, the field of clustering branches into a vast number of technically different approaches, each seeking to formalize this general notion of what a good set of groups might look like.
Clusterings of Maximum Spacing Minimum spanning trees play a role in one of the most basic formalizations, which we describe here. Suppose we are given asetU ofnobjects,labeledp1,p2,...,pn.Foreachpair,pi andpj,wehavea numerical distance d(pi, pj). We require only that d(pi, pi) = 0; that d(pi, pj) > 0 for distinct pi and pj; and that distances are symmetric: d(pi, pj) = d(pj, pi).

Suppose we are seeking to divide the objects in U into k groups, for a given parameter k. We say that a k-clustering of U is a partition of U into k nonempty sets C1, C2, . . . , Ck. We define the spacing of a k-clustering to be the minimum distance between any pair of points lying in different clusters. Given that we want points in different clusters to be far apart from one another, a natural goal is to seek the k-clustering with the maximum possible spacing.

The question now becomes the following. There are exponentially many different k-clusterings of a set U; how can we efficiently find the one that has maximum spacing?

Designing the Algorithm

To find a clustering of maximum spacing, we consider growing a graph on the vertex set U. The connected components will be the clusters, and we will try to bring nearby points together into the same cluster as rapidly as possible. (This way, they don’t end up as points in different clusters that are very close together.) Thus we start by drawing an edge between the closest pair of points. We then draw an edge between the next closest pair of points. We continue adding edges between pairs of points, in order of increasing distance d(pi, pj). In this way, we are growing a graph H on U edge by edge, with connected components corresponding to clusters. Notice that we are only interested in the connected components of the graph H, not the full set of edges; so if we are about to add the edge (pi, pj) and find that pi and pj already belong to the same cluster, we will refrain from adding the edge—it’s not necessary, because it won’t change the set of components. In this way, our graph-growing process will never create a cycle; so H will actually be a union of trees. Each time we add an edge that spans two distinct components, it is as though we have merged the two corresponding clusters. In the clustering literature, the iterative merging of clusters in this way is often termed single-link clustering, a special case of hierarchical agglomerative clustering. (Agglomerative here means that we combine clusters; single-link means that we do so as soon as a single link joins them together.) See Figure 4.14 for an example of an instance with k = 3 clusters where this algorithm partitions the points into an intuitively natural grouping.

What is the connection to minimum spanning trees? It’s very simple: although our graph-growing procedure was motivated by this cluster-merging idea, our procedure is precisely Kruskal’s Minimum Spanning Tree Algorithm. We are doing exactly what Kruskal’s Algorithm would do if given a graph G on U in which there was an edge of cost d(pi, pj) between each pair of nodes (pi,pj). The only difference is that we seek a k-clustering, so we stop the procedure once we obtain k connected components.

In other words, we are running Kruskal’s Algorithm but stopping it just before it adds its last k − 1 edges. This is equivalent to taking the full minimum spanning tree T (as Kruskal’s Algorithm would have produced it), deleting the k − 1 most expensive edges (the ones that we never actually added), and defin- ing the k-clustering to be the resulting connected components C1, C2, . . . , Ck. Thus, iteratively merging clusters is equivalent to computing a minimum span- ning tree and deleting the most expensive edges.

Analyzing the Algorithm

Have we achieved our goal of producing clusters that are as spaced apart as possible? The following claim shows that we have.

4.7 Clustering

159

160

Chapter 4 Greedy Algorithms

Cluster 1

Cluster 3

Cluster 2

Figure 4.14 An example of single-linkage clustering with k = 3 clusters. The clusters are formed by adding edges between points in order of increasing distance.

(4.26) The components C1, C2, . . . , Ck formed by deleting the k − 1 most expensive edges of the minimum spanning tree T constitute a k-clustering of maximum spacing.

Proof. LetCdenotetheclusteringC1,C2,…,Ck.ThespacingofCisprecisely the length d∗ of the (k − 1)st most expensive edge in the minimum spanning tree; this is the length of the edge that Kruskal’s Algorithm would have added next, at the moment we stopped it.

Now consider some other k-clustering C′, which partitions U into non- empty sets C1′,C2′,…,Ck′. We must show that the spacing of C′ is at most d∗.

Since the two clusterings C and C′ are not the same, it must be that one of our clusters Cr is not a subset of any of the k sets Cs′ in C′. Hence there are points pi , pj ∈ Cr that belong to different clusters in C′—say, pi ∈ Cs′ and p j ∈ C t′ ̸ = C s′ .

Now consider the picture in Figure 4.15. Since pi and pj belong to the same component Cr, it must be that Kruskal’s Algorithm added all the edges of a pi-pj path P before we stopped it. In particular, this means that each edge on

Cluster Cr

pi p

Cluster Cs

p

pj

4.8 Huffman Codes and Data Compression

161

Figure 4.15 An illustration of the proof of (4.26), showing that the spacing of any other clustering can be no larger than that of the clustering found by the single-linkage algorithm.

P has length at most d∗. Now, we know that pi ∈Cs′ but pj ̸∈Cs′; so let p′ be the first node on P that does not belong to Cs′, and let p be the node on P that comes just before p′. We have just argued that d(p, p′) ≤ d∗, since the edge (p, p′) was added by Kruskal’s Algorithm. But p and p′ belong to different sets in the clustering C′, and hence the spacing of C′ is at most d(p, p′) ≤ d∗. This completes the proof.

4.8 Huffman Codes and Data Compression

In the Shortest-Path and Minimum Spanning Tree Problems, we’ve seen how greedy algorithms can be used to commit to certain parts of a solution (edges in a graph, in these cases), based entirely on relatively short-sighted consid- erations. We now consider a problem in which this style of “committing” is carried out in an even looser sense: a greedy rule is used, essentially, to shrink the size of the problem instance, so that an equivalent smaller problem can then be solved by recursion. The greedy operation here is proved to be “safe,” in the sense that solving the smaller instance still leads to an optimal solu- tion for the original instance, but the global consequences of the initial greedy decision do not become fully apparent until the full recursion is complete.

The problem itself is one of the basic questions in the area of data com- pression, an area that forms part of the foundations for digital communication.

Cluster Ct

162

Chapter 4 Greedy Algorithms

The Problem

Encoding Symbols Using Bits Since computers ultimately operate on se- quences of bits (i.e., sequences consisting only of the symbols 0 and 1), one needs encoding schemes that take text written in richer alphabets (such as the alphabets underpinning human languages) and converts this text into long strings of bits.

The simplest way to do this would be to use a fixed number of bits for each symbol in the alphabet, and then just concatenate the bit strings for each symbol to form the text. To take a basic example, suppose we wanted to encode the 26 letters of English, plus the space (to separate words) and five punctuation characters: comma, period, question mark, exclamation point, and apostrophe. This would give us 32 symbols in total to be encoded. Now, you can form 2b different sequences out of b bits, and so if we use 5 bits per symbol, then we can encode 25 = 32 symbols—just enough for our purposes. So, for example, we could let the bit string 00000 represent a, the bit string 00001 represent b, and so forth up to 11111, which could represent the apostrophe. Note that the mapping of bit strings to symbols is arbitrary; the point is simply that five bits per symbol is sufficient. In fact, encoding schemes like ASCII work precisely this way, except that they use a larger number of bits per symbol so as to handle larger character sets, including capital letters, parentheses, and all those other special symbols you see on a typewriter or computer keyboard.

Let’s think about our bare-bones example with just 32 symbols. Is there anything more we could ask for from an encoding scheme? We couldn’t ask to encode each symbol using just four bits, since 24 is only 16—not enough for the number of symbols we have. Nevertheless, it’s not clear that over large stretches of text, we really need to be spending an average of five bits per symbol. If we think about it, the letters in most human alphabets do not get used equally frequently. In English, for example, the letters e, t, a, o, i, and n get used much more frequently than q, j, x, and z (by more than an order of magnitude). So it’s really a tremendous waste to translate them all into the same number of bits; instead we could use a small number of bits for the frequent letters, and a larger number of bits for the less frequent ones, and hope to end up using fewer than five bits per letter when we average over a long string of typical text.

This issue of reducing the average number of bits per letter is a funda- mental problem in the area of data compression. When large files need to be shipped across communication networks, or stored on hard disks, it’s impor- tant to represent them as compactly as possible, subject to the requirement that a subsequent reader of the file should be able to correctly reconstruct it. A huge amount of research is devoted to the design of compression algorithms

4.8 Huffman Codes and Data Compression

163

that can take files as input and reduce their space through efficient encoding schemes.

We now describe one of the fundamental ways of formulating this issue, building up to the question of how we might construct the optimal way to take advantage of the nonuniform frequencies of the letters. In one sense, such an optimal solution is a very appealing answer to the problem of compressing data: it squeezes all the available gains out of nonuniformities in the frequen- cies. At the end of the section, we will discuss how one can make further progress in compression, taking advantage of features other than nonuniform frequencies.

Variable-Length Encoding Schemes Before the Internet, before the digital computer, before the radio and telephone, there was the telegraph. Commu- nicating by telegraph was a lot faster than the contemporary alternatives of hand-delivering messages by railroad or on horseback. But telegraphs were only capable of transmitting pulses down a wire, and so if you wanted to send a message, you needed a way to encode the text of your message as a sequence of pulses.

To deal with this issue, the pioneer of telegraphic communication, Samuel Morse, developed Morse code, translating each letter into a sequence of dots (short pulses) and dashes (long pulses). For our purposes, we can think of dots and dashes as zeros and ones, and so this is simply a mapping of symbols into bit strings, just as in ASCII. Morse understood the point that one could communicate more efficiently by encoding frequent letters with short strings, and so this is the approach he took. (He consulted local printing presses to get frequency estimates for the letters in English.) Thus, Morse code maps e to 0 (a single dot), t to 1 (a single dash), a to 01 (dot-dash), and in general maps more frequent letters to shorter bit strings.

In fact, Morse code uses such short strings for the letters that the encoding of words becomes ambiguous. For example, just using what we know about the encoding of e, t, and a, we see that the string 0101 could correspond to any of the sequences of letters eta, aa, etet, or aet. (There are other possi- bilities as well, involving other letters.) To deal with this ambiguity, Morse code transmissions involve short pauses between letters (so the encoding of aa would actually be dot-dash-pause-dot-dash-pause). This is a reasonable solution—using very short bit strings and then introducing pauses—but it means that we haven’t actually encoded the letters using just 0 and 1; we’ve actually encoded it using a three-letter alphabet of 0, 1, and “pause.” Thus, if we really needed to encode everything using only the bits 0 and 1, there would need to be some further encoding in which the pause got mapped to bits.

164

Chapter 4 Greedy Algorithms

Prefix Codes The ambiguity problem in Morse code arises because there exist pairs of letters where the bit string that encodes one letter is a prefix of the bit string that encodes another. To eliminate this problem, and hence to obtain an encoding scheme that has a well-defined interpretation for every sequence of bits, it is enough to map letters to bit strings in such a way that no encoding is a prefix of any other.

Concretely, we say that a prefix code for a set S of letters is a function γ that maps each letter x ∈ S to some sequence of zeros and ones, in such a way that for distinct x, y ∈ S, the sequence γ (x) is not a prefix of the sequence γ (y).

Now suppose we have a text consisting of a sequence of letters x1x2x3 . . . xn. We can convert this to a sequence of bits by simply encoding each letter as a bit sequence using γ and then concatenating all these bit sequences together: γ (x1)γ (x2) . . . γ (xn). If we then hand this message to a recipient who knows the function γ , they will be able to reconstruct the text according to the following rule.

. Scan the bit sequence from left to right.

. As soon as you’ve seen enough bits to match the encoding of some letter, output this as the first letter of the text. This must be the correct first letter, since no shorter or longer prefix of the bit sequence could encode any other letter.

. Now delete the corresponding set of bits from the front of the message and iterate.

In this way, the recipient can produce the correct set of letters without our having to resort to artificial devices like pauses to separate the letters.

For example, suppose we are trying to encode the set of five letters S = {a, b, c, d, e}. The encoding γ1 specified by

γ1(a)=11 γ1(b) = 01 γ1(c) = 001 γ1(d) = 10 γ1(e) = 000

is a prefix code, since we can check that no encoding is a prefix of any other. Now, for example, the string cecab would be encoded as 0010000011101. A recipient of this message, knowing γ1, would begin reading from left to right. Neither 0 nor 00 encodes a letter, but 001 does, so the recipient concludes that the first letter is c. This is a safe decision, since no longer sequence of bits beginning with 001 could encode a different letter. The recipient now iterates

frequencies sum to 1; that is, f = 1. x∈S x

4.8 Huffman Codes and Data Compression

165

on the rest of the message, 0000011101; next they will conclude that the second letter is e, encoded as 000.

Optimal Prefix Codes We’ve been doing all this because some letters are more frequent than others, and we want to take advantage of the fact that more frequent letters can have shorter encodings. To make this objective precise, we now introduce some notation to express the frequencies of letters.

Suppose that for each letter x ∈ S, there is a frequency fx, representing the

fraction of letters in the text that are equal to x. In other words, assuming

there are n letters total, nfx of these letters are equal to x. We notice that the

Now, if we use a prefix code γ to encode the given text, what is the total length of our encoding? This is simply the sum, over all letters x ∈ S, of the number of times x occurs times the length of the bit string γ (x) used to encode x. Using |γ (x)| to denote the length γ (x), we can write this as

encoding length = nfx|γ (x)| = n fx|γ (x)|. x∈S x∈S

Dropping the leading coefficient of n from the final expression gives us f |γ (x)|, the average number of bits required per letter. We denote this

x∈S x

quantity by ABL(γ ).

To continue the earlier example, suppose we have a text with the letters S = {a, b, c, d, e}, and their frequencies are as follows:

fa =.32, fb =.25, fc =.20, fd =.18, fe =.05.

Then the average number of bits per letter using the prefix code γ1 defined

previously is

.32 · 2 + .25 · 2 + .20 · 3 + .18 · 2 + .05 · 3 = 2.25.

It is interesting to compare this to the average number of bits per letter using a fixed-length encoding. (Note that a fixed-length encoding is a prefix code: if all letters have encodings of the same length, then clearly no encoding can be a prefix of any other.) With a set S of five letters, we would need three bits per letter for a fixed-length encoding, since two bits could only encode four letters. Thus, using the code γ1 reduces the bits per letter from 3 to 2.25, a savings of 25 percent.

And, in fact, γ1 is not the best we can do in this example. Consider the prefix code γ2 given by

166

Chapter 4

Greedy Algorithms

γ2(a)=11 γ2(b) = 10 γ2(c)=01 γ2(d) = 001 γ2(e) = 000

The average number of bits per letter using γ2 is

.32 · 2 + .25 · 2 + .20 · 2 + .18 · 3 + .05 · 3 = 2.23.

So now it is natural to state the underlying question. Given an alphabet

and a set of frequencies for the letters, we would like to produce a prefix

code that is as efficient as possible—namely, a prefix code that minimizes the

average number of bits per letter ABL(γ ) = f |γ (x)|. We will call such a x∈S x

prefix code optimal.

Designing the Algorithm

The search space for this problem is fairly complicated; it includes all possible ways of mapping letters to bit strings, subject to the defining property of prefix codes. For alphabets consisting of an extremely small number of letters, it is feasible to search this space by brute force, but this rapidly becomes infeasible.

We now describe a greedy method to construct an optimal prefix code very efficiently. As a first step, it is useful to develop a tree-based means of representing prefix codes that exposes their structure more clearly than simply the lists of function values we used in our previous examples.

Representing Prefix Codes Using Binary Trees Suppose we take a rooted tree T in which each node that is not a leaf has at most two children; we call such a tree a binary tree. Further suppose that the number of leaves is equal to the size of the alphabet S, and we label each leaf with a distinct letter in S.

Such a labeled binary tree T naturally describes a prefix code, as follows. For each letter x ∈ S, we follow the path from the root to the leaf labeled x; each time the path goes from a node to its left child, we write down a 0, and each time the path goes from a node to its right child, we write down a 1. We take the resulting string of bits as the encoding of x.

Now we observe

(4.27) The encoding of S constructed from T is a prefix code.

Proof. In order for the encoding of x to be a prefix of the encoding of y, the path from the root to x would have to be a prefix of the path from the root

4.8 Huffman Codes and Data Compression

167

to y. But this is the same as saying that x would lie on the path from the root to y, which isn’t possible if x is a leaf.

This relationship between binary trees and prefix codes works in the other direction as well. Given a prefix code γ , we can build a binary tree recursively as follows. We start with a root; all letters x ∈ S whose encodings begin with a 0 will be leaves in the left subtree of the root, and all letters y ∈ S whose encodings begin with a 1 will be leaves in the right subtree of the root. We now build these two subtrees recursively using this rule.

For example, the labeled tree in Figure 4.16(a) corresponds to the prefix code γ0 specified by

γ0(a) = 1 γ0(b)=011 γ0(c) = 010 γ0(d) = 001 γ0(e) = 000

To see this, note that the leaf labeled a is obtained by simply taking the right- hand edge out of the root (resulting in an encoding of 1); the leaf labeled e is obtained by taking three successive left-hand edges starting from the root; and analogous explanations apply for b, c, and d. By similar reasoning, one can see that the labeled tree in Figure 4.16(b) corresponds to the prefix code γ1 defined earlier, and the labeled tree in Figure 4.16(c) corresponds to the prefix code γ2 defined earlier. Note also that the binary trees for the two prefix codes γ1 and γ2 are identical in structure; only the labeling of the leaves is different. The tree for γ0, on the other hand, has a different structure.

Thus the search for an optimal prefix code can be viewed as the search for a binary tree T, together with a labeling of the leaves of T, that minimizes the average number of bits per letter. Moreover, this average quantity has a natural interpretation in the terms of the structure of T: the length of the encoding of a letter x ∈ S is simply the length of the path from the root to the leaf labeled x. We will refer to the length of this path as the depth of the leaf, and we will denote the depth of a leaf v in T simply by depthT (v). (As two bits of notational convenience, we will drop the subscript T when it is clear from context, and we will often use a letter x ∈ S to also denote the leaf that is labeled by it.) Thus we are seeking the labeled tree that minimizes the weighted average of the depths of all leaves, where the average is weighted by the frequencies

of the letters that label the leaves: f · depth (x). We will use ABL(T) to x∈S x T

denote this quantity.

168

Chapter 4

Greedy Algorithms

a

Code: a→1 b→011 c→010 d→001 e→000

Code: a→11 b→01 c→001 d→10 e→000

bda

(b)

edcb ec

(a)

Code: a→11 b→10 c→01 d→001 e→000

ed

cba

(c)

Figure 4.16 Parts (a), (b), and (c) of the figure depict three different prefix codes for the alphabet S = {a, b, c, d, e}.

As a first step in considering algorithms for this problem, let’s note a simple fact about the optimal tree. For this fact, we need a definition: we say that a binary tree is full if each node that is not a leaf has two children. (In other words, there are no nodes with exactly one child.) Note that all three binary trees in Figure 4.16 are full.

(4.28) The binary tree corresponding to the optimal prefix code is full.

Proof. This is easy to prove using an exchange argument. Let T denote the binary tree corresponding to the optimal prefix code, and suppose it contains

4.8 Huffman Codes and Data Compression

169

a node u with exactly one child v. Now convert T into a tree T′ by replacing node u with v.

To be precise, we need to distinguish two cases. If u was the root of the tree, we simply delete node u and use v as the root. If u is not the root, let w be the parent of u in T. Now we delete node u and make v be a child of w in place of u. This change decreases the number of bits needed to encode any leaf in the subtree rooted at node u, and it does not affect other leaves. So the prefix code corresponding to T′ has a smaller average number of bits per letter than the prefix code for T, contradicting the optimality of T.

A First Attempt: The Top-Down Approach Intuitively, our goal is to produce a labeled binary tree in which the leaves are as close to the root as possible. This is what will give us a small average leaf depth.

A natural way to do this would be to try building a tree from the top down

by “packing” the leaves as tightly as possible. So suppose we try to split the

alphabet S into two sets S1 and S2, such that the total frequency of the letters

in each set is exactly 1. If such a perfect split is not possible, then we can try 2

for a split that is as nearly balanced as possible. We then recursively construct prefix codes for S1 and S2 independently, and make these the two subtrees of the root. (In terms of bit strings, this would mean sticking a 0 in front of the encodings we produce for S1, and sticking a 1 in front of the encodings we produce for S2.)

It is not entirely clear how we should concretely define this “nearly balanced” split of the alphabet, but there are ways to make this precise. The resulting encoding schemes are called Shannon-Fano codes, named after Claude Shannon and Robert Fano, two of the major early figures in the area of information theory, which deals with representing and encoding digital information. These types of prefix codes can be fairly good in practice, but for our present purposes they represent a kind of dead end: no version of this top-down splitting strategy is guaranteed to always produce an optimal prefix code. Consider again our example with the five-letter alphabet S = {a, b, c, d, e} and frequencies

fa =.32, fb =.25, fc =.20, fd =.18, fe =.05.

There is a unique way to split the alphabet into two sets of equal frequency: {a, d} and {b, c, e}. For {a, d}, we can use a single bit to encode each. For {b, c, e}, we need to continue recursively, and again there is a unique way to split the set into two subsets of equal frequency. The resulting code corre- sponds to the code γ1, given by the labeled tree in Figure 4.16(b); and we’ve already seen that γ1 is not as efficient as the prefix code γ2 corresponding to the labeled tree in Figure 4.16(c).

170

Chapter 4 Greedy Algorithms

Shannon and Fano knew that their approach did not always yield the optimal prefix code, but they didn’t see how to compute the optimal code without brute-force search. The problem was solved a few years later by David Huffman, at the time a graduate student who learned about the question in a class taught by Fano.

We now describe the ideas leading up to the greedy approach that Huffman discovered for producing optimal prefix codes.

What If We Knew the Tree Structure of the Optimal Prefix Code? A tech- nique that is often helpful in searching for an efficient algorithm is to assume, as a thought experiment, that one knows something partial about the optimal solution, and then to see how one would make use of this partial knowledge in finding the complete solution. (Later, in Chapter 6, we will see in fact that this technique is a main underpinning of the dynamic programming approach to designing algorithms.)

For the current problem, it is useful to ask: What if someone gave us the binary tree T∗ that corresponded to an optimal prefix code, but not the labeling of the leaves? To complete the solution, we would need to figure out which letter should label which leaf of T∗, and then we’d have our code. How hard is this?

In fact, this is quite easy. We begin by formulating the following basic fact.

(4.29) Suppose that u and v are leaves of T∗, such that depth(u) < depth(v). Further, suppose that in a labeling of T∗ corresponding to an optimal prefix code, leaf u is labeled with y∈S and leaf v is labeled with z∈S. Then fy ≥fz.
Proof. This has a quick proof using an exchange argument. If fy < fz, then
consider the code obtained by exchanging the labels at the nodes u and
v. In the expression for the average number of bits per letter, ABL(T∗)=
f depth(x), the effect of this exchange is as follows: the multiplier on f x∈S x
y increases (from depth(u) to depth(v)), and the multiplier on fz decreases by
the same amount (from depth(v) to depth(u)).
Thus the change to the overall sum is (depth(v) − depth(u))(fy − fz). If fy < fz, this change is a negative number, contradicting the supposed optimality of the prefix code that we had before the exchange.
We can see the idea behind (4.29) in Figure 4.16(b): a quick way to see that the code here is not optimal is to notice that it can be improved by exchanging the positions of the labels c and d. Having a lower-frequency letter at a strictly smaller depth than some other higher-frequency letter is precisely what (4.29) rules out for an optimal solution.
4.8 Huffman Codes and Data Compression
171
Statement (4.29) gives us the following intuitively natural, and optimal,
way to label the tree T∗ if someone should give it to us. We first take all leaves
of depth 1 (if there are any) and label them with the highest-frequency letters
in any order. We then take all leaves of depth 2 (if there are any) and label them
with the next-highest-frequency letters in any order. We continue through the
leaves in order of increasing depth, assigning letters in order of decreasing
frequency. The point is that this can’t lead to a suboptimal labeling of T∗,
since any supposedly better labeling would be susceptible to the exchange in
(4.29). It is also crucial to note that, among the labels we assign to a block of
leaves all at the same depth, it doesn’t matter which label we assign to which
leaf. Since the depths are all the same, the corresponding multipliers in the
expression f |γ (x)| are the same, and so the choice of assignment among x∈S x
leaves of the same depth doesn’t affect the average number of bits per letter.
But how is all this helping us? We don’t have the structure of the optimal tree T∗, and since there are exponentially many possible trees (in the size of the alphabet), we aren’t going to be able to perform a brute-force search over all of them.
In fact, our reasoning about T∗ becomes very useful if we think not about the very beginning of this labeling process, with the leaves of minimum depth, but about the very end, with the leaves of maximum depth—the ones that receive the letters with lowest frequency. Specifically, consider a leaf v in T∗ whose depth is as large as possible. Leaf v has a parent u, and by (4.28) T∗ is a full binary tree, so u has another child w. We refer to v and w as siblings, since they have a common parent. Now, we have
(4.30) w is a leaf of T∗.
Proof. If w were not a leaf, there would be some leaf w′ in the subtree below it. But then w′ would have a depth greater than that of v, contradicting our assumption that v is a leaf of maximum depth in T∗.
So v and w are sibling leaves that are as deep as possible in T∗. Thus our level-by-level process of labeling T∗, as justified by (4.29), will get to the level containing v and w last. The leaves at this level will get the lowest-frequency letters. Since we have already argued that the order in which we assign these letters to the leaves within this level doesn’t matter, there is an optimal labeling in which v and w get the two lowest-frequency letters of all.
We sum this up in the following claim.
(4.31) There is an optimal prefix code, with corresponding tree T∗, in which the two lowest-frequency letters are assigned to leaves that are siblings in T∗.
172
Chapter 4 Greedy Algorithms
New merged letter
with sum of frequencies
Two lowest-frequency letters
Figure 4.17 There is an optimal solution in which the two lowest-frequency letters label sibling leaves; deleting them and labeling their parent with a new letter having the combined frequency yields an instance with a smaller alphabet.
An Algorithm to Construct an Optimal Prefix Code Suppose that y∗ and z∗ are the two lowest-frequency letters in S. (We can break ties in the frequencies arbitrarily.) Statement (4.31) is important because it tells us something about where y∗ and z∗ go in the optimal solution; it says that it is safe to “lock them together” in thinking about the solution, because we know they end up as sibling leaves below a common parent. In effect, this common parent acts like a “meta-letter” whose frequency is the sum of the frequencies of y∗ and z∗.
This directly suggests an algorithm: we replace y∗ and z∗ with this meta- letter, obtaining an alphabet that is one letter smaller. We recursively find a prefix code for the smaller alphabet, and then “open up” the meta-letter back into y∗ and z∗ to obtain a prefix code for S. This recursive strategy is depicted in Figure 4.17.
A concrete description of the algorithm is as follows.
To construct a prefix code for an alphabet S, with given frequencies: If S has two letters then
Encode one letter using 0 and the other letter using 1 Else
Let y∗ and z∗ be the two lowest-frequency letters Form a new alphabet S′ by deleting y∗ and z∗ and
replacing them with a new letter ω of frequency fy∗ + fz∗ Recursively construct a prefix code γ′ for S′, with tree T′ Define a prefix code for S as follows:
Start with T′
4.8 Huffman Codes and Data Compression
173
Take the leaf labeled ω and add two children below it labeled y∗ and z∗
Endif
We refer to this as Huffman’s Algorithm, and the prefix code that it produces for a given alphabet is accordingly referred to as a Huffman code. In general, it is clear that this algorithm always terminates, since it simply invokes a recursive call on an alphabet that is one letter smaller. Moreover, using (4.31), it will not be difficult to prove that the algorithm in fact produces an optimal prefix code. Before doing this, however, we pause to note some further observations about the algorithm.
First let’s consider the behavior of the algorithm on our sample instance with S = {a, b, c, d, e} and frequencies
fa =.32, fb =.25, fc =.20, fd =.18, fe =.05.
The algorithm would first merge d and e into a single letter—let’s denote it (de)—of frequency .18 + .05 = .23. We now have an instance of the problem on the four letters S′ = {a, b, c, (de)}. The two lowest-frequency letters in S′ are c and (de), so in the next step we merge these into the single letter (cde) of frequency .20 + .23 = .43. This gives us the three-letter alphabet {a, b, (cde)}. Next we merge a and b, and this gives us a two-letter alphabet, at which point we invoke the base case of the recursion. If we unfold the result back through the recursive calls, we get the tree pictured in Figure 4.16(c).
It is interesting to note how the greedy rule underlying Huffman’s Algorithm—the merging of the two lowest-frequency letters—fits into the structure of the algorithm as a whole. Essentially, at the time we merge these two letters, we don’t know exactly how they will fit into the overall code. Rather, we simply commit to having them be children of the same parent, and this is enough to produce a new, equivalent problem with one less letter.
Moreover, the algorithm forms a natural contrast with the earlier approach that led to suboptimal Shannon-Fano codes. That approach was based on a top-down strategy that worried first and foremost about the top-level split in the binary tree—namely, the two subtrees directly below the root. Huffman’s Algorithm, on the other hand, follows a bottom-up approach: it focuses on the leaves representing the two lowest-frequency letters, and then continues by recursion.
Analyzing the Algorithm
The Optimality of the Algorithm We first prove the optimality of Huffman’s Algorithm. Since the algorithm operates recursively, invoking itself on smaller and smaller alphabets, it is natural to try establishing optimality by induction
174
Chapter 4 Greedy Algorithms
on the size of the alphabet. Clearly it is optimal for all two-letter alphabets (since it uses only one bit per letter). So suppose by induction that it is optimal for all alphabets of size k − 1, and consider an input instance consisting of an alphabet S of size k.
Let’s quickly recap the behavior of the algorithm on this instance. The algorithm merges the two lowest-frequency letters y∗, z∗ ∈ S into a single letter ω, calls itself recursively on the smaller alphabet S′ (in which y∗ and z∗ are replaced by ω), and by induction produces an optimal prefix code for S′, represented by a labeled binary tree T′. It then extends this into a tree T for S, by attaching leaves labeled y∗ and z∗ as children of the node in T′ labeled ω.
There is a close relationship between ABL(T) and ABL(T′). (Note that the former quantity is the average number of bits used to encode letters in S, while the latter quantity is the average number of bits used to encode letters in S′.)
(4.32) ABL(T′) = ABL(T) − fω.
Proof. The depth of each letter x other than y∗, z∗ is the same in both T and T′. Also, the depths of y∗ and z∗ in T are each one greater than the depth of ω in T′. Using this, plus the fact that fω = fy∗ + fz∗, we have
ABL(T ) =
= fy∗ · depthT(y ) + fz∗ · depthT(z ) +
= (fy∗ + fz∗) · (1 + depthT′(ω)) +
x∈S
fx · depthT (x) ∗∗
= fω · (1 + depthT′(ω)) +
= fω + fω · depthT′(ω) +
x̸=y∗,z∗
= fω +
x̸=y∗,z∗
x̸=y∗,z∗ fx · depthT′(x)
x∈S′
= fω + ABL(T′).
x̸=y∗,z∗
fx · depthT′(x) fx · depthT′(x)
fx · depthT(x) fx · depthT′(x)
Using this, we now prove optimality as follows.
(4.33) The Huffman code for a given alphabet achieves the minimum average number of bits per letter of any prefix code.
Proof. SupposebywayofcontradictionthatthetreeTproducedbyourgreedy algorithm is not optimal. This means that there is some labeled binary tree Z
4.8 Huffman Codes and Data Compression
175
such that ABL(Z) < ABL(T); and by (4.31), there is such a tree Z in which the leaves representing y∗ and z∗ are siblings.
It is now easy to get a contradiction, as follows. If we delete the leaves labeled y∗ and z∗ from Z, and label their former parent with ω, we get a tree Z′ that defines a prefix code for S′. In the same way that T is obtained from T′, the tree Z is obtained from Z′ by adding leaves for y∗ and z∗ below ω; thus the identity in (4.32) applies to Z and Z′ as well: ABL(Z′) = ABL(Z) − fω.
But we have assumed that ABL(Z) < ABL(T); subtracting fω from both sides of this inequality we get ABL(Z′) < ABL(T′), which contradicts the optimality of T′ as a prefix code for S′.
Implementation and Running Time It is clear that Huffman’s Algorithm can be made to run in polynomial time in k, the number of letters in the alphabet. The recursive calls of the algorithm define a sequence of k − 1 iterations over smaller and smaller alphabets, and each iteration except the last consists simply of identifying the two lowest-frequency letters and merging them into a single letter that has the combined frequency. Even without being careful about the implementation, identifying the lowest-frequency letters can be done in a single scan of the alphabet, in time O(k), and so summing this over the k − 1 iterations gives O(k2) time.
But in fact Huffman’s Algorithm is an ideal setting in which to use a priority queue. Recall that a priority queue maintains a set of k elements, each with a numerical key, and it allows for the insertion of new elements and the extraction of the element with the minimum key. Thus we can maintain the alphabet S in a priority queue, using each letter’s frequency as its key. In each iteration we just extract the minimum twice (this gives us the two lowest-frequency letters), and then we insert a new letter whose key is the sum of these two minimum frequencies. Our priority queue now contains a representation of the alphabet that we need for the next iteration.
Using an implementation of priority queues via heaps, as in Chapter 2, we can make each insertion and extraction of the minimum run in time O(log k); hence, each iteration—which performs just three of these operations—takes time O(log k). Summing over all k iterations, we get a total running time of O(k log k).
Extensions
The structure of optimal prefix codes, which has been our focus here, stands as a fundamental result in the area of data compression. But it is important to understand that this optimality result does not by any means imply that we have found the best way to compress data under all circumstances.
176
Chapter 4 Greedy Algorithms
What more could we want beyond an optimal prefix code? First, consider an application in which we are transmitting black-and-white images: each image is a 1,000-by-1,000 array of pixels, and each pixel takes one of the two values black or white. Further, suppose that a typical image is almost entirely white: roughly 1,000 of the million pixels are black, and the rest are white. Now, if we wanted to compress such an image, the whole approach of prefix codes has very little to say: we have a text of length one million over the two-letter alphabet {black, white}. As a result, the text is already encoded using one bit per letter—the lowest possible in our framework.
It is clear, though, that such images should be highly compressible. Intuitively, one ought to be able to use a “fraction of a bit” for each white pixel, since they are so overwhelmingly frequent, at the cost of using multiple bits for each black pixel. (In an extreme version, sending a list of (x, y) coordinates for each black pixel would be an improvement over sending the image as a text with a million bits.) The challenge here is to define an encoding scheme where the notion of using fractions of bits is well-defined. There are results in the area of data compression, however, that do just this; arithmetic coding and a range of other techniques have been developed to handle settings like this.
A second drawback of prefix codes, as defined here, is that they cannot adapt to changes in the text. Again let’s consider a simple example. Suppose we are trying to encode the output of a program that produces a long sequence of letters from the set {a, b, c, d}. Further suppose that for the first half of this sequence, the letters a and b occur equally frequently, while c and d do not occur at all; but in the second half of this sequence, the letters c and d occur equally frequently, while a and b do not occur at all. In the framework developed in this section, we are trying to compress a text over the four-letter alphabet {a, b, c, d}, and all letters are equally frequent. Thus each would be encoded with two bits.
But what’s really happening in this example is that the frequency remains stable for half the text, and then it changes radically. So one could get away with just one bit per letter, plus a bit of extra overhead, as follows.
. Begin with an encoding in which the bit 0 represents a and the bit 1 represents b.
. Halfway into the sequence, insert some kind of instruction that says, “We’re changing the encoding now. From now on, the bit 0 represents c and the bit 1 represents d.”
. Use this new encoding for the rest of the sequence.
The point is that investing a small amount of space to describe a new encoding can pay off many times over if it reduces the average number of bits per
4.9 Minimum-Cost Arborescences: A Multi-Phase Greedy Algorithm
177
letter over a long run of text that follows. Such approaches, which change the encoding in midstream, are called adaptive compression schemes, and for many kinds of data they lead to significant improvements over the static method we’ve considered here.
These issues suggest some of the directions in which work on data com- pression has proceeded. In many of these cases, there is a trade-off between the power of the compression technique and its computational cost. In partic- ular, many of the improvements to Huffman codes just described come with a corresponding increase in the computational effort needed both to produce the compressed version of the data and also to decompress it and restore the original text. Finding the right balance among these trade-offs is a topic of active research.
* 4.9 Minimum-Cost Arborescences: A Multi-Phase Greedy Algorithm
As we’ve seen more and more examples of greedy algorithms, we’ve come to appreciate that there can be considerable diversity in the way they operate. Many greedy algorithms make some sort of an initial “ordering” decision on the input, and then process everything in a one-pass fashion. Others make more incremental decisions—still local and opportunistic, but without a global “plan” in advance. In this section, we consider a problem that stresses our intuitive view of greedy algorithms still further.
The Problem
The problem is to compute a minimum-cost arborescence of a directed graph. This is essentially an analogue of the Minimum Spanning Tree Problem for directed, rather than undirected, graphs; we will see that the move to directed graphs introduces significant new complications. At the same time, the style of the algorithm has a strongly greedy flavor, since it still constructs a solution according to a local, myopic rule.
We begin with the basic definitions. Let G = (V , E) be a directed graph in which we’ve distinguished one node r ∈ V as a root. An arborescence (with respect to r) is essentially a directed spanning tree rooted at r. Specifically, it is a subgraph T = (V, F) such that T is a spanning tree of G if we ignore the direction of edges; and there is a path in T from r to each other node v ∈ V if we take the direction of edges into account. Figure 4.18 gives an example of two different arborescences in the same directed graph.
There is a useful equivalent way to characterize arborescences, and this is as follows.
178
Chapter 4 Greedy Algorithms
rrr
(a) (b) (c)
Figure 4.18 A directed graph can have many different arborescences. Parts (b) and (c) depict two different aborescences, both rooted at node r, for the graph in part (a).
(4.34) A subgraph T = (V , F ) of G is an arborescence with respect to root r if and only if T has no cycles, and for each node v ̸= r, there is exactly one edge in F that enters v.
Proof. If T is an arborescence with root r, then indeed every other node v has exactly one edge entering it: this is simply the last edge on the unique r-v path.
Conversely, suppose T has no cycles, and each node v ̸= r has exactly one entering edge. In order to establish that T is an arborescence, we need only show that there is a directed path from r to each other node v. Here is how to construct such a path. We start at v and repeatedly follow edges in the backward direction. Since T has no cycles, we can never return to a node we’ve previously visited, and thus this process must terminate. But r is the only node without incoming edges, and so the process must in fact terminate by reaching r; the sequence of nodes thus visited yields a path (in the reverse direction) from r to v.
It is easy to see that, just as every connected graph has a spanning tree, a directed graph has an arborescence rooted at r provided that r can reach every node. Indeed, in this case, the edges in a breadth-first search tree rooted at r will form an arborescence.
(4.35) A directed graph G has an arborescence rooted at r if and only if there is a directed path from r to each other node.
4.9 Minimum-Cost Arborescences: A Multi-Phase Greedy Algorithm
179
The basic problem we consider here is the following. We are given a directed graph G = (V , E), with a distinguished root node r and with a non- negative cost ce ≥ 0 on each edge, and we wish to compute an arborescence rooted at r of minimum total cost. (We will refer to this as an optimal arbores- cence.) We will assume throughout that G at least has an arborescence rooted at r; by (4.35), this can be easily checked at the outset.
Designing the Algorithm
Given the relationship between arborescences and trees, the minimum-cost arborescence problem certainly has a strong initial resemblance to the Mini- mum Spanning Tree Problem for undirected graphs. Thus it’s natural to start by asking whether the ideas we developed for that problem can be carried over directly to this setting. For example, must the minimum-cost arbores- cence contain the cheapest edge in the whole graph? Can we safely delete the most expensive edge on a cycle, confident that it cannot be in the optimal arborescence?
Clearly the cheapest edge e in G will not belong to the optimal arborescence if e enters the root, since the arborescence we’re seeking is not supposed to have any edges entering the root. But even if the cheapest edge in G belongs to some arborescence rooted at r, it need not belong to the optimal one, as the example of Figure 4.19 shows. Indeed, including the edge of cost 1 in Figure 4.19 would prevent us from including the edge of cost 2 out of the root r (since there can only be one entering edge per node); and this in turn would force us to incur an unacceptable cost of 10 when we included one of
rr
22 10 10
144 22 22
844
(a) (b)
Figure4.19 (a)Adirectedgraphwithcostsonitsedges,and(b)anoptimalarborescence rooted at r for this graph.
180
Chapter 4 Greedy Algorithms
the other edges out of r. This kind of argument never clouded our thinking in the Minimum Spanning Tree Problem, where it was always safe to plunge ahead and include the cheapest edge; it suggests that finding the optimal arborescence may be a significantly more complicated task. (It’s worth noticing that the optimal arborescence in Figure 4.19 also includes the most expensive edge on a cycle; with a different construction, one can even cause the optimal arborescence to include the most expensive edge in the whole graph.)
Despite this, it is possible to design a greedy type of algorithm for this problem; it’s just that our myopic rule for choosing edges has to be a little more sophisticated. First let’s consider a little more carefully what goes wrong with the general strategy of including the cheapest edges. Here’s a particular version of this strategy: for each node v ̸= r, select the cheapest edge entering v (breaking ties arbitrarily), and let F ∗ be this set of n − 1 edges. Now consider the subgraph (V , F ∗). Since we know that the optimal arborescence needs to have exactly one edge entering each node v ̸= r, and (V, F∗) represents the cheapest possible way of making these choices, we have the following fact.
(4.36) If (V , F ∗) is an arborescence, then it is a minimum-cost arborescence.
So the difficulty is that (V , F ∗) may not be an arborescence. In this case, (4.34) implies that (V, F∗) must contain a cycle C, which does not include the root. We now must decide how to proceed in this situation.
To make matters somewhat clearer, we begin with the following observa- tion. Every arborescence contains exactly one edge entering each node v ̸= r; so if we pick some node v and subtract a uniform quantity from the cost of every edge entering v, then the total cost of every arborescence changes by exactly the same amount. This means, essentially, that the actual cost of the cheapest edge entering v is not important; what matters is the cost of all other edges entering v relative to this. Thus let yv denote the minimum cost of any edge entering v. For each edge e = (u, v), with cost ce ≥ 0, we define its modi- fied cost ce′ to be ce − yv. Note that since ce ≥ yv, all the modified costs are still nonnegative. More crucially, our discussion motivates the following fact.
(4.37) T is an optimal arborescence in G subject to costs {ce} if and only if it is an optimal arborescence subject to the modified costs {ce′ }.
Proof. Consider an arbitrary arborescence T. The difference between its cost with costs {c } and {c′ } is exactly y —that is,
e e v̸=r v ′
e∈T
e∈T
ce −
ce = yv. v̸=r
4.9 Minimum-Cost Arborescences: A Multi-Phase Greedy Algorithm
181
This is because an arborescence has exactly one edge entering each node v in the sum. Since the difference between the two costs is independent of the choice of the arborescence T, we see that T has minimum cost subject to {ce} if and only if it has minimum cost subject to {ce′ }.
We now consider the problem in terms of the costs {ce′ }. All the edges in our set F ∗ have cost 0 under these modified costs; and so if (V , F ∗) contains a cycle C, we know that all edges in C have cost 0. This suggests that we can afford to use as many edges from C as we want (consistent with producing an arborescence), since including edges from C doesn’t raise the cost.
Thus our algorithm continues as follows. We contract C into a single supernode, obtaining a smaller graph G′ = (V′, E′). Here, V′ contains the nodes of V −C, plus a single node c∗ representing C. We transform each edge e ∈ E to an edge e′ ∈ E′ by replacing each end of e that belongs to C with the new node c∗. This can result in G′ having parallel edges (i.e., edges with the same ends), which is fine; however, we delete self-loops from E′—edges that have both ends equal to c∗. We recursively find an optimal arborescence in this smaller graph G′, subject to the costs {ce′ }. The arborescence returned by this recursive call can be converted into an arborescence of G by including all but one edge on the cycle C.
In summary, here is the full algorithm.
For each node v̸=r
Let yv be the minimum cost of an edge entering node v Modify the costs of all edges e entering v to ce′ = ce − yv
Choose one 0-cost edge entering each v ̸= r, obtaining a set F∗ If F∗ forms an arborescence, then return it
Else there is a directed cycle C ⊆ F ∗
Contract C to a single supernode, yielding a graph G′ = (V′, E′) Recursively find an optimal arborescence (V′, F′) in G′
with costs {ce′}
Extend (V′, F′) to an arborescence (V, F) in G
by adding all but one edge of C
Analyzing the Algorithm
It is easy to implement this algorithm so that it runs in polynomial time. But does it lead to an optimal arborescence? Before concluding that it does, we need to worry about the following point: not every arborescence in G corresponds to an arborescence in the contracted graph G′. Could we perhaps “miss” the true optimal arborescence in G by focusing on G′? What is true is the following.
182
Chapter 4 Greedy Algorithms
The arborescences of G′ are in one-to-one correspondence with arborescences of G that have exactly one edge entering the cycle C; and these corresponding arborescences have the same cost with respect to {ce′ }, since C consists of 0- cost edges. (We say that an edge e = (u, v) enters C if v belongs to C but u does not.) So to prove that our algorithm finds an optimal arborescence in G, we must prove that G has an optimal arborescence with exactly one edge entering C. We do this now.
(4.38) Let C be a cycle in G consisting of edges of cost 0, such that r ̸∈ C. Then there is an optimal arborescence rooted at r that has exactly one edge entering C.
Proof. Consider an optimal arborescence T in G. Since r has a path in T to every node, there is at least one edge of T that enters C. If T enters C exactly once, then we are done. Otherwise, suppose that T enters C more than once. We show how to modify it to obtain an arborescence of no greater cost that enters C exactly once.
Let e = (a, b) be an edge entering C that lies on as short a path as possible from r; this means in particular that no edges on the path from r to a can enter C. We delete all edges of T that enter C, except for the edge e. We add in all edges of C except for the one edge that enters b, the head of edge e. Let T′ denote the resulting subgraph of G.
We claim that T′ is also an arborescence. This will establish the result, since the cost of T′ is clearly no greater than that of T: the only edges of T′ that do not also belong to T have cost 0. So why is T′ an arborescence? Observe that T′ has exactly one edge entering each node v ̸= r, and no edge entering r. So T′ has exactly n − 1 edges; hence if we can show there is an r-v path in T′ for each v, then T′ must be connected in an undirected sense, and hence a tree. Thus it would satisfy our initial definition of an arborescence.
So consider any node v ̸= r; we must show there is an r-v path in T′. If v∈C, we can use the fact that the path in T from r to e has been preserved in the construction of T′; thus we can reach v by first reaching e and then following the edges of the cycle C. Now suppose that v ̸∈ C, and let P denote the r-v path in T. If P did not touch C, then it still exists in T′. Otherwise, let w be the last node in P ∩ C, and let P′ be the subpath of P from w to v. Observe that all the edges in P′ still exist in T′. We have already argued that w is reachable from r in T′, since it belongs to C. Concatenating this path to w with the subpath P′ gives us a path to v as well.
We can now put all the pieces together to argue that our algorithm is correct.
(4.39) The algorithm finds an optimal arborescence rooted at r in G.
Proof. The proof is by induction on the number of nodes in G. If the edges of F form an arborescence, then the algorithm returns an optimal arborescence by (4.36). Otherwise, we consider the problem with the modified costs {ce′ }, which is equivalent by (4.37). After contracting a 0-cost cycle C to obtain a smaller graph G′, the algorithm produces an optimal arborescence in G′ by the inductive hypothesis. Finally, by (4.38), there is an optimal arborescence in G that corresponds to the optimal arborescence computed for G′.
Solved Exercises
Solved Exercise 1
Suppose that three of your friends, inspired by repeated viewings of the horror-movie phenomenon The Blair Witch Project, have decided to hike the Appalachian Trail this summer. They want to hike as much as possible per day but, for obvious reasons, not after dark. On a map they’ve identified a large set of good stopping points for camping, and they’re considering the following system for deciding when to stop for the day. Each time they come to a potential stopping point, they determine whether they can make it to the next one before nightfall. If they can make it, then they keep hiking; otherwise, they stop.
Despite many significant drawbacks, they claim this system does have one good feature. “Given that we’re only hiking in the daylight,” they claim, “it minimizes the number of camping stops we have to make.”
Is this true? The proposed system is a greedy algorithm, and we wish to determine whether it minimizes the number of stops needed.
To make this question precise, let’s make the following set of simplifying assumptions. We’ll model the Appalachian Trail as a long line segment of length L, and assume that your friends can hike d miles per day (independent of terrain, weather conditions, and so forth). We’ll assume that the potential stopping points are located at distances x1, x2, . . . , xn from the start of the trail. We’ll also assume (very generously) that your friends are always correct when they estimate whether they can make it to the next stopping point before nightfall.
We’ll say that a set of stopping points is valid if the distance between each adjacent pair is at most d, the first is at distance at most d from the start of the trail, and the last is at distance at most d from the end of the trail. Thus a set of stopping points is valid if one could camp only at these places and
Solved Exercises
183
184
Chapter 4 Greedy Algorithms
still make it across the whole trail. We’ll assume, naturally, that the full set of n stopping points is valid; otherwise, there would be no way to make it the whole way.
We can now state the question as follows. Is your friends’ greedy algorithm—hiking as long as possible each day—optimal, in the sense that it finds a valid set whose size is as small as possible?
Solution Often a greedy algorithm looks correct when you first encounter it, so before succumbing too deeply to its intuitive appeal, it’s useful to ask: why might it not work? What should we be worried about?
There’s a natural concern with this algorithm: Might it not help to stop early on some day, so as to get better synchronized with camping opportunities on future days? But if you think about it, you start to wonder whether this could really happen. Could there really be an alternate solution that intentionally lags behind the greedy solution, and then puts on a burst of speed and passes the greedy solution? How could it pass it, given that the greedy solution travels as far as possible each day?
This last consideration starts to look like the outline of an argument based on the “staying ahead” principle from Section 4.1. Perhaps we can show that as long as the greedy camping strategy is ahead on a given day, no other solution can catch up and overtake it the next day.
We now turn this into a proof showing the algorithm is indeed optimal, identifying a natural sense in which the stopping points it chooses “stay ahead” of any other legal set of stopping points. Although we are following the style of proof from Section 4.1, it’s worth noting an interesting contrast with the Interval Scheduling Problem: there we needed to prove that a greedy algorithm maximized a quantity of interest, whereas here we seek to minimize a certain quantity.
Let R = {xp1, . . . , xpk} denote the set of stopping points chosen by the greedy algorithm, and suppose by way of contradiction that there is a smaller valid set of stopping points; let’s call this smaller set S = {xq1 , . . . , xqm }, with m < k.
To obtain a contradiction, we first show that the stopping point reached by the greedy algorithm on each day j is farther than the stopping point reached under the alternate solution. That is,
(4.40) Foreachj=1,2,...,m,wehavexpj ≥xqj.
Proof. We prove this by induction on j. The case j = 1 follows directly from the definition of the greedy algorithm: your friends travel as long as possible
on the first day before stopping. Now let j > 1 and assume that the claim is true for all i < j. Then
xqj −xqj−1 ≤d, since S is a valid set of stopping points, and
xqj − xpj−1 ≤ xqj − xqj−1
since xpj−1 ≥ xqj−1 by the induction hypothesis. Combining these two inequal-
ities, we have
xqj − xpj−1 ≤ d.
This means that your friends have the option of hiking all the way from xpj−1 to xqj in one day; and hence the location xpj at which they finally stop can only be farther along than xqj . (Note the similarity with the corresponding proof for the Interval Scheduling Problem: here too the greedy algorithm is staying ahead because, at each step, the choice made by the alternate solution is one of its valid options.)
Statement (4.40) implies in particular that xqm ≤ xpm . Now, if m < k, then we must have xpm < L − d, for otherwise your friends would never have needed to stop at the location xpm+1. Combining these two inequalities, we have concluded that xqm < L − d; but this contradicts the assumption that S is a valid set of stopping points.
Consequently, we cannot have m < k, and so we have proved that the greedy algorithm produces a valid set of stopping points of minimum possible size.
Solved Exercise 2
Your friends are starting a security company that needs to obtain licenses for n different pieces of cryptographic software. Due to regulations, they can only obtain these licenses at the rate of at most one per month.
Each license is currently selling for a price of $100. However, they are all becoming more expensive according to exponential growth curves: in particular, the cost of license j increases by a factor of rj > 1 each month, where rj is a given parameter. This means that if license j is purchased t months from now, it will cost 100 · rjt. We will assume that all the price growth rates are distinct; that is, ri ̸= rj for licenses i ̸= j (even though they start at the same price of $100).

Solved Exercises

185

186

Chapter 4 Greedy Algorithms

The question is: Given that the company can only buy at most one license a month, in which order should it buy the licenses so that the total amount of money it spends is as small as possible?

Give an algorithm that takes the n rates of price growth r1, r2, . . . , rn, and computes an order in which to buy the licenses so that the total amount of money spent is minimized. The running time of your algorithm should be polynomial in n.

Solution Two natural guesses for a good sequence would be to sort the ri in decreasing order, or to sort them in increasing order. Faced with alternatives like this, it’s perfectly reasonable to work out a small example and see if the example eliminates at least one of them. Here we could try r1 = 2, r2 = 3, and r3 = 4. Buying the licenses in increasing order results in a total cost of

100(2 + 32 + 43) = 7,500,

while buying them in decreasing order results in a total cost of

100(4 + 32 + 23) = 2,100.

This tells us that increasing order is not the way to go. (On the other hand, it doesn’t tell us immediately that decreasing order is the right answer, but our goal was just to eliminate one of the two options.)

Let’s try proving that sorting the ri in decreasing order in fact always gives the optimal solution. When a greedy algorithm works for problems like this, in which we put a set of things in an optimal order, we’ve seen in the text that it’s often effective to try proving correctness using an exchange argument.

To do this here, let’s suppose that there is an optimal solution O that differs from our solution S. (In other words, S consists of the licenses sorted in decreasing order.) So this optimal solution O must contain an inversion—that is, there must exist two neighboring months t and t + 1 such that the price increase rate of the license bought in month t (let us denote it by rt) is less than that bought in month t + 1 (similarly, we use rt+1 to denote this). That is, we have rt < rt+1.
We claim that by exchanging these two purchases, we can strictly improve our optimal solution, which contradicts the assumption that O was optimal. Therefore if we succeed in showing this, we will successfully show that our algorithm is indeed the correct one.
Notice that if we swap these two purchases, the rest of the purchases
are identically priced. In O, the amount paid during the two months involved
in the swap is 100(rt + rt+1). On the other hand, if we swapped these two t t+1
purchases, we would pay 100(rt + rt+1). Since the constant 100 is common t+1 t
to both expressions, we want to show that the second term is less than the first one. So we want to show that
rt +rt+1

194

Chapter 4 Greedy Algorithms

The Problem. Given a set of n streams, each specified by its number of bits bi and its time duration ti, as well as the link parameter r, determine whether there exists a valid schedule.

Example. Suppose we have n = 3 streams, with

(b1, t1) = (2000, 1), (b2, t2) = (6000, 2), (b3, t3) = (2000, 1),

and suppose the link’s parameter is r = 5000. Then the schedule that runs the streams in the order 1, 2, 3, is valid, since the constraint (∗) is satisfied:

t = 1: the whole first stream has been sent, and 2000 < 5000 · 1 t = 2: half of the second stream has also been sent,
and 2000+3000<5000·2
Similar calculations hold for t = 3 and t = 4.
(a) Consider the following claim:
Claim: There exists a valid schedule if and only if each stream i satisfies
bi ≤ rti.
Decide whether you think the claim is true or false, and give a proof
of either the claim or its negation.
(b) Give an algorithm that takes a set of n streams, each specified by its number of bits bi and its time duration ti, as well as the link parameter r, and determines whether there exists a valid schedule. The running time of your algorithm should be polynomial in n.
13. A small business—say, a photocopying service with a single large machine—faces the following scheduling problem. Each morning they get a set of jobs from customers. They want to do the jobs on their single machine in an order that keeps their customers happiest. Customer i’s job will take ti time to complete. Given a schedule (i.e., an ordering of the jobs), let Ci denote the finishing time of job i. For example, if job j is the first to be done, we would have Cj = tj; and if job j is done right after job i, we would have Cj = Ci + tj. Each customer i also has a given weight wi that represents his or her importance to the business. The happiness of customer i is expected to be dependent on the finishing time of i’s job. So the company decides that they want to order the jobs to minimize the weighted sum of the completion times, ni=1 wiCi.
Design an efficient algorithm to solve this problem. That is, you are given a set of n jobs with a processing time ti and a weight wi for each job. You want to order the jobs so as to minimize the weighted sum of the completion times, ni=1 wiCi.
Example. Suppose there are two jobs: the first takes time t1 = 1 and has weight w1 = 10, while the second job takes time t2 = 3 and has weight
w2 = 2. Then doing job 1 first would yield a weighted completion time of 10 · 1 + 2 · 4 = 18, while doing the second job first would yield the larger weighted completion time of 10 · 4 + 2 · 3 = 46.
14. You’re working with a group of security consultants who are helping to monitor a large computer system. There’s particular interest in keeping track of processes that are labeled “sensitive.” Each such process has a designated start time and finish time, and it runs continuously between these times; the consultants have a list of the planned start and finish times of all sensitive processes that will be run that day.
As a simple first step, they’ve written a program called status_check that, when invoked, runs for a few seconds and records various pieces of logging information about all the sensitive processes running on the system at that moment. (We’ll model each invocation of status_check as lasting for only this single point in time.) What they’d like to do is to run status_check as few times as possible during the day, but enough that for each sensitive process P, status_check is invoked at least once during the execution of process P.
(a) Give an efficient algorithm that, given the start and finish times of all the sensitive processes, finds as small a set of times as possi- ble at which to invoke status_check, subject to the requirement that status_check is invoked at least once during each sensitive process P.
(b) While you were designing your algorithm, the security consultants were engaging in a little back-of-the-envelope reasoning. “Suppose we can find a set of k sensitive processes with the property that no two are ever running at the same time. Then clearly your algorithm will need to invoke status_check at least k times: no one invocation of status_check can handle more than one of these processes.”
This is true, of course, and after some further discussion, you all begin wondering whether something stronger is true as well, a kind of converse to the above argument. Suppose that k∗ is the largest value of k such that one can find a set of k sensitive processes with no two ever running at the same time. Is it the case that there must be a set of k∗ times at which you can run status_check so that some invocation occurs during the execution of each sensitive process? (In other words, the kind of argument in the previous paragraph is really the only thing forcing you to need a lot of invocations of status_ check.) Decide whether you think this claim is true or false, and give a proof or a counterexample.
Exercises
195
196
Chapter 4 Greedy Algorithms
15. The manager of a large student union on campus comes to you with the following problem. She’s in charge of a group of n students, each of whom is scheduled to work one shift during the week. There are different jobs associated with these shifts (tending the main desk, helping with package delivery, rebooting cranky information kiosks, etc.), but we can view each shift as a single contiguous interval of time. There can be multiple shifts going on at once.
She’s trying to choose a subset of these n students to form a super- vising committee that she can meet with once a week. She considers such a committee to be complete if, for every student not on the committee, that student’s shift overlaps (at least partially) the shift of some student who is on the committee. In this way, each student’s performance can be observed by at least one person who’s serving on the committee.
Give an efficient algorithm that takes the schedule of n shifts and produces a complete supervising committee containing as few students as possible.
Example. Suppose n = 3, and the shifts are
Monday 4 P.M.–Monday 8 P.M., Monday 6 P.M.–Monday 10 P.M., Monday 9 P.M.–Monday 11 P.M..
Then the smallest complete supervising committee would consist of just the second student, since the second shift overlaps both the first and the third.
16. Some security consultants working in the financial domain are cur- rently advising a client who is investigating a potential money-laundering scheme. The investigation thus far has indicated that n suspicious trans- actions took place in recent days, each involving money transferred into a single account. Unfortunately, the sketchy nature of the evidence to date means that they don’t know the identity of the account, the amounts of the transactions, or the exact times at which the transactions took place. What they do have is an approximate time-stamp for each transaction; the evidence indicates that transaction i took place at time ti ± ei, for some “margin of error” ei. (In other words, it took place sometime between ti − ei and ti + ei.) Note that different transactions may have different margins of error.
In the last day or so, they’ve come across a bank account that (for other reasons we don’t need to go into here) they suspect might be the one involved in the crime. There are n recent events involving the account, which took place at times x1, x2, . . . , xn. To see whether it’s plausible that this really is the account they’re looking for, they’re wondering
whether it’s possible to associate each of the account’s n events with a distinct one of the n suspicious transactions in such a way that, if the account event at time xi is associated with the suspicious transaction that occurred approximately at time tj, then |tj − xi| ≤ ej. (In other words, they want to know if the activity on the account lines up with the suspicious transactions to within the margin of error; the tricky part here is that they don’t know which account event to associate with which suspicious transaction.)
Give an efficient algorithm that takes the given data and decides whether such an association exists. If possible, you should make the running time be at most O(n2).
17. ConsiderthefollowingvariationontheIntervalSchedulingProblem.You have a processor that can operate 24 hours a day, every day. People submit requests to run daily jobs on the processor. Each such job comes with a start time and an end time; if the job is accepted to run on the processor, it must run continuously, every day, for the period between its start and end times. (Note that certain jobs can begin before midnight and end after midnight; this makes for a type of situation different from what we saw in the Interval Scheduling Problem.)
Given a list of n such jobs, your goal is to accept as many jobs as possible (regardless of their length), subject to the constraint that the processor can run at most one job at any given point in time. Provide an algorithm to do this with a running time that is polynomial in n. You may assume for simplicity that no two jobs have the same start or end times.
Example. Consider the following four jobs, specified by (start-time, end- time) pairs.
(6 P.M., 6 A.M.), (9 P.M., 4 A.M.), (3 A.M., 2 P.M.), (1 P.M., 7 P.M.).
The optimal solution would be to pick the two jobs (9 P.M., 4 A.M.) and (1 P.M., 7 P.M.), which can be scheduled without overlapping.
18. YourfriendsareplanninganexpeditiontoasmalltowndeepintheCana- dian north next winter break. They’ve researched all the travel options and have drawn up a directed graph whose nodes represent intermediate destinations and edges represent the roads between them.
In the course of this, they’ve also learned that extreme weather causes roads in this part of the world to become quite slow in the winter and may cause large travel delays. They’ve found an excellent travel Web site that can accurately predict how fast they’ll be able to travel along the roads; however, the speed of travel depends on the time of year. More precisely, the Web site answers queries of the following form: given an
Exercises
197
198
Chapter 4 Greedy Algorithms
edge e = (v, w) connecting two sites v and w, and given a proposed starting time t from location v, the site will return a value fe(t), the predicted arrival time at w. The Web site guarantees that fe(t) ≥ t for all edges e and all times t (you can’t travel backward in time), and that fe(t) is a monotone increasing function of t (that is, you do not arrive earlier by starting later). Other than that, the functions fe(t) may be arbitrary. For example, in areas where the travel time does not vary with the season, we would have fe(t) = t + le, where le is the time needed to travel from the beginning to the end of edge e.
Your friends want to use the Web site to determine the fastest way to travel through the directed graph from their starting point to their intended destination. (You should assume that they start at time 0, and that all predictions made by the Web site are completely correct.) Give a polynomial-time algorithm to do this, where we treat a single query to the Web site (based on a specific edge e and a time t) as taking a single computational step.
19. A group of network designers at the communications company CluNet find themselves facing the following problem. They have a connected graph G = (V , E), in which the nodes represent sites that want to com- municate. Each edge e is a communication link, with a given available bandwidth be.
For each pair of nodes u, v ∈ V, they want to select a single u-v path P on which this pair will communicate. The bottleneck rate b(P) of this path P is the minimum bandwidth of any edge it contains; that is, b(P) = mine∈P be. The best achievable bottleneck rate for the pair u, v in G is simply the maximum, over all u-v paths P in G, of the value b(P).
It’s getting to be very complicated to keep track of a path for each pair of nodes, and so one of the network designers makes a bold suggestion: Maybe one can find a spanning tree T of G so that for every pair of nodes u, v, the unique u-v path in the tree actually attains the best achievable bottleneck rate for u, v in G. (In other words, even if you could choose any u-v path in the whole graph, you couldn’t do better than the u-v path in T.)
This idea is roundly heckled in the offices of CluNet for a few days, and there’s a natural reason for the skepticism: each pair of nodes might want a very different-looking path to maximize its bottleneck rate; why should there be a single tree that simultaneously makes everybody happy? But after some failed attempts to rule out the idea, people begin to suspect it could be possible.
Show that such a tree exists, and give an efficient algorithm to find one. That is, give an algorithm constructing a spanning tree T in which, for each u, v ∈ V, the bottleneck rate of the u-v path in T is equal to the best achievable bottleneck rate for the pair u, v in G.
20. Every September, somewhere in a far-away mountainous part of the world, the county highway crews get together and decide which roads to keep clear through the coming winter. There are n towns in this county, and the road system can be viewed as a (connected) graph G = (V , E) on this set of towns, each edge representing a road joining two of them. In the winter, people are high enough up in the mountains that they stop worrying about the length of roads and start worrying about their altitude—this is really what determines how difficult the trip will be.
So each road—each edge e in the graph—is annotated with a number ae that gives the altitude of the highest point on the road. We’ll assume that no two edges have exactly the same altitude value ae. The height of a path P in the graph is then the maximum of ae over all edges e on P. Finally, a path between towns i and j is declared to be winter-optimal if it achieves the minimum possible height over all paths from i to j.
The highway crews are going to select a set E′ ⊆ E of the roads to keep clear through the winter; the rest will be left unmaintained and kept off limits to travelers. They all agree that whichever subset of roads E′ they decide to keep clear, it should have the property that (V , E′) is a connected subgraph; and more strongly, for every pair of towns i and j, the height of the winter-optimal path in (V , E′) should be no greater than it is in the full graph G = (V , E). We’ll say that (V , E′) is a minimum-altitude connected subgraph if it has this property.
Given that they’re going to maintain this key property, however, they otherwise want to keep as few roads clear as possible. One year, they hit upon the following conjecture:
The minimum spanning tree of G, with respect to the edge weights ae, is a minimum-altitude connected subgraph.
(In an earlier problem, we claimed that there is a unique minimum span- ning tree when the edge weights are distinct. Thus, thanks to the assump- tion that all ae are distinct, it is okay for us to speak of the minimum spanning tree.)
Initially, this conjecture is somewhat counterintuitive, since the min- imum spanning tree is trying to minimize the sum of the values ae, while the goal of minimizing altitude seems to be asking for a fairly different thing. But lacking an argument to the contrary, they begin considering an even bolder second conjecture:
Exercises
199
200
Chapter 4 Greedy Algorithms
A subgraph (V , E′) is a minimum-altitude connected subgraph if and only if it contains the edges of the minimum spanning tree.
Note that this second conjecture would immediately imply the first one, since a minimum spanning tree contains its own edges.
So here’s the question.
(a) Is the first conjecture true, for all choices of G and distinct altitudes
ae? Give a proof or a counterexample with explanation.
(b) Is the second conjecture true, for all choices of G and distinct alti-
tudes ae? Give a proof or a counterexample with explanation.
21. LetussaythatagraphG=(V,E)isanear-treeifitisconnectedandhasat most n + 8 edges, where n = |V|. Give an algorithm with running time O(n) that takes a near-tree G with costs on its edges, and returns a minimum spanning tree of G. You may assume that all the edge costs are distinct.
22. Consider the Minimum Spanning Tree Problem on an undirected graph G=(V,E), with a cost ce ≥0 on each edge, where the costs may not all be different. If the costs are not all distinct, there can in general be many distinct minimum-cost solutions. Suppose we are given a spanning tree T ⊆ E with the guarantee that for every e ∈ T, e belongs to some minimum-cost spanning tree in G. Can we conclude that T itself must be a minimum-cost spanning tree in G? Give a proof or a counterexample with explanation.
23. Recall the problem of computing a minimum-cost arborescence in a directed graph G = (V , E), with a cost ce ≥ 0 on each edge. Here we will consider the case in which G is a directed acyclic graph—that is, it contains no directed cycles.
As in general directed graphs, there can be many distinct minimum- cost solutions. Suppose we are given a directed acyclic graph G = (V , E), and an arborescence A ⊆ E with the guarantee that for every e ∈ A, e belongs to some minimum-cost arborescence in G. Can we conclude that A itself must be a minimum-cost arborescence in G? Give a proof or a counterexample with explanation.
24. Timing circuits are a crucial component of VLSI chips. Here’s a simple model of such a timing circuit. Consider a complete balanced binary tree with n leaves, where n is a power of two. Each edge e of the tree has an associated length le, which is a positive number. The distance from the root to a given leaf is the sum of the lengths of all the edges on the path from the root to the leaf.
Exercises
201
v
21
v 2121
abcd
Figure 4.20 An instance of the zero-skew problem, described in Exercise 23.
The root generates a clock signal which is propagated along the edges to the leaves. We’ll assume that the time it takes for the signal to reach a given leaf is proportional to the distance from the root to the leaf.
Now, if all leaves do not have the same distance from the root, then the signal will not reach the leaves at the same time, and this is a big problem. We want the leaves to be completely synchronized, and all to receive the signal at the same time. To make this happen, we will have to increase the lengths of certain edges, so that all root-to-leaf paths have the same length (we’re not able to shrink edge lengths). If we achieve this, then the tree (with its new edge lengths) will be said to have zero skew. Our goal is to achieve zero skew in a way that keeps the sum of all the edge lengths as small as possible.
Give an algorithm that increases the lengths of certain edges so that the resulting tree has zero skew and the total edge length is as small as possible.
Example. Consider the tree in Figure 4.20, in which letters name the nodes and numbers indicate the edge lengths.
The unique optimal solution for this instance would be to take the three length-1 edges and increase each of their lengths to 2. The resulting tree has zero skew, and the total edge length is 12, the smallest possible.
25. Suppose we are given a set of points P = {p1, p2, . . . , pn}, together with a distance function d on the set P; d is simply a function on pairs of points in P with the properties that d(pi, pj) = d(pj, pi) > 0 if i ̸= j, and that d(pi, pi) = 0 for each i.

We define a hierarchical metric on P to be any distance function τ that can be constructed as follows. We build a rooted tree T with n leaves, and we associate with each node v of T (both leaves and internal nodes) a height hv. These heights must satisfy the properties that h(v) = 0 for each

v

202

Chapter 4 Greedy Algorithms

leaf v, and if u is the parent of v in T, then h(u) ≥ h(v). We place each point in P at a distinct leaf in T. Now, for any pair of points pi and pj, their distance τ(pi, pj) is defined as follows. We determine the least common ancestor v in T of the leaves containing pi and pj, and define τ(pi, pj) = hv.

We say that a hierarchical metric τ is consistent with our distance function d if, for all pairs i, j, we have τ(pi, pj) ≤ d(pi, pj).

Give a polynomial-time algorithm that takes the distance function d and produces a hierarchical metric τ with the following properties.

(i) τ is consistent with d, and

(ii) ifτ′isanyotherhierarchicalmetricconsistentwithd,thenτ′(pi,pj)≤

τ(pi, pj) for each pair of points pi and pj.

26. One of the first things you learn in calculus is how to minimize a dif- ferentiable function such as y = ax2 + bx + c, where a > 0. The Minimum Spanning Tree Problem, on the other hand, is a minimization problem of a very different flavor: there are now just a finite number of possibilities for how the minimum might be achieved—rather than a continuum of possibilities—and we are interested in how to perform the computation without having to exhaust this (huge) finite number of possibilities.

One can ask what happens when these two minimization issues are brought together, and the following question is an example of this. Suppose we have a connected graph G = (V , E). Each edge e now has a time- varying edge cost given by a function fe :R→R. Thus, at time t, it has cost fe(t). We’ll assume that all these functions are positive over their entire range. Observe that the set of edges constituting the minimum spanning tree of G may change over time. Also, of course, the cost of the minimum spanning tree of G becomes a function of the time t; we’ll denote this function cG(t). A natural problem then becomes: find a value of t at which cG(t) is minimized.

Suppose each function fe is a polynomial of degree 2: fe(t) = aet2 + be t + ce , where ae > 0. Give an algorithm that takes the graph G and the values {(ae,be,ce):e∈E} and returns a value of the time t at which the minimum spanning tree has minimum cost. Your algorithm should run in time polynomial in the number of nodes and edges of the graph G. You may assume that arithmetic operations on the numbers {(ae , be , ce )} can be done in constant time per operation.

27. In trying to understand the combinatorial structure of spanning trees, we can consider the space of all possible spanning trees of a given graph and study the properties of this space. This is a strategy that has been applied to many similar problems as well.

Here is one way to do this. Let G be a connected graph, and T and T′ two different spanning trees of G. We say that T and T′ are neighbors if T contains exactly one edge that is not in T′, and T′ contains exactly one edge that is not in T.

Now, from any graph G, we can build a (large) graph H as follows. The nodes of H are the spanning trees of G, and there is an edge between two nodes of H if the corresponding spanning trees are neighbors.

Is it true that, for any connected graph G, the resulting graph H is connected? Give a proof that H is always connected, or provide an example (with explanation) of a connected graph G for which H is not connected.

28. Suppose you’re a consultant for the networking company CluNet, and they have the following problem. The network that they’re currently working on is modeled by a connected graph G = (V , E) with n nodes. Each edge e is a fiber-optic cable that is owned by one of two companies— creatively named X and Y—and leased to CluNet.

Their plan is to choose a spanning tree T of G and upgrade the links corresponding to the edges of T. Their business relations people have already concluded an agreement with companies X and Y stipulating a number k so that in the tree T that is chosen, k of the edges will be owned byX andn−k−1oftheedgeswillbeownedbyY.

CluNet management now faces the following problem. It is not at all clear to them whether there even exists a spanning tree T meeting these conditions, or how to find one if it exists. So this is the problem they put to you: Give a polynomial-time algorithm that takes G, with each edge labeled X or Y, and either (i) returns a spanning tree with exactly k edges labeled X, or (ii) reports correctly that no such tree exists.

29. Given a list of n natural numbers d1, d2, . . . , dn, show how to decide in polynomial time whether there exists an undirected graph G = (V , E) whose node degrees are precisely the numbers d1, d2, . . . , dn. (That is, if V = {v1, v2, . . . , vn}, then the degree of vi should be exactly di.) G should not contain multiple edges between the same pair of nodes, or “loop” edges with both endpoints equal to the same node.

30. Let G = (V, E) be a graph with n nodes in which each pair of nodes is joined by an edge. There is a positive weight wij on each edge (i, j); and we will assume these weights satisfy the triangle inequality wik ≤ wij + wjk . For a subset V′ ⊆ V, we will use G[V′] to denote the subgraph (with edge weights) induced on the nodes in V′.

Exercises

203

204

Chapter 4 Greedy Algorithms

We are given a set X ⊆ V of k terminals that must be connected by edges. We say that a Steiner tree on X is a set Z so that X ⊆ Z ⊆ V, together with a spanning subtree T of G[Z]. The weight of the Steiner tree is the weight of the tree T.

Show that the problem of finding a minimum-weight Steiner tree on X can be solved in time O(nO(k)).

31. Let’s go back to the original motivation for the Minimum Spanning Tree Problem. We are given a connected, undirected graph G = (V , E) with positive edge lengths {le}, and we want to find a spanning subgraph of it. Now suppose we are willing to settle for a subgraph H = (V , F ) that is “denser” than a tree, and we are interested in guaranteeing that, for each pair of vertices u, v ∈ V, the length of the shortest u-v path in H is not much longer than the length of the shortest u-v path in G. By the length ofapathPhere,wemeanthesumofle overalledgeseinP.

Here’s a variant of Kruskal’s Algorithm designed to produce such a subgraph.

. First we sort all the edges in order of increasing length. (You may assume all edge lengths are distinct.)

. We then construct a subgraph H = (V , F ) by considering each edge in order.

. Whenwecometoedgee=(u,v),weaddetothesubgraphHifthere is currently no u-v path in H. (This is what Kruskal’s Algorithm would do as well.) On the other hand, if there is a u-v path in H, we let duv denote the length of the shortest such path; again, length is with respect to the values {le}. We add e to H if 3le < duv.
In other words, we add an edge even when u and v are already in the same connected component, provided that the addition of the edge reduces their shortest-path distance by a sufficient amount.
Let H = (V , F ) be the subgraph of G returned by the algorithm.
(a) Prove that for every pair of nodes u, v ∈ V, the length of the shortest u-v path in H is at most three times the length of the shortest u-v path in G.
(b) Despite its ability to approximately preserve shortest-path distances, the subgraph H produced by the algorithm cannot be too dense. Let f(n) denote the maximum number of edges that can possibly be produced as the output of this algorithm, over all n-node input graphs with edge lengths. Prove that
lim f(n)=0. n→∞ n2
32. Consider a directed graph G = (V , E) with a root r ∈ V and nonnegative costs on the edges. In this problem we consider variants of the minimum- cost arborescence algorithm.
(a) The algorithm discussed in Section 4.9 works as follows. We modify the costs, consider the subgraph of zero-cost edges, look for a directed cycle in this subgraph, and contract it (if one exists). Argue briefly that instead of looking for cycles, we can instead identify and contract strong components of this subgraph.
(b) In the course of the algorithm, we defined yv to be the minimum
cost of an edge entering v, and we modified the costs of all edges e
entering node v to be ce′ = ce − yv. Suppose we instead use the follow-
ing modified cost: c′′ = max(0, c − 2y ). This new change is likely to eev
turn more edges to 0 cost. Suppose now we find an arborescence T of 0 cost. Prove that this T has cost at most twice the cost of the minimum-cost arborescence in the original graph.
(c) Assume you do not find an arborescence of 0 cost. Contract all 0- cost strong components and recursively apply the same procedure on the resulting graph until an arborescence is found. Prove that this T has cost at most twice the cost of the minimum-cost arborescence in the original graph.
33. SupposeyouaregivenadirectedgraphG=(V,E)inwhicheachedgehas a cost of either 0 or 1. Also suppose that G has a node r such that there is a path from r to every other node in G. You are also given an integer k. Give a polynomial-time algorithm that either constructs an arborescence rooted at r of cost exactly k, or reports (correctly) that no such arborescence exists.
Notes and Further Reading
Due to their conceptual cleanness and intuitive appeal, greedy algorithms have a long history and many applications throughout computer science. In this chapter we focused on cases in which greedy algorithms find the optimal solution. Greedy algorithms are also often used as simple heuristics even when they are not guaranteed to find the optimal solution. In Chapter 11 we will discuss greedy algorithms that find near-optimal approximate solutions.
As discussed in Chapter 1, Interval Scheduling can be viewed as a special case of the Independent Set Problem on a graph that represents the overlaps among a collection of intervals. Graphs arising this way are called interval graphs, and they have been extensively studied; see, for example, the book by Golumbic (1980). Not just Independent Set but many hard computational
Notes and Further Reading
205
206
Chapter 4 Greedy Algorithms
problems become much more tractable when restricted to the special case of interval graphs.
Interval Scheduling and the problem of scheduling to minimize the max- imum lateness are two of a range of basic scheduling problems for which a simple greedy algorithm can be shown to produce an optimal solution. A wealth of related problems can be found in the survey by Lawler, Lenstra, Rinnooy Kan, and Shmoys (1993).
The optimal algorithm for caching and its analysis are due to Belady (1966). As we mentioned in the text, under real operating conditions caching algorithms must make eviction decisions in real time without knowledge of future requests. We will discuss such caching strategies in Chapter 13.
The algorithm for shortest paths in a graph with nonnegative edge lengths is due to Dijkstra (1959). Surveys of approaches to the Minimum Spanning Tree Problem, together with historical background, can be found in the reviews by Graham and Hell (1985) and Nesetril (1997).
The single-link algorithm is one of the most widely used approaches to the general problem of clustering; the books by Anderberg (1973), Duda, Hart, and Stork (2001), and Jain and Dubes (1981) survey a variety of clustering techniques.
The algorithm for optimal prefix codes is due to Huffman (1952); the ear- lier approaches mentioned in the text appear in the books by Fano (1949) and Shannon and Weaver (1949). General overviews of the area of data compres- sion can be found in the book by Bell, Cleary, and Witten (1990) and the survey by Lelewer and Hirschberg (1987). More generally, this topic belongs to the area of information theory, which is concerned with the representation and encoding of digital information. One of the founding works in this field is the book by Shannon and Weaver (1949), and the more recent textbook by Cover and Thomas (1991) provides detailed coverage of the subject.
The algorithm for finding minimum-cost arborescences is generally cred- ited to Chu and Liu (1965) and to Edmonds (1967) independently. As discussed in the chapter, this multi-phase approach stretches our notion of what consti- tutes a greedy algorithm. It is also important from the perspective of linear programming, since in that context it can be viewed as a fundamental ap- plication of the pricing method, or the primal-dual technique, for designing algorithms. The book by Nemhauser and Wolsey (1988) develops these con- nections to linear programming. We will discuss this method in Chapter 11 in the context of approximation algorithms.
More generally, as we discussed at the outset of the chapter, it is hard to find a precise definition of what constitutes a greedy algorithm. In the search for such a definition, it is not even clear that one can apply the analogue
of U.S. Supreme Court Justice Potter Stewart’s famous test for obscenity— “I know it when I see it”—since one finds disagreements within the research community on what constitutes the boundary, even intuitively, between greedy and nongreedy algorithms. There has been research aimed at formalizing classes of greedy algorithms: the theory of matroids is one very influential example (Edmonds 1971; Lawler 2001); and the paper of Borodin, Nielsen, and Rackoff (2002) formalizes notions of greedy and “greedy-type” algorithms, as well as providing a comparison to other formal work on this question.
Notes on the Exercises Exercise 24 is based on results of M. Edahiro, T. Chao, Y. Hsu, J. Ho, K. Boese, and A. Kahng; Exercise 31 is based on a result of Ingo Althofer, Gautam Das, David Dobkin, and Deborah Joseph.
Notes and Further Reading
207
This page intentionally left blank
Chapter 5 Divide and Conquer
Divide and conquer refers to a class of algorithmic techniques in which one breaks the input into several parts, solves the problem in each part recursively, and then combines the solutions to these subproblems into an overall solution. In many cases, it can be a simple and powerful method.
Analyzing the running time of a divide and conquer algorithm generally involves solving a recurrence relation that bounds the running time recursively in terms of the running time on smaller instances. We begin the chapter with a general discussion of recurrence relations, illustrating how they arise in the analysis and describing methods for working out upper bounds from them.
We then illustrate the use of divide and conquer with applications to a number of different domains: computing a distance function on different rankings of a set of objects; finding the closest pair of points in the plane; multiplying two integers; and smoothing a noisy signal. Divide and conquer will also come up in subsequent chapters, since it is a method that often works well when combined with other algorithm design techniques. For example, in Chapter 6 we will see it combined with dynamic programming to produce a space-efficient solution to a fundamental sequence comparison problem, and in Chapter 13 we will see it combined with randomization to yield a simple and efficient algorithm for computing the median of a set of numbers.
One thing to note about many settings in which divide and conquer is applied, including these, is that the natural brute-force algorithm may already be polynomial time, and the divide and conquer strategy is serving to reduce the running time to a lower polynomial. This is in contrast to most of the problems in the previous chapters, for example, where brute force was exponential and the goal in designing a more sophisticated algorithm was to achieve any kind of polynomial running time. For example, we discussed in
210
Chapter 5 Divide and Conquer
Chapter 2 that the natural brute-force algorithm for finding the closest pair among n points in the plane would simply measure all (n2) distances, for a (polynomial) running time of (n2). Using divide and conquer, we will improve the running time to O(n log n). At a high level, then, the overall theme of this chapter is the same as what we’ve been seeing earlier: that improving on brute-force search is a fundamental conceptual hurdle in solving a problem efficiently, and the design of sophisticated algorithms can achieve this. The difference is simply that the distinction between brute-force search and an improved solution here will not always be the distinction between exponential and polynomial.
5.1 A First Recurrence: The Mergesort Algorithm
To motivate the general approach to analyzing divide-and-conquer algorithms, we begin with the Mergesort Algorithm. We discussed the Mergesort Algorithm briefly in Chapter 2, when we surveyed common running times for algorithms. Mergesort sorts a given list of numbers by first dividing them into two equal halves, sorting each half separately by recursion, and then combining the results of these recursive calls—in the form of the two sorted halves—using the linear-time algorithm for merging sorted lists that we saw in Chapter 2.
To analyze the running time of Mergesort, we will abstract its behavior into the following template, which describes many common divide-and-conquer algorithms.
(†) Divide the input into two pieces of equal size; solve the two subproblems on these pieces separately by recursion; and then combine the two results into an overall solution, spending only linear time for the initial division and final recombining.
In Mergesort, as in any algorithm that fits this style, we also need a base case for the recursion, typically having it “bottom out” on inputs of some constant size. In the case of Mergesort, we will assume that once the input has been reduced to size 2, we stop the recursion and sort the two elements by simply comparing them to each other.
Consider any algorithm that fits the pattern in (†), and let T(n) denote its worst-case running time on input instances of size n. Supposing that n is even, the algorithm spends O(n) time to divide the input into two pieces of size n/2 each; it then spends time T(n/2) to solve each one (since T(n/2) is the worst- case running time for an input of size n/2); and finally it spends O(n) time to combine the solutions from the two recursive calls. Thus the running time T(n) satisfies the following recurrence relation.
(5.1) For some constant c,
when n > 2, and

T(n) ≤ 2T(n/2) + cn

T(2)≤c.

5.1 A First Recurrence: The Mergesort Algorithm

211

The structure of (5.1) is typical of what recurrences will look like: there’s an inequality or equation that bounds T(n) in terms of an expression involving T(k) for smaller values k; and there is a base case that generally says that T(n) is equal to a constant when n is a constant. Note that one can also write (5.1) more informally as T(n) ≤ 2T(n/2) + O(n), suppressing the constant c. However, it is generally useful to make c explicit when analyzing the recurrence.

To keep the exposition simpler, we will generally assume that parameters like n are even when needed. This is somewhat imprecise usage; without this assumption, the two recursive calls would be on problems of size ⌈n/2⌉ and ⌊n/2⌋, and the recurrence relation would say that

T(n) ≤ T(⌈n/2⌉) + T(⌊n/2⌋) + cn

for n ≥ 2. Nevertheless, for all the recurrences we consider here (and for most that arise in practice), the asymptotic bounds are not affected by the decision to ignore all the floors and ceilings, and it makes the symbolic manipulation much cleaner.

Now (5.1) does not explicitly provide an asymptotic bound on the growth rate of the function T; rather, it specifies T(n) implicitly in terms of its values on smaller inputs. To obtain an explicit bound, we need to solve the recurrence relation so that T appears only on the left-hand side of the inequality, not the right-hand side as well.

Recurrence solving is a task that has been incorporated into a number of standard computer algebra systems, and the solution to many standard recurrences can now be found by automated means. It is still useful, however, to understand the process of solving recurrences and to recognize which recurrences lead to good running times, since the design of an efficient divide- and-conquer algorithm is heavily intertwined with an understanding of how a recurrence relation determines a running time.

Approaches to Solving Recurrences

There are two basic ways one can go about solving a recurrence, each of which we describe in more detail below.

212

Chapter 5 Divide and Conquer

. The most intuitively natural way to search for a solution to a recurrence is to “unroll” the recursion, accounting for the running time across the first few levels, and identify a pattern that can be continued as the recursion expands. One then sums the running times over all levels of the recursion (i.e., until it “bottoms out” on subproblems of constant size) and thereby arrives at a total running time.

. A second way is to start with a guess for the solution, substitute it into the recurrence relation, and check that it works. Formally, one justifies this plugging-in using an argument by induction on n. There is a useful variant of this method in which one has a general form for the solution, but does not have exact values for all the parameters. By leaving these parameters unspecified in the substitution, one can often work them out as needed.

We now discuss each of these approaches, using the recurrence in (5.1) as an example.

Unrolling the Mergesort Recurrence

Let’s start with the first approach to solving the recurrence in (5.1). The basic argument is depicted in Figure 5.1.

. Analyzing the first few levels: At the first level of recursion, we have a single problem of size n, which takes time at most cn plus the time spent in all subsequent recursive calls. At the next level, we have two problems each of size n/2. Each of these takes time at most cn/2, for a total of at most cn, again plus the time in subsequent recursive calls. At the third level, we have four problems each of size n/4, each taking time at most cn/4, for a total of at most cn.

cn/2

cn/4 cn/4

cn/2

cn

Level 0: cn

Level 1: cn/2 + cn/2 = cn total

cn/4 Level 2: 4(cn/4) = cn total

Figure 5.1

Unrolling the recurrence T(n) ≤ 2T(n/2) + O(n).

cn/4

5.1 A First Recurrence: The Mergesort Algorithm

213

. Identifying a pattern: What’s going on in general? At level j of the recursion, the number of subproblems has doubled j times, so there are now a total of 2j. Each has correspondingly shrunk in size by a factor of two j times, and so each has size n/2j, and hence each takes time at most cn/2j. Thus level j contributes a total of at most 2j(cn/2j) = cn to the total running time.

. Summing over all levels of recursion: We’ve found that the recurrence in (5.1) has the property that the same upper bound of cn applies to total amount of work performed at each level. The number of times the input must be halved in order to reduce its size from n to 2 is log2 n. So summing the cn work over log n levels of recursion, we get a total running time of O(n log n).

We summarize this in the following claim.

(5.2) Any function T(·) satisfying (5.1) is bounded by O(n log n), when

n > 1.

Substituting a Solution into the Mergesort Recurrence

The argument establishing (5.2) can be used to determine that the function T(n) is bounded by O(n log n). If, on the other hand, we have a guess for the running time that we want to verify, we can do so by plugging it into the recurrence as follows.

Suppose we believe that T(n) ≤ cn log2 n for all n ≥ 2, and we want to check whether this is indeed true. This clearly holds for n = 2, since in this case cn log2 n = 2c, and (5.1) explicitly tells us that T(2) ≤ c. Now suppose, by induction, that T(m) ≤ cm log2 m for all values of m less than n, and we want to establish this for T(n). We do this by writing the recurrence for T(n) and plugging in the inequality T(n/2) ≤ c(n/2) log2(n/2). We then simplify the resulting expression by noticing that log2(n/2) = (log2 n) − 1. Here is the full calculation.

T(n) ≤ 2T(n/2) + cn

≤ 2c(n/2) log2(n/2) + cn = cn[(log2 n) − 1] + cn =(cnlog2 n)−cn+cn = cn log2 n.

This establishes the bound we want for T(n), assuming it holds for smaller values m < n, and thus it completes the induction argument.
214
Chapter 5 Divide and Conquer
An Approach Using Partial Substitution
There is a somewhat weaker kind of substitution one can do, in which one guesses the overall form of the solution without pinning down the exact values of all the constants and other parameters at the outset.
Specifically, suppose we believe that T(n) = O(n log n), but we’re not sure of the constant inside the O(·) notation. We can use the substitution method even without being sure of this constant, as follows. We first write T(n) ≤ kn logb n for some constant k and base b that we’ll determine later. (Actually, the base and the constant we’ll end up needing are related to each other, since we saw in Chapter 2 that one can change the base of the logarithm by simply changing the multiplicative constant in front.)
Now we’d like to know whether there is any choice of k and b that will work in an inductive argument. So we try out one level of the induction as follows.
T(n) ≤ 2T(n/2) + cn ≤ 2k(n/2) logb(n/2) + cn.
It’s now very tempting to choose the base b = 2 for the logarithm, since we see that this will let us apply the simplification log2(n/2) = (log2 n) − 1. Proceeding with this choice, we have
T(n) ≤ 2k(n/2) log2(n/2) + cn
= 2k(n/2)[(log2 n) − 1] + cn = kn[(log2 n) − 1] + cn =(knlog2 n)−kn+cn.
Finally, we ask: Is there a choice of k that will cause this last expression to be bounded by kn log2 n? The answer is clearly yes; we just need to choose any k that is at least as large as c, and we get
T(n)≤(knlog2 n)−kn+cn≤knlog2 n, which completes the induction.
Thus the substitution method can actually be useful in working out the exact constants when one has some guess of the general form of the solution.
5.2 Further Recurrence Relations
We’ve just worked out the solution to a recurrence relation, (5.1), that will come up in the design of several divide-and-conquer algorithms later in this chapter. As a way to explore this issue further, we now consider a class of recurrence relations that generalizes (5.1), and show how to solve the recurrences in this class. Other members of this class will arise in the design of algorithms both in this and in later chapters.
This more general class of algorithms is obtained by considering divide- and-conquer algorithms that create recursive calls on q subproblems of size n/2 each and then combine the results in O(n) time. This corresponds to the Mergesort recurrence (5.1) when q = 2 recursive calls are used, but other algorithms find it useful to spawn q > 2 recursive calls, or just a single (q = 1) recursive call. In fact, we will see the case q > 2 later in this chapter when we design algorithms for integer multiplication; and we will see a variant on the case q = 1 much later in the book, when we design a randomized algorithm for median finding in Chapter 13.

If T(n) denotes the running time of an algorithm designed in this style, then T(n) obeys the following recurrence relation, which directly generalizes (5.1) by replacing 2 with q:

(5.3) For some constant c,

when n > 2, and

T(n) ≤ qT(n/2) + cn

T(2)≤c.

We now describe how to solve (5.3) by the methods we’ve seen above: unrolling, substitution, and partial substitution. We treat the cases q > 2 and q = 1 separately, since they are qualitatively different from each other—and different from the case q = 2 as well.

The Case of q > 2 Subproblems

We begin by unrolling (5.3) in the case q > 2, following the style we used

earlier for (5.1). We will see that the punch line ends up being quite different.

. Analyzing the first few levels: We show an example of this for the case q = 3 in Figure 5.2. At the first level of recursion, we have a single problem of size n, which takes time at most cn plus the time spent in all subsequent recursive calls. At the next level, we have q problems, each of size n/2. Each of these takes time at most cn/2, for a total of at most (q/2)cn, again plus the time in subsequent recursive calls. The next level yields q2 problems of size n/4 each, for a total time of (q2/4)cn. Since q > 2, we see that the total work per level is increasing as we proceed through the recursion.

. Identifying a pattern: At an arbitrary level j, we have qj distinct instances, each of size n/2j. Thus the total work performed at level j is qj(cn/2j) = (q/2)jcn.

5.2 Further Recurrence Relations

215

216

Chapter 5 Divide and Conquer

cn time, plus recursive calls

Level 0: cn total

Level 1: cn/2 + cn/2 + cn/2 = (3/2)cn total

Level 2: 9(cn/4) = (9/4)cn total

cn/2

cn/4 cn/4 cn/4

cn/4

cn/2 cn/2

cn/4 cn/4 cn/4 cn/4 cn/4

Figure 5.2 Unrolling the recurrence T(n) ≤ 3T(n/2) + O(n).

. Summing over all levels of recursion: As before, there are log2 n levels of recursion, and the total amount of work performed is the sum over all these:

log n−1 log n−1 2qj2qj

T(n) ≤ cn = cn .

j=0 2

j=0 2

This is a geometric sum, consisting of powers of r = q/2. We can use the formula for a geometric sum when r > 1, which gives us the formula

rlog2 n − 1 rlog2 n T(n) ≤ cn r − 1 ≤ cn r − 1 .

Since we’re aiming for an asymptotic upper bound, it is useful to figure out what’s simply a constant; we can pull out the factor of r − 1 from the denominator, and write the last expression as

T(n) ≤ c nrlog2 n. r−1

Finally, we need to figure out what rlog2 n is. Here we use a very handy identity, which says that, for any a > 1 and b > 1, we have alog b = blog a. Thus

rlog2 n = nlog2 r = nlog2(q/2) = n(log2 q)−1.

Thus we have

T(n) ≤ c n · n(log2 q)−1 ≤ c nlog2 q = O(nlog2 q). r−1 r−1

We sum this up as follows.

(5.4) Any function T(·) satisfying (5.3) with q > 2 is bounded by O(nlog2 q).

So we find that the running time is more than linear, since log2 q > 1, but still polynomial in n. Plugging in specific values of q, the running time is O(nlog2 3) = O(n1.59) when q = 3; and the running time is O(nlog2 4) = O(n2) when q = 4. This increase in running time as q increases makes sense, of course, since the recursive calls generate more work for larger values of q.

Applying Partial Substitution The appearance of log2 q in the exponent followed naturally from our solution to (5.3), but it’s not necessarily an expression one would have guessed at the outset. We now consider how an approach based on partial substitution into the recurrence yields a different way of discovering this exponent.

Suppose we guess that the solution to (5.3), when q > 2, has the form T(n) ≤ knd for some constants k > 0 and d > 1. This is quite a general guess, since we haven’t even tried specifying the exponent d of the polynomial. Now let’s try starting the inductive argument and seeing what constraints we need on k and d. We have

T(n) ≤ qT(n/2) + cn,

and applying the inductive hypothesis to T(n/2), this expands to

nd T(n)≤qk +cn

2

= qknd+cn.

2d

This is remarkably close to something that works: if we choose d so that q/2d = 1, then we have T(n) ≤ knd + cn, which is almost right except for the extra term cn. So let’s deal with these two issues: first, how to choose d so we get q/2d = 1; and second, how to get rid of the cn term.

Choosing d is easy: we want 2d =q, and so d=log2 q. Thus we see that the exponent log2 q appears very naturally once we decide to discover which value of d works when substituted into the recurrence.

But we still have to get rid of the cn term. To do this, we change the form of our guess for T(n) so as to explicitly subtract it off. Suppose we try the form T(n) ≤ knd − ln, where we’ve now decided that d = log2 q but we haven’t fixed the constants k or l. Applying the new formula to T(n/2), this expands to

5.2 Further Recurrence Relations

217

218

Chapter 5

Divide and Conquer

nd n T(n)≤qk −ql +cn

22

= qknd−qln+cn 2d 2

=knd − qln+cn 2

=knd −(ql −c)n. 2

This now works completely, if we simply choose l so that ( ql − c) = l: in other 2

words, l = 2c/(q − 2). This completes the inductive step for n. We also need to handle the base case n = 2, and this we do using the fact that the value of k has not yet been fixed: we choose k large enough so that the formula is a valid upper bound for the case n = 2.

The Case of One Subproblem

We now consider the case of q = 1 in (5.3), since this illustrates an outcome of yet another flavor. While we won’t see a direct application of the recurrence for q = 1 in this chapter, a variation on it comes up in Chapter 13, as we mentioned earlier.

We begin by unrolling the recurrence to try constructing a solution.

. Analyzing the first few levels: We show the first few levels of the recursion in Figure 5.3. At the first level of recursion, we have a single problem of size n, which takes time at most cn plus the time spent in all subsequent recursive calls. The next level has one problem of size n/2, which contributes cn/2, and the level after that has one problem of size n/4, which contributes cn/4. So we see that, unlike the previous case, the total work per level when q = 1 is actually decreasing as we proceed through the recursion.

. Identifying a pattern: At an arbitrary level j, we still have just one instance; it has size n/2j and contributes cn/2j to the running time.

. Summing over all levels of recursion: There are log2 n levels of recursion, and the total amount of work performed is the sum over all these:

.

j=0 j=0

This geometric sum is very easy to work out; even if we continued it to

infinity, it would converge to 2. Thus we have T(n) ≤ 2cn = O(n).

log n−1 log n−1

2 cn 2 1 T(n)≤ 2j =cn 2j

cn time, plus recursive calls

Level 0: cn total

cn/2 Level 1: cn/2 total

5.2 Further Recurrence Relations

219

cn/4 Level 2: cn/4 total

Figure 5.3 Unrolling the recurrence T(n) ≤ T(n/2) + O(n).

We sum this up as follows.

(5.5) Any function T(·) satisfying (5.3) with q = 1 is bounded by O(n).

This is counterintuitive when you first see it. The algorithm is performing log n levels of recursion, but the overall running time is still linear in n. The point is that a geometric series with a decaying exponent is a powerful thing: fully half the work performed by the algorithm is being done at the top level of the recursion.

It is also useful to see how partial substitution into the recurrence works very well in this case. Suppose we guess, as before, that the form of the solution is T(n) ≤ knd. We now try to establish this by induction using (5.3), assuming that the solution holds for the smaller value n/2:

T(n) ≤ T(n/2) + cn nd

≤k +cn 2

= k nd + cn. 2d

If we now simply choose d=1and k=2c, we have T(n)≤ kn+cn=(k +c)n=kn,

22 which completes the induction.

The Effect of the Parameter q. It is worth reflecting briefly on the role of the parameter q in the class of recurrences T(n) ≤ qT(n/2) + O(n) defined by (5.3). When q = 1, the resulting running time is linear; when q = 2, it’s O(n log n); and when q > 2, it’s a polynomial bound with an exponent larger than 1 that grows with q. The reason for this range of different running times lies in where

220

Chapter 5 Divide and Conquer

most of the work is spent in the recursion: when q = 1, the total running time is dominated by the top level, whereas when q > 2 it’s dominated by the work done on constant-size subproblems at the bottom of the recursion. Viewed this way, we can appreciate that the recurrence for q = 2 really represents a “knife- edge”—the amount of work done at each level is exactly the same, which is what yields the O(n log n) running time.

A Related Recurrence: T(n) ≤ 2T(n/2) + O(n2)

We conclude our discussion with one final recurrence relation; it is illustrative both as another application of a decaying geometric sum and as an interesting contrast with the recurrence (5.1) that characterized Mergesort. Moreover, we will see a close variant of it in Chapter 6, when we analyze a divide-and- conquer algorithm for solving the Sequence Alignment Problem using a small amount of working memory.

The recurrence is based on the following divide-and-conquer structure.

Divide the input into two pieces of equal size; solve the two subproblems on these pieces separately by recursion; and then combine the two results into an overall solution, spending quadratic time for the initial division and final recombining.

For our purposes here, we note that this style of algorithm has a running time T(n) that satisfies the following recurrence.

(5.6) For some constant c,

T(n) ≤ 2T(n/2) + cn2

when n > 2, and

T(2)≤c.

One’s first reaction is to guess that the solution will be T(n) = O(n2 log n), since it looks almost identical to (5.1) except that the amount of work per level is larger by a factor equal to the input size. In fact, this upper bound is correct (it would need a more careful argument than what’s in the previous sentence), but it will turn out that we can also show a stronger upper bound.

We’ll do this by unrolling the recurrence, following the standard template for doing this.

. Analyzing the first few levels: At the first level of recursion, we have a single problem of size n, which takes time at most cn2 plus the time spent in all subsequent recursive calls. At the next level, we have two problems, each of size n/2. Each of these takes time at most c(n/2)2 = cn2/4, for a

total of at most cn2/2, again plus the time in subsequent recursive calls. At the third level, we have four problems each of size n/4, each taking time at most c(n/4)2 = cn2/16, for a total of at most cn2/4. Already we see that something is different from our solution to the analogous recurrence (5.1); whereas the total amount of work per level remained the same in that case, here it’s decreasing.

. Identifying a pattern: At an arbitrary level j of the recursion, there are 2j

subproblems, each of size n/2j, and hence the total work at this level is

bounded by 2jc( n )2 = cn2/2j. 2j

. Summing over all levels of recursion: Having gotten this far in the calcu- lation, we’ve arrived at almost exactly the same sum that we had for the case q = 1 in the previous recurrence. We have

log n−1 log n−1 2

2 cn 2 1

= cn2

where the second inequality follows from the fact that we have a con-

vergent geometric sum.

In retrospect, our initial guess of T(n) = O(n2 log n), based on the analogy

to (5.1), was an overestimate because of how quickly n2 decreases as we

replace it with ( n )2, ( n )2, ( n )2, and so forth in the unrolling of the recurrence. 248

This means that we get a geometric sum, rather than one that grows by a fixed amount over all n levels (as in the solution to (5.1)).

5.3 Counting Inversions

We’ve spent some time discussing approaches to solving a number of common recurrences. The remainder of the chapter will illustrate the application of divide-and-conquer to problems from a number of different domains; we will use what we’ve seen in the previous sections to bound the running times of these algorithms. We begin by showing how a variant of the Mergesort technique can be used to solve a problem that is not directly related to sorting numbers.

The Problem

We will consider a problem that arises in the analysis of rankings, which are becoming important to a number of current applications. For example, a number of sites on the Web make use of a technique known as collaborative filtering, in which they try to match your preferences (for books, movies, restaurants) with those of other people out on the Internet. Once the Web site has identified people with “similar” tastes to yours—based on a comparison

T(n) ≤

j=0

2j

2j

≤ 2cn2 = O(n2),

5.3 Counting Inversions

221

j=0

222

Chapter 5 Divide and Conquer

2 4

1 3

5

of how you and they rate various things—it can recommend new things that these other people have liked. Another application arises in meta-search tools on the Web, which execute the same query on many different search engines and then try to synthesize the results by looking for similarities and differences among the various rankings that the search engines return.

A core issue in applications like this is the problem of comparing two rankings. You rank a set of n movies, and then a collaborative filtering system consults its database to look for other people who had “similar” rankings. But what’s a good way to measure, numerically, how similar two people’s rankings are? Clearly an identical ranking is very similar, and a completely reversed ranking is very different; we want something that interpolates through the middle region.

Let’s consider comparing your ranking and a stranger’s ranking of the same set of n movies. A natural method would be to label the movies from 1 to n according to your ranking, then order these labels according to the stranger’s ranking, and see how many pairs are “out of order.” More concretely, we will consider the following problem. We are given a sequence of n numbers a1, . . . , an; we will assume that all the numbers are distinct. We want to define a measure that tells us how far this list is from being in ascending order; the value of the measure should be 0 if a1 < a2 < . . . < an, and should increase as the numbers become more scrambled.
A natural way to quantify this notion is by counting the number of inversions. We say that two indices i < j form an inversion if ai > aj, that is, if the two elements ai and aj are “out of order.” We will seek to determine the number of inversions in the sequence a1,…,an.

Just to pin down this definition, consider an example in which the se- quence is 2, 4, 1, 3, 5. There are three inversions in this sequence: (2, 1), (4, 1), and (4, 3). There is also an appealing geometric way to visualize the inver- sions, pictured in Figure 5.4: we draw the sequence of input numbers in the order they’re provided, and below that in ascending order. We then draw a line segment between each number in the top list and its copy in the lower list. Each crossing pair of line segments corresponds to one pair that is in the opposite order in the two lists—in other words, an inversion.

12345

Figure 5.4 Counting the number of inversions in the sequence 2, 4, 1, 3, 5. Each crossing pair of line segments corresponds to one pair that is in the opposite order in the input list and the ascend- ing list—in other words, an inversion.

Note how the number of inversions is a measure that smoothly interpolates between complete agreement (when the sequence is in ascending order, then there are no inversions) and complete disagreement (if the sequence is in

descending order, then every pair forms an inversion, and so there are n of 2

them).

5.3 Counting Inversions

223

Designing and Analyzing the Algorithm

What is the simplest algorithm to count inversions? Clearly, we could look at every pair of numbers (ai,aj) and determine whether they constitute an inversion; this would take O(n2) time.

We now show how to count the number of inversions much more quickly, in O(n log n) time. Note that since there can be a quadratic number of inver- sions, such an algorithm must be able to compute the total number without ever looking at each inversion individually. The basic idea is to follow the strategy (†) defined in Section 5.1. We set m = ⌈n/2⌉ and divide the list into the two pieces a1,…,am and am+1,…,an. We first count the number of inversions in each of these two halves separately. Then we count the number of inversions (ai, aj), where the two numbers belong to different halves; the trick is that we must do this part in O(n) time, if we want to apply (5.2). Note that these first-half/second-half inversions have a particularly nice form: they are precisely the pairs (ai, aj), where ai is in the first half, aj is in the second half, and ai > aj.

To help with counting the number of inversions between the two halves, we will make the algorithm recursively sort the numbers in the two halves as well. Having the recursive step do a bit more work (sorting as well as counting inversions) will make the “combining” portion of the algorithm easier.

So the crucial routine in this process is Merge-and-Count. Suppose we have recursively sorted the first and second halves of the list and counted the inversions in each. We now have two sorted lists A and B, containing the first and second halves, respectively. We want to produce a single sorted list C from their union, while also counting the number of pairs (a, b) with a ∈ A, b ∈ B, and a > b. By our previous discussion, this is precisely what we will need for the “combining” step that computes the number of first-half/second-half inversions.

This is closely related to the simpler problem we discussed in Chapter 2, which formed the corresponding “combining” step for Mergesort: there we had two sorted lists A and B, and we wanted to merge them into a single sorted list in O(n) time. The difference here is that we want to do something extra: not only should we produce a single sorted list from A and B, but we should also count the number of “inverted pairs” (a, b) where a ∈ A, b ∈ B, and a > b.

It turns out that we will be able to do this in very much the same style that we used for merging. Our Merge-and-Count routine will walk through the sorted lists A and B, removing elements from the front and appending them to the sorted list C. In a given step, we have a Current pointer into each list, showing our current position. Suppose that these pointers are currently

224

Chapter 5

Divide and Conquer

Elements inverted with bj < ai
A B
Figure 5.5 Merging two sorted lists while also counting the number of inversions between them.
at elements ai and bj. In one step, we compare the elements ai and bj being pointed to in each list, remove the smaller one from its list, and append it to the end of list C.
This takes care of merging. How do we also count the number of inver- sions? Because A and B are sorted, it is actually very easy to keep track of the number of inversions we encounter. Every time the element ai is appended to C, no new inversions are encountered, since ai is smaller than everything left in list B, and it comes before all of them. On the other hand, if bj is appended to list C, then it is smaller than all the remaining items in A, and it comes after all of them, so we increase our count of the number of inversions by the number of elements remaining in A. This is the crucial idea: in constant time, we have accounted for a potentially large number of inversions. See Figure 5.5 for an illustration of this process.
To summarize, we have the following algorithm.
Merge-and-Count(A,B)
Maintain a Current pointer into each list, initialized to
point to the front elements
Maintain a variable Count for the number of inversions,
initialized to 0
While both lists are nonempty:
Let ai and bj be the elements pointed to by the Current pointer Append the smaller of these two to the output list
If bj is the smaller element then
Increment Count by the number of elements remaining in A Endif
Advance the Current pointer in the list from which the smaller element was selected.
EndWhile
//////
ai
Merged result
///
bj
Once one list is empty, append the remainder of the other list
to the output
Return Count and the merged list
The running time of Merge-and-Count can be bounded by the analogue of the argument we used for the original merging algorithm at the heart of Mergesort: each iteration of the While loop takes constant time, and in each iteration we add some element to the output that will never be seen again. Thus the number of iterations can be at most the sum of the initial lengths of A and B, and so the total running time is O(n).
We use this Merge-and-Count routine in a recursive procedure that simultaneously sorts and counts the number of inversions in a list L.
Sort-and-Count(L)
If the list has one element then
there are no inversions
Else
Divide the list into two halves:
A contains the first ⌈n/2⌉ elements
B contains the remaining ⌊n/2⌋ elements
(rA, A) = Sort-and-Count(A) (rB, B) = Sort-and-Count(B) (r,L) = Merge-and-Count(A,B)
Endif
Return r =rA +rB +r, and the sorted list L
Since our Merge-and-Count procedure takes O(n) time, the running time T(n) of the full Sort-and-Count procedure satisfies the recurrence (5.1). By (5.2), we have
(5.7) The Sort-and-Count algorithm correctly sorts the input list and counts the number of inversions; it runs in O(n log n) time for a list with n elements.
5.4 Finding the Closest Pair of Points
We now describe another problem that can be solved by an algorithm in the style we’ve been discussing; but finding the right way to “merge” the solutions to the two subproblems it generates requires quite a bit of ingenuity.
5.4 Finding the Closest Pair of Points
225
226
Chapter 5 Divide and Conquer
The Problem
The problem we consider is very simple to state: Given n points in the plane, find the pair that is closest together.
The problem was considered by M. I. Shamos and D. Hoey in the early 1970s, as part of their project to work out efficient algorithms for basic com- putational primitives in geometry. These algorithms formed the foundations of the then-fledgling field of computational geometry, and they have found their way into areas such as graphics, computer vision, geographic informa- tion systems, and molecular modeling. And although the closest-pair problem is one of the most natural algorithmic problems in geometry, it is surprisingly hard to find an efficient algorithm for it. It is immediately clear that there is an O(n2) solution—compute the distance between each pair of points and take the minimum—and so Shamos and Hoey asked whether an algorithm asymp- totically faster than quadratic could be found. It took quite a long time before they resolved this question, and the O(n log n) algorithm we give below is essentially the one they discovered. In fact, when we return to this problem in Chapter 13, we will see that it is possible to further improve the running time to O(n) using randomization.
Designing the Algorithm
We begin with a bit of notation. Let us denote the set of points by P = {p1, . . . , pn}, where pi has coordinates (xi, yi); and for two points pi, pj ∈ P, we use d(pi, pj) to denote the standard Euclidean distance between them. Our goal is to find a pair of points pi, pj that minimizes d(pi, pj).
We will assume that no two points in P have the same x-coordinate or the same y-coordinate. This makes the discussion cleaner; and it’s easy to eliminate this assumption either by initially applying a rotation to the points that makes it true, or by slightly extending the algorithm we develop here.
It’s instructive to consider the one-dimensional version of this problem for a minute, since it is much simpler and the contrasts are revealing. How would we find the closest pair of points on a line? We’d first sort them, in O(n log n) time, and then we’d walk through the sorted list, computing the distance from each point to the one that comes after it. It is easy to see that one of these distances must be the minimum one.
In two dimensions, we could try sorting the points by their y-coordinate (or x-coordinate) and hoping that the two closest points were near one another in the order of this sorted list. But it is easy to construct examples in which they are very far apart, preventing us from adapting our one-dimensional approach.
Instead, our plan will be to apply the style of divide and conquer used in Mergesort: we find the closest pair among the points in the “left half” of
P and the closest pair among the points in the “right half” of P; and then we use this information to get the overall solution in linear time. If we develop an algorithm with this structure, then the solution of our basic recurrence from (5.1) will give us an O(n log n) running time.
It is the last, “combining” phase of the algorithm that’s tricky: the distances that have not been considered by either of our recursive calls are precisely those that occur between a point in the left half and a point in the right half; there are (n2) such distances, yet we need to find the smallest one in O(n) time after the recursive calls return. If we can do this, our solution will be complete: it will be the smallest of the values computed in the recursive calls and this minimum “left-to-right” distance.
Setting Up the Recursion Let’s get a few easy things out of the way first. It will be very useful if every recursive call, on a set P′ ⊆ P, begins with two lists: a list Px′ in which all the points in P′ have been sorted by increasing x- coordinate, and a list Py′ in which all the points in P′ have been sorted by increasing y-coordinate. We can ensure that this remains true throughout the algorithm as follows.
First, before any of the recursion begins, we sort all the points in P by x- coordinate and again by y-coordinate, producing lists Px and Py. Attached to each entry in each list is a record of the position of that point in both lists.
The first level of recursion will work as follows, with all further levels working in a completely analogous way. We define Q to be the set of points in the first ⌈n/2⌉ positions of the list Px (the “left half”) and R to be the set of points in the final ⌊n/2⌋ positions of the list Px (the “right half”). See Figure 5.6. By a single pass through each of Px and Py, in O(n) time, we can create the
Q
Line L R
5.4 Finding the Closest Pair of Points
227
δ
Figure 5.6 The first level of recursion: The point set P is divided evenly into Q and R by the line L, and the closest pair is found on each side recursively.
228
Chapter 5 Divide and Conquer
following four lists: Qx, consisting of the points in Q sorted by increasing x- coordinate; Qy, consisting of the points in Q sorted by increasing y-coordinate; and analogous lists Rx and Ry. For each entry of each of these lists, as before, we record the position of the point in both lists it belongs to.
We now recursively determine a closest pair of points in Q (with access to the lists Qx and Qy). Suppose that q0∗ and q1∗ are (correctly) returned as a closest pair of points in Q. Similarly, we determine a closest pair of points in R, obtaining r0∗ and r1∗.
Combining the Solutions The general machinery of divide and conquer has gotten us this far, without our really having delved into the structure of the closest-pair problem. But it still leaves us with the problem that we saw looming originally: How do we use the solutions to the two subproblems as part of a linear-time “combining” operation?
Let δ be the minimum of d(q0∗,q1∗) and d(r0∗,r1∗). The real question is: Are there points q ∈ Q and r ∈ R for which d(q, r) < δ? If not, then we have already found the closest pair in one of our recursive calls. But if there are, then the closest such q and r form the closest pair in P.
Let x∗ denote the x-coordinate of the rightmost point in Q, and let L denote the vertical line described by the equation x = x∗. This line L “separates” Q from R. Here is a simple fact.
(5.8) Ifthereexistsq∈Qandr∈Rforwhichd(q,r)<δ,theneachofqand r lies within a distance δ of L.
Proof. Suppose such q and r exist; we write q = (qx, qy) and r = (rx, ry). By the definition of x∗, we know that qx ≤ x∗ ≤ rx. Then we have
and
x∗ −qx ≤rx −qx ≤d(q,r)<δ rx −x∗ ≤rx −qx ≤d(q,r)<δ,
so each of q and r has an x-coordinate within δ of x∗ and hence lies within distance δ of the line L.
So if we want to find a close q and r, we can restrict our search to the narrow band consisting only of points in P within δ of L. Let S ⊆ P denote this set, and let Sy denote the list consisting of the points in S sorted by increasing y-coordinate. By a single pass through the list Py, we can construct Sy in O(n) time.
We can restate (5.8) as follows, in terms of the set S.
(5.9) There exist q∈Q and r∈R for which d(q,r)<δ if and only if there exist s,s′ ∈S for which d(s,s′)<δ.
It’s worth noticing at this point that S might in fact be the whole set P, in which case (5.8) and (5.9) really seem to buy us nothing. But this is actually far from true, as the following amazing fact shows.
(5.10) If s, s′ ∈ S have the property that d(s, s′) < δ, then s and s′ are within 15 positions of each other in the sorted list Sy.
Proof. Consider the subset Z of the plane consisting of all points within distance δ of L. We partition Z into boxes: squares with horizontal and vertical sides of length δ/2. One row of Z will consist of four boxes whose horizontal sides have the same y-coordinates. This collection of boxes is depicted in Figure 5.7.
Suppose two points of S lie in the same box. Since all points in this box lie on the same side of L, these two points either both belong to Q or both belong
Each box can contain at most one input point.
Line L
5.4 Finding the Closest Pair of Points
229
δ/2 δ/2
Boxes
2/2 < δ, which contradicts our definition of δ as the minimum distance between any
to R. But any two points in the same box are within distance δ ·
pair of points in Q or in R. Thus each box contains at most one point of S.
Figure 5.7 The portion of the plane close to the dividing line L, as analyzed in the proof of (5.10).
Now suppose that s, s′ ∈ S have the property that d(s, s′) < δ, and that they are at least 16 positions apart in Sy. Assume without loss of generality that s has the smaller y-coordinate. Then, since there can be at most one point per box, there are at least three rows of Z lying between s and s′. But any two points in Z separated by at least three rows must be a distance of at least 3δ/2 apart—a contradiction.
We note that the value of 15 can be reduced; but for our purposes at the moment, the important thing is that it is an absolute constant.
In view of (5.10), we can conclude the algorithm as follows. We make one pass through Sy, and for each s ∈ Sy, we compute its distance to each of the next 15 points in Sy. Statement (5.10) implies that in doing so, we will have computed the distance of each pair of points in S (if any) that are at distance less than δ from each other. So having done this, we can compare the smallest such distance to δ, and we can report one of two things: (i) the closest pair of points in S, if their distance is less than δ; or (ii) the (correct) conclusion that no pairs of points in S are within δ of each other. In case (i), this pair is the closest pair in P; in case (ii), the closest pair found by our recursive calls is the closest pair in P.
Note the resemblance between this procedure and the algorithm we re- jected at the very beginning, which tried to make one pass through P in order
√
δδ
230
Chapter 5 Divide and Conquer
of y-coordinate. The reason such an approach works now is due to the ex- tra knowledge (the value of δ) we’ve gained from the recursive calls, and the special structure of the set S.
This concludes the description of the “combining” part of the algorithm, since by (5.9) we have now determined whether the minimum distance between a point in Q and a point in R is less than δ, and if so, we have found the closest such pair.
A complete description of the algorithm and its proof of correctness are implicitly contained in the discussion so far, but for the sake of concreteness, we now summarize both.
Summary of the Algorithm A high-level description of the algorithm is the following, using the notation we have developed above.
Closest-Pair(P )
Construct Px and Py (O(n log n) time) (p0∗, p1∗) = Closest-Pair-Rec(Px,Py)
Closest-Pair-Rec(Px, Py) If |P| ≤ 3 then
find closest pair by measuring all pairwise distances
Endif
Construct Qx, Qy, Rx, Ry (O(n) time) (q0∗,q1∗) = Closest-Pair-Rec(Qx, Qy) (r0∗,r1∗) = Closest-Pair-Rec(Rx, Ry)
δ = min(d(q0∗,q1∗), d(r0∗,r1∗))
x∗ = maximum x-coordinate of a point in set Q L = {(x,y) : x = x∗}
S = points in P within distance δ of L.
Construct Sy (O(n) time)
For each point s ∈ Sy, compute distance from s
to each of next 15 points in Sy
Let s, s′ be pair achieving minimum of these distances (O(n) time)
If d(s,s′) < δ then Return (s,s′)
Else if d(q0∗,q1∗) < d(r0∗,r1∗) then Return (q0∗,q1∗)
Else
Return (r0∗,r1∗)
Endif
Analyzing the Algorithm
We first prove that the algorithm produces a correct answer, using the facts we’ve established in the process of designing it.
(5.11) The algorithm correctly outputs a closest pair of points in P.
Proof. As we’ve noted, all the components of the proof have already been
worked out, so here we just summarize how they fit together.
We prove the correctness by induction on the size of P, the case of |P| ≤ 3 being clear. For a given P, the closest pair in the recursive calls is computed correctly by induction. By (5.10) and (5.9), the remainder of the algorithm correctly determines whether any pair of points in S is at distance less than δ, and if so returns the closest such pair. Now the closest pair in P either has both elements in one of Q or R, or it has one element in each. In the former case, the closest pair is correctly found by the recursive call; in the latter case, this pair is at distance less than δ, and it is correctly found by the remainder of the algorithm.
We now bound the running time as well, using (5.2). (5.12) The running time of the algorithm is O(n log n).
Proof. The initial sorting of P by x- and y-coordinate takes time O(n log n). The running time of the remainder of the algorithm satisfies the recurrence (5.1), and hence is O(n log n) by (5.2).
5.5 Integer Multiplication
We now discuss a different application of divide and conquer, in which the “default” quadratic algorithm is improved by means of a different recurrence. The analysis of the faster algorithm will exploit one of the recurrences con- sidered in Section 5.2, in which more than two recursive calls are spawned at each level.
The Problem
The problem we consider is an extremely basic one: the multiplication of two integers. In a sense, this problem is so basic that one may not initially think of it
5.5 Integer Multiplication
231
232
Chapter 5
Divide and Conquer
1100 × 1101 12 1100
× 13 0000 36 1100
12 1100 156 10011100
(a) (b)
Figure 5.8 The elementary-school algorithm for multiplying two integers, in (a) decimal
and (b) binary representation.
even as an algorithmic question. But, in fact, elementary schoolers are taught a concrete (and quite efficient) algorithm to multiply two n-digit numbers x and y. You first compute a “partial product” by multiplying each digit of y separately by x, and then you add up all the partial products. (Figure 5.8 should help you recall this algorithm. In elementary school we always see this done in base- 10, but it works exactly the same way in base-2 as well.) Counting a single operation on a pair of bits as one primitive step in this computation, it takes O(n) time to compute each partial product, and O(n) time to combine it in with the running sum of all partial products so far. Since there are n partial products, this is a total running time of O(n2).
If you haven’t thought about this much since elementary school, there’s something initially striking about the prospect of improving on this algorithm. Aren’t all those partial products “necessary” in some way? But, in fact, it is possible to improve on O(n2) time using a different, recursive way of performing the multiplication.
Designing the Algorithm
The improved algorithm is based on a more clever way to break up the product into partial sums. Let’s assume we’re in base-2 (it doesn’t really matter), and start by writing x as x1 · 2n/2 + x0. In other words, x1 corresponds to the “high- order” n/2 bits, and x0 corresponds to the “low-order” n/2 bits. Similarly, we write y = y1 · 2n/2 + y0. Thus, we have
xy = (x1 · 2n/2 + x0)(y1 · 2n/2 + y0)
= x1y1 · 2n + (x1y0 + x0y1) · 2n/2 + x0y0. (5.1)
Equation (5.1) reduces the problem of solving a single n-bit instance (multiplying the two n-bit numbers x and y) to the problem of solving four n/2- bit instances (computing the products x1y1, x1y0, x0y1, and x0y0). So we have a first candidate for a divide-and-conquer solution: recursively compute the results for these four n/2-bit instances, and then combine them using Equation
(5.1). The combining of the solution requires a constant number of additions of O(n)-bit numbers, so it takes time O(n); thus, the running time T(n) is bounded by the recurrence
T(n) ≤ 4T(n/2) + cn
for a constant c. Is this good enough to give us a subquadratic running time?
We can work out the answer by observing that this is just the case q = 4 of the class of recurrences in (5.3). As we saw earlier in the chapter, the solution to this is T(n) ≤ O(nlog2 q) = O(n2).
So, in fact, our divide-and-conquer algorithm with four-way branching was just a complicated way to get back to quadratic time! If we want to do better using a strategy that reduces the problem to instances on n/2 bits, we should try to get away with only three recursive calls. This will lead to the case q = 3 of (5.3), which we saw had the solution T(n) ≤ O(nlog2 q) = O(n1.59).
Recall that our goal is to compute the expression x1y1 · 2n + (x1y0 + x0y1) · 2n/2 + x0y0 in Equation (5.1). It turns out there is a simple trick that lets us determine all of the terms in this expression using just three recursive calls. The trick is to consider the result of the single multiplication (x1 + x0)(y1 + y0) = x1y1 + x1y0 + x0y1 + x0y0. This has the four products above added together, at the cost of a single recursive multiplication. If we now also determine x1y1 and x0y0 by recursion, then we get the outermost terms explicitly, and we get the middle term by subtracting x1y1 and x0y0 away from (x1 + x0)(y1 + y0).
Thus, in full, our algorithm is
Recursive-Multiply(x,y): Write x=x1·2n/2+x0
y = y1 · 2n/2 + y0
Compute x1 + x0 and y1 + y0
p = Recursive-Multiply(x1 + x0,
x1y1 = Recursive-Multiply(x1, y1)
x0y0 = Recursive-Multiply(x0 , y0)
Return x1y1·2n+(p−x1y1−x0y0)·2n/2+x0y0
Analyzing the Algorithm
We can determine the running time of this algorithm as follows. Given two n- bit numbers, it performs a constant number of additions on O(n)-bit numbers, in addition to the three recursive calls. Ignoring for now the issue that x1 + x0 and y1 + y0 may have n/2 + 1 bits (rather than just n/2), which turns out not to affect the asymptotic results, each of these recursive calls is on an instance of size n/2. Thus, in place of our four-way branching recursion, we now have
y1 + y0)
5.5 Integer Multiplication
233
234
Chapter 5 Divide and Conquer
a three-way branching one, with a running time that satisfies T(n) ≤ 3T(n/2) + cn
for a constant c.
This is the case q = 3 of (5.3) that we were aiming for. Using the solution
to that recurrence from earlier in the chapter, we have
(5.13) The running time of Recursive-Multiply on two n-bit factors is O(nlog2 3) = O(n1.59).
5.6 Convolutions and the Fast Fourier Transform
As a final topic in this chapter, we show how our basic recurrence from (5.1) is used in the design of the Fast Fourier Transform, an algorithm with a wide range of applications.
The Problem
Given two vectors a = (a0, a1, . . . , an−1) and b = (b0, b1, . . . , bn−1), there are a number of common ways of combining them. For example, one can compute the sum, producing the vector a+b=(a0 +b0,a1+b1,...,an−1+bn−1); or one can compute the inner product, producing the real number a · b = a0b0 + a1b1 + . . . + an−1bn−1. (For reasons that will emerge shortly, it is useful to write vectors in this section with coordinates that are indexed starting from 0 rather than 1.)
A means of combining vectors that is very important in applications, even if it doesn’t always show up in introductory linear algebra courses, is the convolution a ∗ b. The convolution of two vectors of length n (as a and b are) is a vector with 2n − 1 coordinates, where coordinate k is equal to
aibj. (i , j):i+j=k
i,j

To see the connection with the convolution operation, we picture this smoothing operation as follows. We first define a “mask”

w = (w−k, w−(k−1), . . . , w−1, w0, w1, . . . , wk−1, wk) consisting of the weights we want to use for averaging each point with

its neighbors. (For example, w = 1 (e−k2, e−(k−1)2, . . . , e−1, 1, e−1, . . . , Z

e−(k−1)2 , e−k2) in the Gaussian case above.) We then iteratively position this mask so it is centered at each possible point in the sequence a; and for each positioning, we compute the weighted average. In other words, we replace ai with ai′ = ks=−k wsai+s.

This last expression is essentially a convolution; we just have to warp the notation a bit so that this becomes clear. Let’s define b = (b0, b1, . . . , b2k) by setting bl = wk−l. Then it’s not hard to check that with this definition we have the smoothed value

′ ai =

(j,l):j+l=i+k

ajbl.

In other words, the smoothed sequence is just the convolution of the original signal and the reverse of the mask (with some meaningless coordinates at the beginning and end).

5.6 Convolutions and the Fast Fourier Transform

237

. We mention one final application: the problem of combining histograms. Suppose we’re studying a population of people, and we have the follow- ing two histograms: One shows the annual income of all the men in the population, and one shows the annual income of all the women. We’d now like to produce a new histogram, showing for each k the number of pairs (M , W ) for which man M and woman W have a combined income of k.

This is precisely a convolution. We can write the first histogram as a vector a = (a0, . . . , am−1), to indicate that there are ai men with annual income equal to i. We can similarly write the second histogram as a vector b = (b0, . . . , bn−1). Now, let ck denote the number of pairs (m, w) with combined income k; this is the number of ways of choosing a man with income ai and a woman with income bj, for any pair (i, j) where i+j=k. In other words,

ck =

(Using terminology from probability that we will develop in Chap- ter 13, one can view this example as showing how convolution is the underlying means for computing the distribution of the sum of two in- dependent random variables.)

Computing the Convolution Having now motivated the notion of convolu- tion, let’s discuss the problem of computing it efficiently. For simplicity, we will consider the case of equal length vectors (i.e., m = n), although everything we say carries over directly to the case of vectors of unequal lengths.

Computing the convolution is a more subtle question than it may first appear. The definition of convolution, after all, gives us a perfectly valid way to compute it: for each k, we just calculate the sum

aibj (i,j):i+j=k

and use this as the value of the kth coordinate. The trouble is that this direct way of computing the convolution involves calculating the product aibj for every pair (i, j) (in the process of distributing over the sums in the different terms) and this is (n2) arithmetic operations. Spending O(n2) time on computing the convolution seems natural, as the definition involves O(n2) multiplications aibj. However, it’s not inherently clear that we have to spend quadratic time to compute a convolution, since the input and output both only have size O(n).

aibj.

so the combined histogram c = (c0, . . . , cm+n−2) is simply the convolu-

tion of a and b.

(i,j):i+j=k

238

Chapter 5 Divide and Conquer

Could one design an algorithm that bypasses the quadratic-size definition of convolution and computes it in some smarter way?

In fact, quite surprisingly, this is possible. We now describe a method that computes the convolution of two vectors using only O(n log n) arithmetic operations. The crux of this method is a powerful technique known as the Fast Fourier Transform (FFT). The FFT has a wide range of further applications in analyzing sequences of numerical values; computing convolutions quickly, which we focus on here, is just one of these applications.

Designing and Analyzing the Algorithm

To break through the quadratic time barrier for convolutions, we are going to exploit the connection between the convolution and the multiplication of two polynomials, as illustrated in the first example discussed previously. But rather than use convolution as a primitive in polynomial multiplication, we are going to exploit this connection in the opposite direction.

Suppose we are given the vectors a=(a0,a1,…,an−1) and b=(b0, b1, . . . , bn−1). We will view them as the polynomials A(x) = a0 + a1x + a2x2 + . . . an−1xn−1 and B(x) = b0 + b1x + b2x2 + . . . bn−1xn−1, and we’ll seek to com- pute their product C(x) = A(x)B(x) in O(n log n) time. If c = (c0, c1, . . . , c2n−2) is the vector of coefficients of C, then we recall from our earlier discussion that c is exactly the convolution a ∗ b, and so we can then read off the desired answer directly from the coefficients of C(x).

Now, rather than multiplying A and B symbolically, we can treat them as functions of the variable x and multiply them as follows.

(i) Firstwechoose2nvaluesx1,x2,…,x2nandevaluateA(xj)andB(xj)for each of j=1,2,…,2n.

(ii) We can now compute C(xj) for each j very easily: C(xj) is simply the product of the two numbers A(xj) and B(xj).

(iii) Finally,wehavetorecoverCfromitsvaluesonx1,x2,…,x2n.Herewe take advantage of a fundamental fact about polynomials: any polynomial of degree d can be reconstructed from its values on any set of d + 1 or more points. This is known as polynomial interpolation, and we’ll discuss the mechanics of performing interpolation in more detail later. For the moment, we simply observe that since A and B each have degree at most n−1, their product C has degree at most 2n−2, and so it can be reconstructed from the values C(x1), C(x2), . . . , C(x2n) that we computed in step (ii).

This approach to multiplying polynomials has some promising aspects

and some problematic ones. First, the good news: step (ii) requires only

5.6 Convolutions and the Fast Fourier Transform

239

O(n) arithmetic operations, since it simply involves the multiplication of O(n) numbers. But the situation doesn’t look as hopeful with steps (i) and (iii). In particular, evaluating the polynomials A and B on a single value takes (n) operations, and our plan calls for performing 2n such evaluations. This seems to bring us back to quadratic time right away.

The key idea that will make this all work is to find a set of 2n values x1, x2, . . . , x2n that are intimately related in some way, such that the work in evaluating A and B on all of them can be shared across different evaluations. A set for which this will turn out to work very well is the complex roots of unity.

The Complex Roots of Unity At this point, we’re going to need to recall a few facts about complex numbers and their role as solutions to polynomial equations.

Recall that complex numbers can be viewed as lying in the “complex plane,” with axes representing their real and imaginary parts. We can write a complex number using polar coordinates with respect to this plane as reθi, where eπ i = −1 (and e2π i = 1). Now, for a positive integer k, the polynomial equation xk = 1 has k distinct complex roots, and it is easy to identify them. Eachofthecomplexnumbersωj,k =e2πji/k (forj=0,1,2,…,k−1)satisfies the equation, since

(e2πji/k)k = e2πji = (e2πi)j = 1j = 1,

and each of these numbers is distinct, so these are all the roots. We refer to these numbers as the kth roots of unity. We can picture these roots as a set of k equally spaced points lying on the unit circle in the complex plane, as shown in Figure 5.9 for the case k = 8.

For our numbers x1, . . . , x2n on which to evaluate A and B, we will choose the (2n)th roots of unity. It’s worth mentioning (although it’s not necessary for understanding the algorithm) that the use of the complex roots of unity is the basis for the name Fast Fourier Transform: the representation of a degree-d

i

–1 1

–i

Figure 5.9 The 8th roots of unity in the complex plane.

240

Chapter 5 Divide and Conquer

polynomial P by its values on the (d + 1)st roots of unity is sometimes referred to as the discrete Fourier transform of P; and the heart of our procedure is a method for making this computation fast.

A Recursive Procedure for Polynomial Evaluation We want to design an algorithm for evaluating A on each of the (2n)th roots of unity recursively, so as to take advantage of the familiar recurrence from (5.1)—namely, T(n) ≤ 2T(n/2) + O(n) where T(n) in this case denotes the number of operations required to evaluate a polynomial of degree n − 1 on all the (2n)th roots of unity. For simplicity in describing this algorithm, we will assume that n is a power of 2.

How does one break the evaluation of a polynomial into two equal-sized subproblems? A useful trick is to define two polynomials, Aeven(x) and Aodd(x), that consist of the even and odd coefficients of A, respectively. That is,

Aeven(x) = a0 + a2x + a4x2 + . . . + an−2x(n−2)/2,

and

Aodd(x) = a1 + a3x + a5x2 + . . . + a(n−1)x(n−2)/2. Simple algebra shows us that

A(x) = Aeven(x2) + xAodd(x2),

and so this gives us a way to compute A(x) in a constant number of operations, given the evaluation of the two constituent polynomials that each have half the degree of A.

Now suppose that we evaluate each of Aeven and Aodd on the nth roots of unity. This is exactly a version of the problem we face with A and the (2n)th roots of unity, except that the input is half as large: the degree is (n − 2)/2 rather than n − 1, and we have n roots of unity rather than 2n. Thus we can perform these evaluations in time T(n/2) for each of Aeven and Aodd, for a total time of 2T(n/2).

We’re now very close to having a recursive algorithm that obeys (5.1) and

gives us the running time we want; we just have to produce the evaluations

of A on the (2n)th roots of unity using O(n) additional operations. But this is

easy, given the results from the recursive calls on Aeven and Aodd. Consider

one of these roots of unity ωj,2n = e2πji/2n. The quantity ω2 is equal to j,2n

(e2πji/2n)2 = e2πji/n, and hence ω2 is an nth root of unity. So when we go to compute j,2n

A(ωj,2n) = Aeven(ω2 ) + ωj,2nAodd(ω2 ), j,2n j,2n

we discover that both of the evaluations on the right-hand side have been performed in the recursive step, and so we can determine A(ωj,2n) using a

5.6 Convolutions and the Fast Fourier Transform

241

constant number of operations. Doing this for all 2n roots of unity is therefore O(n) additional operations after the two recursive calls, and so the bound T(n) on the number of operations indeed satisfies T(n) ≤ 2T(n/2) + O(n). We run the same procedure to evaluate the polynomial B on the (2n)th roots of unity as well, and this gives us the desired O(n log n) bound for step (i) of our algorithm outline.

Polynomial Interpolation We’ve now seen how to evaluate A and B on the set of all (2n)th roots of unity using O(n log n) operations and, as noted above, we can clearly compute the products C(ωj,n) = A(ωj,2n)B(ωj,2n) in O(n) more operations. Thus, to conclude the algorithm for multiplying A and B, we need to execute step (iii) in our earlier outline using O(n log n) operations, reconstructing C from its values on the (2n)th roots of unity.

In describing this part of the algorithm, it’s worth keeping track of the following top-level point: it turns out that the reconstruction of C can be achieved simply by defining an appropriate polynomial (the polynomial D below) and evaluating it at the (2n)th roots of unity. This is exactly what we’ve just seen how to do using O(n log n) operations, so we do it again here, spending an additional O(n log n) operations and concluding the algorithms.

Consider a polynomial C(x) = 2n−1 csxs that we want to reconstruct s=0

from its values C(ωs,2n) at the (2n)th roots of unity. Define a new polynomial D(x) = 2n−1 dsxs, where ds = C(ωs,2n). We now consider the values of D(x)

s=0

at the (2n)th roots of unity.

2n−1

D(ωj,2n) = C(ωs,2n)ωs

s=0

2n−1 2n−1

= ( ctωt )ωs s,2n j,2n

s=0 t=0 2n−1 2n−1

= ct( ωt ωs ), s,2n j,2n

t=0 s=0

by definition. Now recall that ωs,2n = (e2πi/2n)s. Using this fact and extending

the notation to ωs,2n = (e2πi/2n)s even when s ≥ 2n, we get that 2n−1 2n−1

D(ωj,2n)= ct( e(2πi)(st+js)/2n)

t=0 s=0 2n−1 2n−1

= ct( ωs ). t+j,2n

t=0 s=0

j,2n

242

Chapter 5 Divide and Conquer

To analyze the last line, we use the fact that for any (2n)th root of unity ω ̸= 1, we have 2n−1 ωs = 0. This is simply because ω is by definition a root of

s=0

x2n −1=0; since x2n −1=(x−1)(

also a root of ( 2n−1 xt). t=0

2n−1xt) and ω̸=1, it follows that ω is t=0

Thus the only term of the last line’s outer sum that is not equal to 0 is

for ct such that ωt+j,2n = 1; and this happens if t + j is a multiple of 2n, that

is, if t=2n−j. For this value, 2n−1ωs =2n−11=2n. So we get that s=0 t+j,2n s=0

D(ωj,2n) = 2nc2n−j. Evaluating the polynomial D(x) at the (2n)th roots of unity thus gives us the coeffients of the polynomial C(x) in reverse order (multiplied by 2n each). We sum this up as follows.

(5.14) For any polynomial C(x) = 2n−1 csxs, and corresponding polynomial s=0

D(x) = 2n−1 C(ωs,2n)xs, we have that cs = 1 D(ω2n−s,2n). s=0 2n

We can do all the evaluations of the values D(ω2n−s,2n) in O(n log n) operations using the divide-and-conquer approach developed for step (i).

And this wraps everything up: we reconstruct the polynomial C from its values on the (2n)th roots of unity, and then the coefficients of C are the coordinates in the convolution vector c = a ∗ b that we were originally seeking.

In summary, we have shown the following.

(5.15) Using the Fast Fourier Transform to determine the product polynomial C(x), we can compute the convolution of the original vectors a and b in O(n log n) time.

Solved Exercises

Solved Exercise 1

Suppose you are given an array A with n entries, with each entry holding a distinct number. You are told that the sequence of values A[1], A[2], . . . , A[n] is unimodal: For some index p between 1 and n, the values in the array entries increase up to position p in A and then decrease the remainder of the way until position n. (So if you were to draw a plot with the array position j on the x-axis and the value of the entry A[j] on the y-axis, the plotted points would rise until x-value p, where they’d achieve their maximum, and then fall from there on.)

You’d like to find the “peak entry” p without having to read the entire array—in fact, by reading as few entries of A as possible. Show how to find the entry p by reading at most O(log n) entries of A.

Solution Let’s start with a general discussion on how to achieve a running time of O(log n) and then come back to the specific problem here. If one needs to compute something using only O(log n) operations, a useful strategy that we discussed in Chapter 2 is to perform a constant amount of work, throw away half the input, and continue recursively on what’s left. This was the idea, for example, behind the O(log n) running time for binary search.

We can view this as a divide-and-conquer approach: for some constant c > 0, we perform at most c operations and then continue recursively on an input of size at most n/2. As in the chapter, we will assume that the recursion “bottoms out” when n = 2, performing at most c operations to finish the computation. If T(n) denotes the running time on an input of size n, then we have the recurrence

(5.16)

when n > 2, and

T(n) ≤ T(n/2) + c

T(2)≤c.

It is not hard to solve this recurrence by unrolling it, as follows.

. Analyzing the first few levels: At the first level of recursion, we have a single problem of size n, which takes time at most c plus the time spent in all subsequent recursive calls. The next level has one problem of size at most n/2, which contributes another c, and the level after that has one problem of size at most n/4, which contributes yet another c.

. Identifying a pattern: No matter how many levels we continue, each level will have just one problem: level j has a single problem of size at most n/2j, which contributes c to the running time, independent of j.

. Summing over all levels of recursion: Each level of the recursion is contributing at most c operations, and it takes log2 n levels of recursion to reduce n to 2. Thus the total running time is at most c times the number of levels of recursion, which is at most c log2 n = O(log n).

We can also do this by partial substitution. Suppose we guess that T(n) ≤ k logb n, where we don’t know k or b. Assuming that this holds for smaller values of n in an inductive argument, we would have

T(n) ≤ T(n/2) + c

≤ k logb(n/2) + c

= k logb n − k logb 2 + c.

Solved Exercises

243

244

Chapter 5 Divide and Conquer

The first term on the right is exactly what we want, so we just need to choose k and b to negate the added c at the end. This we can do by setting b=2 and k = c, so that k logb 2 = c log2 2 = c. Hence we end up with the solution T(n) ≤ c log2 n, which is exactly what we got by unrolling the recurrence.

Finally, we should mention that one can get an O(log n) running time, by essentially the same reasoning, in the more general case when each level of the recursion throws away any constant fraction of the input, transforming an instance of size n to one of size at most an, for some constant a < 1. It now takes at most log1/a n levels of recursion to reduce n down to a constant size, and each level of recursion involves at most c operations.
Now let’s get back to the problem at hand. If we wanted to set ourselves up to use (5.16), we could probe the midpoint of the array and try to determine whether the “peak entry” p lies before or after this midpoint.
So suppose we look at the value A[n/2]. From this value alone, we can’t tell whether p lies before or after n/2, since we need to know whether entry n/2 is sitting on an “up-slope” or on a “down-slope.” So we also look at the values A[n/2 − 1] and A[n/2 + 1]. There are now three possibilities.
. If A[n/2 − 1] < A[n/2] < A[n/2 + 1], then entry n/2 must come strictly before p, and so we can continue recursively on entries n/2 + 1 through n.
. If A[n/2 − 1] > A[n/2] > A[n/2 + 1], then entry n/2 must come strictly after p, and so we can continue recursively on entries 1 through n/2 − 1.

. Finally, if A[n/2] is larger than both A[n/2 − 1] and A[n/2 + 1], we are done: the peak entry is in fact equal to n/2 in this case.

In all these cases, we perform at most three probes of the array A and reduce the problem to one of at most half the size. Thus we can apply (5.16) to conclude that the running time is O(log n).

Solved Exercise 2

You’re consulting for a small computation-intensive investment company, and they have the following type of problem that they want to solve over and over. A typical instance of the problem is the following. They’re doing a simulation in which they look at n consecutive days of a given stock, at some point in the past. Let’s number the days i=1,2,…,n; for each day i, they have a price p(i) per share for the stock on that day. (We’ll assume for simplicity that the price was fixed during each day.) Suppose during this time period, they wanted to buy 1,000 shares on some day and sell all these shares on some (later) day. They want to know: When should they have bought and when should they have sold in order to have made as much money as possible? (If

there was no way to make money during the n days, you should report this instead.)

For example, suppose n = 3, p(1) = 9, p(2) = 1, p(3) = 5. Then you should return “buy on 2, sell on 3” (buying on day 2 and selling on day 3 means they would have made $4 per share, the maximum possible for that period).

Clearly, there’s a simple algorithm that takes time O(n2): try all possible pairs of buy/sell days and see which makes them the most money. Your investment friends were hoping for something a little better.

Show how to find the correct numbers i and j in time O(n log n).

Solution We’ve seen a number of instances in this chapter where a brute- force search over pairs of elements can be reduced to O(n log n) by divide and conquer. Since we’re faced with a similar issue here, let’s think about how we might apply a divide-and-conquer strategy.

A natural approach would be to consider the first n/2 days and the final

n/2 days separately, solving the problem recursively on each of these two

sets, and then figure out how to get an overall solution from this in O(n) time.

This would give us the usual recurrence T (n) ≤ 2T n + O(n), and hence 2

Solved Exercises

245

O(n log n) by (5.1).

Also, to make things easier, we’ll make the usual assumption that n is a power of 2. This is no loss of generality: if n′ is the next power of 2 greater than n, we can set p(i) = p(n) for all i between n and n′. In this way, we do not change the answer, and we at most double the size of the input (which will not affect the O() notation).

Now, let S be the set of days 1,…,n/2, and S′ be the set of days n/2+ 1, . . . , n. Our divide-and-conquer algorithm will be based on the following observation: either there is an optimal solution in which the investors are holding the stock at the end of day n/2, or there isn’t. Now, if there isn’t, then the optimal solution is the better of the optimal solutions on the sets S and S′. If there is an optimal solution in which they hold the stock at the end of day n/2, then the value of this solution is p(j) − p(i) where i ∈ S and j ∈ S′. But this value is maximized by simply choosing i ∈ S which minimizes p(i), and choosing j ∈ S′ which maximizes p(j).

Thus our algorithm is to take the best of the following three possible solutions.

. The optimal solution on S.

. The optimal solution on S′.

. The maximum of p(j)−p(i), over i∈S and j∈S′.

The first two alternatives are computed in time T(n/2), each by recursion, and the third alternative is computed by finding the minimum in S and the

246

Chapter 5 Divide and Conquer

maximum in S′, which takes time O(n). Thus the running time T(n) satisfies

T(n) ≤ 2T n + O(n), 2

as desired.

We note that this is not the best running time achievable for this problem. In fact, one can find the optimal pair of days in O(n) time using dynamic programming, the topic of the next chapter; at the end of that chapter, we will pose this question as Exercise 7.

Exercises

1. Youareinterestedinanalyzingsomehard-to-obtaindatafromtwosepa- rate databases. Each database contains n numerical values—so there are 2n values total—and you may assume that no two values are the same. You’d like to determine the median of this set of 2n values, which we will define here to be the nth smallest value.

However, the only way you can access these values is through queries to the databases. In a single query, you can specify a value k to one of the two databases, and the chosen database will return the kth smallest value that it contains. Since queries are expensive, you would like to compute the median using as few queries as possible.

Give an algorithm that finds the median value using at most O(log n) queries.

2. Recall the problem of finding the number of inversions. As in the text, we are given a sequence of n numbers a1, . . . , an, which we assume are all distinct, and we define an inversion to be a pair i < j such that ai > aj.

We motivated the problem of counting inversions as a good measure of how different two orderings are. However, one might feel that this measure is too sensitive. Let’s call a pair a significant inversion if i < j and ai > 2aj. Give an O(n log n) algorithm to count the number of significant inversions between two orderings.

3. Suppose you’re consulting for a bank that’s concerned about fraud de- tection, and they come to you with the following problem. They have a collection of n bank cards that they’ve confiscated, suspecting them of being used in fraud. Each bank card is a small plastic object, contain- ing a magnetic stripe with some encrypted data, and it corresponds to a unique account in the bank. Each account can have many bank cards

corresponding to it, and we’ll say that two bank cards are equivalent if they correspond to the same account.

It’s very difficult to read the account number off a bank card directly, but the bank has a high-tech “equivalence tester” that takes two bank cards and, after performing some computations, determines whether they are equivalent.

Their question is the following: among the collection of n cards, is there a set of more than n/2 of them that are all equivalent to one another? Assume that the only feasible operations you can do with the cards are to pick two of them and plug them in to the equivalence tester. Show how to decide the answer to their question with only O(n log n) invocations of the equivalence tester.

4. You’vebeenworkingwithsomephysicistswhoneedtostudy,aspartof their experimental design, the interactions among large numbers of very small charged particles. Basically, their setup works as follows. They have an inert lattice structure, and they use this for placing charged particles at regular spacing along a straight line. Thus we can model their structure as consisting of the points {1, 2, 3, . . . , n} on the real line; and at each of these points j, they have a particle with charge qj. (Each charge can be either positive or negative.)

They want to study the total force on each particle, by measuring it and then comparing it to a computational prediction. This computational part is where they need your help. The total net force on particle j, by Coulomb’s Law, is equal to

Fj = Cqiqj − Cqiqj

Exercises

247

(j − i)2 (j − i)2 i>j

i

Add−Cqiqj toF (j − i)2 j

Endif Endfor

Output Fj Endfor

248

Chapter 5 Divide and Conquer

It’s not hard to analyze the running time of this program: each invocation of the inner loop, over i, takes O(n) time, and this inner loop is invoked O(n) times total, so the overall running time is O(n2).

The trouble is, for the large values of n they’re working with, the pro- gram takes several minutes to run. On the other hand, their experimental setup is optimized so that they can throw down n particles, perform the measurements, and be ready to handle n more particles within a few sec- onds. So they’d really like it if there were a way to compute all the forces Fj much more quickly, so as to keep up with the rate of the experiment.

Help them out by designing an algorithm that computes all the forces Fj in O(n log n) time.

5. Hidden surface removal is a problem in computer graphics that scarcely needs an introduction: when Woody is standing in front of Buzz, you should be able to see Woody but not Buzz; when Buzz is standing in front of Woody, . . . well, you get the idea.

The magic of hidden surface removal is that you can often compute things faster than your intuition suggests. Here’s a clean geometric ex- ample to illustrate a basic speed-up that can be achieved. You are given n nonvertical lines in the plane, labeled L1, . . . , Ln, with the ith line specified by the equation y = aix + bi. We will make the assumption that no three of the lines all meet at a single point. We say line Li is uppermost at a given x-coordinate x0 if its y-coordinate at x0 is greater than the y-coordinates ofalltheotherlinesatx0:aix0+bi>ajx0+bj forallj̸=i.WesaylineLi is visible if there is some x-coordinate at which it is uppermost—intuitively, some portion of it can be seen if you look down from “y = ∞.”

Give an algorithm that takes n lines as input and in O(n log n) time returns all of the ones that are visible. Figure 5.10 gives an example.

6. Consider an n-node complete binary tree T, where n = 2d − 1 for some d. Each node v of T is labeled with a real number xv. You may assume that the real numbers labeling the nodes are all distinct. A node v of T is a local minimum if the label xv is less than the label xw for all nodes w that are joined to v by an edge.

You are given such a complete binary tree T, but the labeling is only specified in the following implicit way: for each node v, you can determine the value xv by probing the node v. Show how to find a local minimum of T using only O(log n) probes to the nodes of T.

7. Supposenowthatyou’regivenann×ngridgraphG.(Ann×ngridgraph is just the adjacency graph of an n × n chessboard. To be completely precise, it is a graph whose node set is the set of all ordered pairs of

2 3

15

4

Notes and Further Reading

249

Figure 5.10 An instance of hidden surface removal with five lines (labeled 1–5 in the figure). All the lines except for 2 are visible.

natural numbers (i, j), where 1≤ i ≤ n and 1≤ j ≤ n; the nodes (i, j) and (k, l) are joined by an edge if and only if |i − k| + |j − l| = 1.)

We use some of the terminology of the previous question. Again, each node v is labeled by a real number xv; you may assume that all these labels are distinct. Show how to find a local minimum of G using only O(n) probes to the nodes of G. (Note that G has n2 nodes.)

Notes and Further Reading

The militaristic coinage “divide and conquer” was introduced somewhat after the technique itself. Knuth (1998) credits John von Neumann with one early explicit application of the approach, the development of the Mergesort Algo- rithm in 1945. Knuth (1997b) also provides further discussion of techniques for solving recurrences.

The algorithm for computing the closest pair of points in the plane is due to Michael Shamos, and is one of the earliest nontrivial algorithms in the field of computational geometry; the survey paper by Smid (1999) discusses a wide range of results on closest-point problems. A faster randomized algorithm for this problem will be discussed in Chapter 13. (Regarding the nonobviousness of the divide-and-conquer algorithm presented here, Smid also makes the in- teresting historical observation that researchers originally suspected quadratic time might be the best one could do for finding the closest pair of points in the plane.) More generally, the divide-and-conquer approach has proved very useful in computational geometry, and the books by Preparata and Shamos

250

Chapter 5 Divide and Conquer

(1985) and de Berg et al. (1997) give many further examples of this technique in the design of geometric algorithms.

The algorithm for multiplying two n-bit integers in subquadratic time is due to Karatsuba and Ofman (1962). Further background on asymptotically fast multiplication algorithms is given by Knuth (1997b). Of course, the number of bits in the input must be sufficiently large for any of these subquadratic methods to improve over the standard algorithm.

Press et al. (1988) provide further coverage of the Fast Fourier Transform, including background on its applications in signal processing and related areas.

Notes on the Exercises Exercise 7 is based on a result of Donna Llewellyn, Craig Tovey, and Michael Trick.

Chapter 6 Dynamic Programming

We began our study of algorithmic techniques with greedy algorithms, which in some sense form the most natural approach to algorithm design. Faced with a new computational problem, we’ve seen that it’s not hard to propose multiple possible greedy algorithms; the challenge is then to determine whether any of these algorithms provides a correct solution to the problem in all cases.

The problems we saw in Chapter 4 were all unified by the fact that, in the end, there really was a greedy algorithm that worked. Unfortunately, this is far from being true in general; for most of the problems that one encounters, the real difficulty is not in determining which of several greedy strategies is the right one, but in the fact that there is no natural greedy algorithm that works. For such problems, it is important to have other approaches at hand. Divide and conquer can sometimes serve as an alternative approach, but the versions of divide and conquer that we saw in the previous chapter are often not strong enough to reduce exponential brute-force search down to polynomial time. Rather, as we noted in Chapter 5, the applications there tended to reduce a running time that was unnecessarily large, but already polynomial, down to a faster running time.

We now turn to a more powerful and subtle design technique, dynamic programming. It will be easier to say exactly what characterizes dynamic pro- gramming after we’ve seen it in action, but the basic idea is drawn from the intuition behind divide and conquer and is essentially the opposite of the greedy strategy: one implicitly explores the space of all possible solutions, by carefully decomposing things into a series of subproblems, and then build- ing up correct solutions to larger and larger subproblems. In a way, we can thus view dynamic programming as operating dangerously close to the edge of

252

Chapter 6 Dynamic Programming

brute-force search: although it’s systematically working through the exponen- tially large set of possible solutions to the problem, it does this without ever examining them all explicitly. It is because of this careful balancing act that dynamic programming can be a tricky technique to get used to; it typically takes a reasonable amount of practice before one is fully comfortable with it.

With this in mind, we now turn to a first example of dynamic program- ming: the Weighted Interval Scheduling Problem that we defined back in Section 1.2. We are going to develop a dynamic programming algorithm for this problem in two stages: first as a recursive procedure that closely resembles brute-force search; and then, by reinterpreting this procedure, as an iterative algorithm that works by building up solutions to larger and larger subproblems.

6.1 Weighted Interval Scheduling: A Recursive Procedure

We have seen that a particular greedy algorithm produces an optimal solution to the Interval Scheduling Problem, where the goal is to accept as large a set of nonoverlapping intervals as possible. The Weighted Interval Scheduling Problem is a strictly more general version, in which each interval has a certain value (or weight), and we want to accept a set of maximum value.

Designing a Recursive Algorithm

Since the original Interval Scheduling Problem is simply the special case in which all values are equal to 1, we know already that most greedy algorithms will not solve this problem optimally. But even the algorithm that worked before (repeatedly choosing the interval that ends earliest) is no longer optimal in this more general setting, as the simple example in Figure 6.1 shows.

Indeed, no natural greedy algorithm is known for this problem, which is what motivates our switch to dynamic programming. As discussed above, we will begin our introduction to dynamic programming with a recursive type of algorithm for this problem, and then in the next section we’ll move to a more iterative method that is closer to the style we use in the rest of this chapter.

Index 1

2 3

Value = 1

Value = 3

Value = 1

Figure 6.1 A simple instance of weighted interval scheduling.

Index 1

2

3

4

5

6

v1 = 2

v2 = 4

p(1) = 0 p(2) = 0 p(3) = 1 p(4) = 0 p(5) = 3 p(6) = 3

6.1 Weighted Interval Scheduling: A Recursive Procedure

253

We use the notation from our discussion of Interval Scheduling in Sec-

tion 1.2. We have n requests labeled 1, . . . , n, with each request i specifying a

start time si and a finish time fi. Each interval i now also has a value, or weight

vi. Two intervals are compatible if they do not overlap. The goal of our current

problem is to select a subset S ⊆ {1, . . . , n} of mutually compatible intervals,

i∈S vi. Let’s suppose that the requests are sorted in order of nondecreasing finish time: f1 ≤ f2 ≤ . . . ≤ fn. We’ll say a request i comes before a request j if i < j. This will be the natural left-to-right order in which we’ll consider intervals. To help in talking about this order, we define p(j), for an interval j, to be the largest index i < j such that intervals i and j are disjoint. In other words, i is the leftmost interval that ends before j begins. We define p(j) = 0 if no request i < j is disjoint from j. An example of the definition of p(j) is shown
in Figure 6.2.
Now, given an instance of the Weighted Interval Scheduling Problem, let’s consider an optimal solution O, ignoring for now that we have no idea what it is. Here’s something completely obvious that we can say about O: either interval n (the last one) belongs to O, or it doesn’t. Suppose we explore both sides of this dichotomy a little further. If n ∈ O, then clearly no interval indexed strictly between p(n) and n can belong to O, because by the definition of p(n), we know that intervals p(n)+1,p(n)+2,...,n−1 all overlap interval n. Moreover, if n ∈ O, then O must include an optimal solution to the problem consisting of requests {1, . . . , p(n)}—for if it didn’t, we could replace O’s choice of requests from {1, . . . , p(n)} with a better one, with no danger of overlapping request n.
so as to maximize the sum of the values of the selected intervals,
v3 = 4
v4 = 7
v5 = 2
v6 = 1
An instance of weighted interval scheduling with the functions p(j) defined for each interval j.
Figure 6.2
254
Chapter 6 Dynamic Programming
On the other hand, if n ̸∈ O, then O is simply equal to the optimal solution to the problem consisting of requests {1, . . . , n − 1}. This is by completely analogous reasoning: we’re assuming that O does not include request n; so if it does not choose the optimal set of requests from {1, . . . , n − 1}, we could replace it with a better one.
All this suggests that finding the optimal solution on intervals {1, 2, . . . , n} involves looking at the optimal solutions of smaller problems of the form {1, 2, . . . , j}. Thus, for any value of j between 1 and n, let Oj denote the optimal solution to the problem consisting of requests {1, . . . , j}, and let OPT(j) denote the value of this solution. (We define OPT(0) = 0, based on the convention that this is the optimum over an empty set of intervals.) The optimal solution we’re seeking is precisely On, with value OPT(n). For the optimal solution Oj on {1, 2, . . . , j}, our reasoning above (generalizing from the case in which j = n) says that either j ∈ Oj, in which case OPT(j) = vj + OPT(p(j)), or j ̸∈ Oj, in which case OPT(j) = OPT(j − 1). Since these are precisely the two possible choices (j ∈ Oj or j ̸∈ Oj), we can further say that
(6.1) OPT(j) = max(vj + OPT(p(j)), OPT(j − 1)).
And how do we decide whether n belongs to the optimal solution Oj? This too is easy: it belongs to the optimal solution if and only if the first of the options above is at least as good as the second; in other words,
(6.2) Request j belongs to an optimal solution on the set {1, 2, . . . , j} if and only if
vj + OPT(p(j)) ≥ OPT(j − 1).
These facts form the first crucial component on which a dynamic pro- gramming solution is based: a recurrence equation that expresses the optimal solution (or its value) in terms of the optimal solutions to smaller subproblems.
Despite the simple reasoning that led to this point, (6.1) is already a significant development. It directly gives us a recursive algorithm to compute OPT(n), assuming that we have already sorted the requests by finishing time and computed the values of p(j) for each j.
Compute-Opt(j) If j=0 then
Return 0 Else
Return max(vj+Compute-Opt(p(j)), Compute-Opt(j − 1)) Endif
6.1 Weighted Interval Scheduling: A Recursive Procedure
255
The correctness of the algorithm follows directly by induction on j: (6.3) Compute-Opt(j) correctly computes OPT(j) for each j = 1, 2, . . . , n.
Proof. By definition OPT(0) = 0. Now, take some j > 0, and suppose by way of induction that Compute-Opt(i) correctly computes OPT(i) for all i < j. By the induction hypothesis, we know that Compute-Opt(p(j)) = OPT(p(j)) and Compute-Opt(j − 1) = OPT(j − 1); and hence from (6.1) it follows that
OPT(j) = max(vj + Compute-Opt(p(j)), Compute-Opt(j − 1)) = Compute-Opt(j).
Unfortunately, if we really implemented the algorithm Compute-Opt as just written, it would take exponential time to run in the worst case. For example, see Figure 6.3 for the tree of calls issued for the instance of Figure 6.2: the tree widens very quickly due to the recursive branching. To take a more extreme example, on a nicely layered instance like the one in Figure 6.4, where p(j) = j − 2 for each j = 2, 3, 4, . . . , n, we see that Compute-Opt(j) generates separate recursive calls on problems of sizes j − 1 and j − 2. In other words, the total number of calls made to Compute-Opt on this instance will grow
OPT(5)
OPT(4) OPT(3)
OPT(3)
OPT(6)
OPT(2)
OPT(1)
OPT(3)
OPT(2)
OPT(1)
OPT(2) OPT(1)
OPT(1) OPT(1)
OPT(1)
The tree of subproblems grows very quickly.
Figure 6.3 The tree of subproblems called by Compute-Opt on the problem instance of Figure 6.2.
256
Chapter 6 Dynamic Programming
Figure 6.4 An instance of weighted interval scheduling on which the simple Compute- Opt recursion will take exponential time. The values of all intervals in this instance are 1.
like the Fibonacci numbers, which increase exponentially. Thus we have not achieved a polynomial-time solution.
Memoizing the Recursion
In fact, though, we’re not so far from having a polynomial-time algorithm. A fundamental observation, which forms the second crucial component of a dynamic programming solution, is that our recursive algorithm Compute- Opt is really only solving n + 1 different subproblems: Compute-Opt(0), Compute-Opt(1), . . . , Compute-Opt(n). The fact that it runs in exponential time as written is simply due to the spectacular redundancy in the number of times it issues each of these calls.
How could we eliminate all this redundancy? We could store the value of Compute-Opt in a globally accessible place the first time we compute it and then simply use this precomputed value in place of all future recursive calls. This technique of saving values that have already been computed is referred to as memoization.
We implement the above strategy in the more “intelligent” procedure M- Compute-Opt. This procedure will make use of an array M[0 . . . n]; M[j] will start with the value “empty,” but will hold the value of Compute-Opt(j) as soon as it is first determined. To determine OPT(n), we invoke M-Compute- Opt(n).
M-Compute-Opt(j) If j=0 then Return 0
Else if M[j] is not empty then Return M[j]
Else
6.1 Weighted Interval Scheduling: A Recursive Procedure
257
Define M[j] = max(vj+M-Compute-Opt(p(j)), M-Compute-Opt(j − 1)) Return M[j]
Endif
Analyzing the Memoized Version
Clearly, this looks very similar to our previous implementation of the algo- rithm; however, memoization has brought the running time way down.
(6.4) The running time of M-Compute-Opt(n) is O(n) (assuming the input intervals are sorted by their finish times).
Proof. ThetimespentinasinglecalltoM-Compute-OptisO(1),excludingthe time spent in recursive calls it generates. So the running time is bounded by a constant times the number of calls ever issued to M-Compute-Opt. Since the implementation itself gives no explicit upper bound on this number of calls, we try to find a bound by looking for a good measure of “progress.”
The most useful progress measure here is the number of entries in M that are not “empty.” Initially this number is 0; but each time the procedure invokes the recurrence, issuing two recursive calls to M-Compute-Opt, it fills in a new entry, and hence increases the number of filled-in entries by 1. Since M has only n + 1 entries, it follows that there can be at most O(n) calls to M-Compute- Opt, and hence the running time of M-Compute-Opt(n) is O(n), as desired.
Computing a Solution in Addition to Its Value
So far we have simply computed the value of an optimal solution; presumably we want a full optimal set of intervals as well. It would be easy to extend M-Compute-Opt so as to keep track of an optimal solution in addition to its value: we could maintain an additional array S so that S[i] contains an optimal set of intervals among {1, 2, . . . , i}. Naively enhancing the code to maintain the solutions in the array S, however, would blow up the running time by an additional factor of O(n): while a position in the M array can be updated in O(1) time, writing down a set in the S array takes O(n) time. We can avoid this O(n) blow-up by not explicitly maintaining S, but rather by recovering the optimal solution from values saved in the array M after the optimum value has been computed.
We know from (6.2) that j belongs to an optimal solution for the set of intervals {1, . . . , j} if and only if vj + OPT(p(j)) ≥ OPT(j − 1). Using this observation, we get the following simple procedure, which “traces back” through the array M to find the set of intervals in an optimal solution.
258
Chapter 6 Dynamic Programming
Find-Solution(j) If j=0 then
Output nothing
Else
If vj+M[p(j)]≥M[j−1] then
Output j together with the result of Find-Solution(p(j))
Else
Output the result of Find-Solution(j − 1)
Endif Endif
Since Find-Solution calls itself recursively only on strictly smaller val- ues, it makes a total of O(n) recursive calls; and since it spends constant time per call, we have
(6.5) Given the array M of the optimal values of the sub-problems, Find- Solution returns an optimal solution in O(n) time.
6.2 Principles of Dynamic Programming: Memoization or Iteration over Subproblems
We now use the algorithm for the Weighted Interval Scheduling Problem developed in the previous section to summarize the basic principles of dynamic programming, and also to offer a different perspective that will be fundamental to the rest of the chapter: iterating over subproblems, rather than computing solutions recursively.
In the previous section, we developed a polynomial-time solution to the Weighted Interval Scheduling Problem by first designing an exponential-time recursive algorithm and then converting it (by memoization) to an efficient recursive algorithm that consulted a global array M of optimal solutions to subproblems. To really understand what is going on here, however, it helps to formulate an essentially equivalent version of the algorithm. It is this new formulation that most explicitly captures the essence of the dynamic program- ming technique, and it will serve as a general template for the algorithms we develop in later sections.
Designing the Algorithm
The key to the efficient algorithm is really the array M. It encodes the notion that we are using the value of optimal solutions to the subproblems on intervals {1,2,...,j} for each j, and it uses (6.1) to define the value of M[j]based on
values that come earlier in the array. Once we have the array M, the problem is solved: M[n]contains the value of the optimal solution on the full instance, and Find-Solution can be used to trace back through M efficiently and return an optimal solution itself.
The point to realize, then, is that we can directly compute the entries in M by an iterative algorithm, rather than using memoized recursion. We just start with M[0]= 0 and keep incrementing j; each time we need to determine a value M[j], the answer is provided by (6.1). The algorithm looks as follows.
Iterative-Compute-Opt
M[0]=0
For j=1,2,...,n
M[j]= max(vj + M[p(j)], M[j − 1]) Endfor
Analyzing the Algorithm
By exact analogy with the proof of (6.3), we can prove by induction on j that this algorithm writes OPT(j) in array entry M[j]; (6.1) provides the induction step. Also, as before, we can pass the filled-in array M to Find-Solution to get an optimal solution in addition to the value. Finally, the running time of Iterative-Compute-Opt is clearly O(n), since it explicitly runs for n iterations and spends constant time in each.
An example of the execution of Iterative-Compute-Opt is depicted in Figure 6.5. In each iteration, the algorithm fills in one additional entry of the array M, by comparing the value of vj + M[p(j)] to the value of M[j − 1].
A Basic Outline of Dynamic Programming
This, then, provides a second efficient algorithm to solve the Weighted In- terval Scheduling Problem. The two approaches clearly have a great deal of conceptual overlap, since they both grow from the insight contained in the recurrence (6.1). For the remainder of the chapter, we will develop dynamic programming algorithms using the second type of approach—iterative build- ing up of subproblems—because the algorithms are often simpler to express this way. But in each case that we consider, there is an equivalent way to formulate the algorithm as a memoized recursion.
Most crucially, the bulk of our discussion about the particular problem of selecting intervals can be cast more generally as a rough template for designing dynamic programming algorithms. To set about developing an algorithm based on dynamic programming, one needs a collection of subproblems derived from the original problem that satisfies a few basic properties.
6.2 Principles of Dynamic Programming
259
260
Chapter 6 Dynamic Programming
Index 0123456
0
2
1 2 3 4 5 6
w1 = 2
p(1) p(2) p(3) p(4) p(5) p(6)
=0 =0 =1 =0 =3 =3
M=
w2 = 4
0
2
4
w3 = 4
w4 = 7
w5 = 2 w6 = 1
0
2
4
6
0
2
4
6
7
0
2
4
6
7
8
0
2
4
6
7
8
8
(a)
(b)
Figure 6.5 Part (b) shows the iterations of Iterative-Compute-Opt on the sample instance of Weighted Interval Scheduling depicted in part (a).
(i) There are only a polynomial number of subproblems.
(ii) The solution to the original problem can be easily computed from the solutions to the subproblems. (For example, the original problem may actually be one of the subproblems.)
(iii) Thereisanaturalorderingonsubproblemsfrom“smallest”to“largest,” together with an easy-to-compute recurrence (as in (6.1) and (6.2)) that allows one to determine the solution to a subproblem from the solutions to some number of smaller subproblems.
Naturally, these are informal guidelines. In particular, the notion of “smaller” in part (iii) will depend on the type of recurrence one has.
We will see that it is sometimes easier to start the process of designing such an algorithm by formulating a set of subproblems that looks natural, and then figuring out a recurrence that links them together; but often (as happened in the case of weighted interval scheduling), it can be useful to first define a recurrence by reasoning about the structure of an optimal solution, and then determine which subproblems will be necessary to unwind the recurrence. This chicken-and-egg relationship between subproblems and recurrences is a subtle issue underlying dynamic programming. It’s never clear that a collection of subproblems will be useful until one finds a recurrence linking them together; but it can be difficult to think about recurrences in the absence of the “smaller” subproblems that they build on. In subsequent sections, we will develop further practice in managing this design trade-off.
Error(L, P) =
n
(yi − axi − b)2.
i=1
6.3 Segmented Least Squares: Multi-way Choices
261
6.3 Segmented Least Squares: Multi-way Choices
We now discuss a different type of problem, which illustrates a slightly more complicated style of dynamic programming. In the previous section, we developed a recurrence based on a fundamentally binary choice: either the interval n belonged to an optimal solution or it didn’t. In the problem we consider here, the recurrence will involve what might be called “multi- way choices”: at each step, we have a polynomial number of possibilities to consider for the structure of the optimal solution. As we’ll see, the dynamic programming approach adapts to this more general situation very naturally.
As a separate issue, the problem developed in this section is also a nice illustration of how a clean algorithmic definition can formalize a notion that initially seems too fuzzy and nonintuitive to work with mathematically.
The Problem
Often when looking at scientific or statistical data, plotted on a two- dimensional set of axes, one tries to pass a “line of best fit” through the data, as in Figure 6.6.
This is a foundational problem in statistics and numerical analysis, formu- lated as follows. Suppose our data consists of a set P of n points in the plane, denoted (x1, y1), (x2, y2), . . . , (xn, yn); and suppose x1 < x2 < . . . < xn. Given a line L defined by the equation y=ax+b, we say that the error of L with respect to P is the sum of its squared “distances” to the points in P:
Figure 6.6 A “line of best fit.”
262
Chapter 6 Dynamic Programming
Figure 6.7 A set of points that lie approximately on two lines.
A natural goal is then to find the line with minimum error; this turns out to have a nice closed-form solution that can be easily derived using calculus. Skipping the derivation here, we simply state the result: The line of minimum error is y=ax+b, where
n x y − x y y − a x a=iii iiiiandb=ii ii.
n x2 − x 2 n ii ii
Now, here’s a kind of issue that these formulas weren’t designed to cover. Often we have data that looks something like the picture in Figure 6.7. In this case, we’d like to make a statement like: “The points lie roughly on a sequence of two lines.” How could we formalize this concept?
Essentially, any single line through the points in the figure would have a terrible error; but if we use two lines, we could achieve quite a small error. So we could try formulating a new problem as follows: Rather than seek a single line of best fit, we are allowed to pass an arbitrary set of lines through the points, and we seek a set of lines that minimizes the error. But this fails as a good problem formulation, because it has a trivial solution: if we’re allowed to fit the points with an arbitrarily large set of lines, we could fit the points perfectly by having a different line pass through each pair of consecutive points in P.
At the other extreme, we could try “hard-coding” the number two into the problem; we could seek the best fit using at most two lines. But this too misses a crucial feature of our intuition: We didn’t start out with a preconceived idea that the points lay approximately on two lines; we concluded that from looking at the picture. For example, most people would say that the points in Figure 6.8 lie approximately on three lines.
6.3 Segmented Least Squares: Multi-way Choices
263
Figure 6.8 A set of points that lie approximately on three lines.
Thus, intuitively, we need a problem formulation that requires us to fit the points well, using as few lines as possible. We now formulate a problem— the Segmented Least Squares Problem—that captures these issues quite cleanly. The problem is a fundamental instance of an issue in data mining and statistics known as change detection: Given a sequence of data points, we want to identify a few points in the sequence at which a discrete change occurs (in this case, a change from one linear approximation to another).
Formulating the Problem As in the discussion above, we are given a set of points P = {(x1, y1), (x2, y2), . . . , (xn , yn)}, with x1 < x2 < . . . < xn. We will use pi to denote the point (xi,yi). We must first partition P into some number of segments. Each segment is a subset of P that represents a contiguous set of x-coordinates; that is, it is a subset of the form {pi, pi+1, . . . , pj−1, pj} for some indices i ≤ j. Then, for each segment S in our partition of P, we compute the line minimizing the error with respect to the points in S, according to the formulas above.
The penalty of a partition is defined to be a sum of the following terms. (i) The number of segments into which we partition P, times a fixed, given
multiplier C > 0.

(ii) For each segment, the error value of the optimal line through that

segment.

Our goal in the Segmented Least Squares Problem is to find a partition of minimum penalty. This minimization captures the trade-offs we discussed earlier. We are allowed to consider partitions into any number of segments; as we increase the number of segments, we reduce the penalty terms in part (ii) of the definition, but we increase the term in part (i). (The multiplier C is provided

264

Chapter 6 Dynamic Programming

with the input, and by tuning C, we can penalize the use of additional lines to a greater or lesser extent.)

There are exponentially many possible partitions of P, and initially it is not clear that we should be able to find the optimal one efficiently. We now show how to use dynamic programming to find a partition of minimum penalty in time polynomial in n.

Designing the Algorithm

To begin with, we should recall the ingredients we need for a dynamic program- ming algorithm, as outlined at the end of Section 6.2.We want a polynomial number of subproblems, the solutions of which should yield a solution to the original problem; and we should be able to build up solutions to these subprob- lems using a recurrence. As with the Weighted Interval Scheduling Problem, it helps to think about some simple properties of the optimal solution. Note, however, that there is not really a direct analogy to weighted interval sched- uling: there we were looking for a subset of n objects, whereas here we are seeking to partition n objects.

For segmented least squares, the following observation is very useful: The last point pn belongs to a single segment in the optimal partition, and that segment begins at some earlier point pi. This is the type of observation that can suggest the right set of subproblems: if we knew the identity of the last segment pi , . . . , pn (see Figure 6.9), then we could remove those points from consideration and recursively solve the problem on the remaining points p1, . . . , pi−1.

OPT(i – 1)

i

n

Figure 6.9 A possible solution: a single line segment fits points pi , pi+1, . . . , pn , and then an optimal solution is found for the remaining points p1, p2, . . . , pi−1.

6.3 Segmented Least Squares: Multi-way Choices

265

Suppose we let OPT(i) denote the optimum solution for the points p1, . . . , pi, and we let ei,j denote the minimum error of any line with re- spect to pi, pi+1, . . . , pj. (We will write OPT(0) = 0 as a boundary case.) Then our observation above says the following.

(6.6) If the last segment of the optimal partition is pi, . . . , pn, then the value of the optimal solution is OPT(n) = ei,n + C + OPT(i − 1).

Using the same observation for the subproblem consisting of the points p1,…,pj, we see that to get OPT(j) we should find the best way to produce a final segment pi , . . . , pj —paying the error plus an additive C for this segment— together with an optimal solution OPT(i − 1) for the remaining points. In other words, we have justified the following recurrence.

(6.7) For the subproblem on the points p1, . . . , pj, OPT(j) = min(ei,j + C + OPT(i − 1)),

1≤i≤j

and the segment pi , . . . , pj is used in an optimum solution for the subproblem

if and only if the minimum is obtained using index i.

The hard part in designing the algorithm is now behind us. From here, we simply build up the solutions OPT(i) in order of increasing i.

Segmented-Least-Squares(n) Array M[0…n]

Set M[0]=0

For all pairs i≤j

Compute the least

Endfor

squares

error

ei , j

for the

segment

pi , . . . , pj

For j=1,2,…,n

Use the recurrence (6.7) to compute M[j]

Endfor Return M[n]

By analogy with the arguments for weighted interval scheduling, the correctness of this algorithm can be proved directly by induction, with (6.7) providing the induction step.

And as in our algorithm for weighted interval scheduling, we can trace back through the array M to compute an optimum partition.

266

Chapter 6 Dynamic Programming

Find-Segments(j) If j=0 then

Output nothing

Else

Find an i that minimizes ei,j+C+M[i−1]

Output the segment {pi, . . . , pj} and the result of

Find-Segments(i − 1)

Endif

Analyzing the Algorithm

Finally, we consider the running time of Segmented-Least-Squares. First we need to compute the values of all the least-squares errors ei,j. To perform a simple accounting of the running time for this, we note that there are O(n2) pairs (i, j) for which this computation is needed; and for each pair (i, j), we can use the formula given at the beginning of this section to compute ei,j in O(n) time. Thus the total running time to compute all ei,j values is O(n3).

Following this, the algorithm has n iterations, for values j = 1, . . . , n. For each value of j, we have to determine the minimum in the recurrence (6.7) to fill in the array entry M[j]; this takes time O(n) for each j, for a total of O(n2). Thus the running time is O(n2) once all the ei,j values have been determined.1

6.4 SubsetSumsandKnapsacks:AddingaVariable

We’re seeing more and more that issues in scheduling provide a rich source of practically motivated algorithmic problems. So far we’ve considered problems in which requests are specified by a given interval of time on a resource, as well as problems in which requests have a duration and a deadline but do not mandate a particular interval during which they need to be done.

In this section, we consider a version of the second type of problem, with durations and deadlines, which is difficult to solve directly using the techniques we’ve seen so far. We will use dynamic programming to solve the problem, but with a twist: the “obvious” set of subproblems will turn out not to be enough, and so we end up creating a richer collection of subproblems. As

1 In this analysis, the running time is dominated by the O(n3) needed to compute all ei,j values. But, in fact, it is possible to compute all these values in O(n2) time, which brings the running time of the full algorithm down to O(n2). The idea, whose details we will leave as an exercise for the reader, is to firstcomputeei,j forallpairs(i,j)wherej−i=1,thenforallpairswherej−i=2,thenj−i=3,and so forth. This way, when we get to a particular ei,j value, we can use the ingredients of the calculation for ei,j−1 to determine ei,j in constant time.

6.4 Subset Sums and Knapsacks: Adding a Variable

267

we will see, this is done by adding a new variable to the recurrence underlying the dynamic program.

The Problem

In the scheduling problem we consider here, we have a single machine that can process jobs, and we have a set of requests {1, 2, . . . , n}. We are only able to use this resource for the period between time 0 and time W, for some number W. Each request corresponds to a job that requires time wi to process. If our goal is to process jobs so as to keep the machine as busy as possible up to the “cut-off” W, which jobs should we choose?

More formally, we are given n items {1, . . . , n}, and each has a given

nonnegative weight wi (for i = 1, . . . , n). We are also given a bound W. We

would like to select a subset S of the items so that w ≤ W and, subject i∈S i

to this restriction, i∈S wi is as large as possible. We will call this the Subset Sum Problem.

This problem is a natural special case of a more general problem called the Knapsack Problem, where each request i has both a value vi and a weight wi. The goal in this more general problem is to select a subset of maximum total value, subject to the restriction that its total weight not exceed W. Knapsack problems often show up as subproblems in other, more complex problems. The name knapsack refers to the problem of filling a knapsack of capacity W as full as possible (or packing in as much value as possible), using a subset of the items {1, . . . , n}. We will use weight or time when referring to the quantities wi and W.

Since this resembles other scheduling problems we’ve seen before, it’s natural to ask whether a greedy algorithm can find the optimal solution. It appears that the answer is no—at least, no efficient greedy rule is known that always constructs an optimal solution. One natural greedy approach to try would be to sort the items by decreasing weight—or at least to do this for all items of weight at most W—and then start selecting items in this order as long as the total weight remains below W. But if W is a multiple of 2, and we have three items with weights {W/2 + 1, W/2, W/2}, then we see that this greedy algorithm will not produce the optimal solution. Alternately, we could sort by increasing weight and then do the same thing; but this fails on inputs like {1, W/2, W/2}.

The goal of this section is to show how to use dynamic programming to solve this problem. Recall the main principles of dynamic programming: We have to come up with a small number of subproblems so that each subproblem can be solved easily from “smaller” subproblems, and the solution to the original problem can be obtained easily once we know the solutions to all

268

Chapter 6 Dynamic Programming

the subproblems. The tricky issue here lies in figuring out a good set of subproblems.

Designing the Algorithm

A False Start One general strategy, which worked for us in the case of Weighted Interval Scheduling, is to consider subproblems involving only the first i requests. We start by trying this strategy here. We use the notation OPT(i), analogously to the notation used before, to denote the best possible solution using a subset of the requests {1, . . . , i}. The key to our method for the Weighted Interval Scheduling Problem was to concentrate on an optimal solution O to our problem and consider two cases, depending on whether or not the last request n is accepted or rejected by this optimum solution. Just as in that case, we have the first part, which follows immediately from the definition of OPT(i).

. If n̸∈O, then OPT(n)=OPT(n−1).

Next we have to consider the case in which n ∈ O. What we’d like here is a simple recursion, which tells us the best possible value we can get for solutions that contain the last request n. For Weighted Interval Scheduling this was easy, as we could simply delete each request that conflicted with request n. In the current problem, this is not so simple. Accepting request n does not immediately imply that we have to reject any other request. Instead, it means that for the subset of requests S ⊆ {1, . . . , n − 1} that we will accept, we have less available weight left: a weight of wn is used on the accepted request n, and we only have W − wn weight left for the set S of remaining requests that we accept. See Figure 6.10.

A Better Solution This suggests that we need more subproblems: To find out the value for OPT(n) we not only need the value of OPT(n − 1), but we also need to know the best solution we can get using a subset of the first n − 1 items and total allowed weight W − wn. We are therefore going to use many more subproblems: one for each initial set {1, . . . , i} of the items, and each possible

W

wn

Figure 6.10 After item n is included in the solution, a weight of wn is used up and there is W − wn available weight left.

6.4 Subset Sums and Knapsacks: Adding a Variable

269

value for the remaining available weight w. Assume that W is an integer, and all requests i = 1, . . . , n have integer weights wi. We will have a subproblem for each i=0,1,…,n and each integer 0≤w≤W. We will use OPT(i,w) to denote the value of the optimal solution using a subset of the items {1, . . . , i} with maximum allowed weight w, that is,

OPT(i, w) as a simple expression in terms of values from smaller problems. Moreover, OPT(n, W) is the quantity we’re looking for in the end. As before, let O denote an optimum solution for the original problem.

. If n ̸∈ O, then OPT(n, W) = OPT(n − 1, W), since we can simply ignore item n.

. Ifn∈O,thenOPT(n,W)=wn+OPT(n−1,W−wn),sincewenowseek to use the remaining capacity of W − wn in an optimal way across items 1, 2, . . . , n − 1.

When the nth item is too big, that is, W < wn, then we must have OPT(n, W) = OPT(n − 1, W). Otherwise, we get the optimum solution allowing all n requests by taking the better of these two options. Using the same line of argument for the subproblem for items {1, . . . , i}, and maximum allowed weight w, gives us the following recurrence.
(6.8) If w < wi then OPT(i, w) = OPT(i − 1, w). Otherwise OPT(i,w)=max(OPT(i−1,w),wi +OPT(i−1,w−wi)).
As before, we want to design an algorithm that builds up a table of all OPT(i, w) values while computing each of them at most once.
Subset-Sum(n, W)
Array M[0...n,0...W]
Initialize M[0,w]=0 for each w=0,1,...,W For i=1,2,...,n
For w=0,...,W
Use the recurrence (6.8) to compute M[i, w]
Endfor
Endfor
Return M[n, W]
OPT(i, w) = max S
wj ,
where the maximum is over subsets S⊆{1,...,i} that satisfy
j∈S
Using this new set of subproblems, we will be able to express the value
j∈S
w ≤w. j
270
Chapter 6
Dynamic Programming
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
n
i i –1
2 1 0
Figure 6.11 The two-dimensional table of OPT values. The leftmost column and bottom row is always 0. The entry for OPT(i, w) is computed from the two other entries OPT(i − 1, w) and OPT(i − 1, w − wi), as indicated by the arrows.
Using (6.8) one can immediately prove by induction that the returned value M[n, W] is the optimum solution value for the requests 1, . . . , n and available weight W.
Analyzing the Algorithm
Recall the tabular picture we considered in Figure 6.5, associated with weighted interval scheduling, where we also showed the way in which the ar- ray M for that algorithm was iteratively filled in. For the algorithm we’ve just designed, we can use a similar representation, but we need a two- dimensional table, reflecting the two-dimensional array of subproblems that is being built up. Figure 6.11 shows the building up of subproblems in this case: the value M[i, w] is computed from the two other values M[i − 1, w] and M[i − 1, w − wi].
As an example of this algorithm executing, consider an instance with weight limit W = 6, and n = 3 items of sizes w1 = w2 = 2 and w3 = 3. We find that the optimal value OPT(3, 6) = 5 (which we get by using the third item and one of the first two items). Figure 6.12 illustrates the way the algorithm fills in the two-dimensional table of OPT values row by row.
Next we will worry about the running time of this algorithm. As before in the case of weighted interval scheduling, we are building up a table of solutions M, and we compute each of the values M[i, w] in O(1) time using the previous values. Thus the running time is proportional to the number of entries in the table.
012w–wiw W
6.4 Subset Sums and Knapsacks: Adding a Variable
271
Knapsack size W = 6, items w1 = 2, w2 = 2, w3 = 3
33 22 11 00
0123456 0123456
Initial values Filling in values for i = 1
33 22 11 00
0123456 0123456
Filling in values for i = 2 Filling in values for i = 3
Figure 6.12 The iterations of the algorithm on a sample instance of the Subset Sum
Problem.
(6.9) The Subset-Sum(n , W ) Algorithm correctly computes the optimal value of the problem, and runs in O(nW) time.
Note that this method is not as efficient as our dynamic program for the Weighted Interval Scheduling Problem. Indeed, its running time is not a polynomial function of n; rather, it is a polynomial function of n and W, the largest integer involved in defining the problem. We call such algorithms pseudo-polynomial. Pseudo-polynomial algorithms can be reasonably efficient when the numbers {wi} involved in the input are reasonably small; however, they become less practical as these numbers grow large.
To recover an optimal set S of items, we can trace back through the array M by a procedure similar to those we developed in the previous sections.
(6.10) Given a table M of the optimal values of the subproblems, the optimal set S can be found in O(n) time.
Extension: The Knapsack Problem
The Knapsack Problem is a bit more complex than the scheduling problem we discussed earlier. Consider a situation in which each item i has a nonnegative weight wi as before, and also a distinct value vi. Our goal is now to find a
0
0
0
0
0
0
0
0
0
2
2
2
2
2
0
0
0
0
0
0
0
0
0
2
2
4
4
4
0
0
2
2
2
2
2
0
0
0
0
0
0
0
0
0
2
3
4
5
5
0
0
2
2
4
4
4
0
0
2
2
2
2
2
0
0
0
0
0
0
0
272
Chapter 6 Dynamic Programming
weight of the set should not exceed W: w ≤ W.
subset S of maximum value
i∈S vi, subject to the restriction that the total
i∈S i
It is not hard to extend our dynamic programming algorithm to this more
general problem. We use the analogous set of subproblems, OPT(i, w), to denote the value of the optimal solution using a subset of the items {1, . . . , i} and maximum available weight w. We consider an optimal solution O, and identify two cases depending on whether or not n ∈ O.
. If n̸∈O, then OPT(n,W)=OPT(n−1,W).
. If n∈O, then OPT(n,W)=vn +OPT(n−1,W −wn).
Using this line of argument for the subproblems implies the following analogue of (6.8).
(6.11) If w < wi then OPT(i, w) = OPT(i − 1, w). Otherwise OPT(i,w)=max(OPT(i−1,w),vi +OPT(i−1,w−wi)).
Using this recurrence, we can write down a completely analogous dynamic programming algorithm, and this implies the following fact.
(6.12) The Knapsack Problem can be solved in O(nW) time.
6.5 RNA Secondary Structure: Dynamic Programming over Intervals
In the Knapsack Problem, we were able to formulate a dynamic programming algorithm by adding a new variable. A different but very common way by which one ends up adding a variable to a dynamic program is through the following scenario. We start by thinking about the set of subproblems on {1, 2, . . . , j}, for all choices of j, and find ourselves unable to come up with a natural recurrence. We then look at the larger set of subproblems on {i,i+1,...,j} for all choices of i and j (where i≤j), and find a natural recurrence relation on these subproblems. In this way, we have added the second variable i; the effect is to consider a subproblem for every contiguous interval in {1,2,...,n}.
There are a few canonical problems that fit this profile; those of you who have studied parsing algorithms for context-free grammars have probably seen at least one dynamic programming algorithm in this style. Here we focus on the problem of RNA secondary structure prediction, a fundamental issue in computational biology.
6.5 RNA Secondary Structure: Dynamic Programming over Intervals
273
GC AA
UA
AU
CG UAGCGC ACGAUAAG
GGCUAUUA CGAGCG GC
AU CG C A
Figure 6.13 An RNA secondary structure. Thick lines connect adjacent elements of the sequence; thin lines indicate pairs of elements that are matched.
The Problem
As one learns in introductory biology classes, Watson and Crick posited that double-stranded DNA is “zipped” together by complementary base-pairing. Each strand of DNA can be viewed as a string of bases, where each base is drawn from the set {A, C, G, T}.2 The bases A and T pair with each other, and the bases C and G pair with each other; it is these A-T and C-G pairings that hold the two strands together.
Now, single-stranded RNA molecules are key components in many of the processes that go on inside a cell, and they follow more or less the same structural principles. However, unlike double-stranded DNA, there’s no “second strand” for the RNA to stick to; so it tends to loop back and form base pairs with itself, resulting in interesting shapes like the one depicted in Figure 6.13. The set of pairs (and resulting shape) formed by the RNA molecule through this process is called the secondary structure, and understanding the secondary structure is essential for understanding the behavior of the molecule.
2 Adenine, cytosine, guanine, and thymine, the four basic units of DNA.
274
Chapter 6 Dynamic Programming
For our purposes, a single-stranded RNA molecule can be viewed as a sequence of n symbols (bases) drawn from the alphabet {A, C, G, U}.3 Let B = b1b2 . . . bn be a single-stranded RNA molecule, where each bi ∈ {A, C, G, U}. To a first approximation, one can model its secondary structure as follows. As usual, we require that A pairs with U, and C pairs with G; we also require that each base can pair with at most one other base—in other words, the set of base pairs forms a matching. It also turns out that secondary structures are (again, to a first approximation) “knot-free,” which we will formalize as a kind of noncrossing condition below.
Thus, concretely, we say that a secondary structure on B is a set of pairs S = {(i, j)}, where i, j ∈ {1, 2, . . . , n}, that satisfies the following conditions.
(i) (Nosharpturns.)TheendsofeachpairinSareseparatedbyatleastfour intervening bases; that is, if (i, j) ∈ S, then i < j − 4.
(ii) TheelementsofanypairinSconsistofeither{A,U}or{C,G}(ineither order).
(iii) S is a matching: no base appears in more than one pair.
(iv) (The noncrossing condition.) If (i, j) and (k, l) are two pairs in S, then
we cannot have i < k < j < l. (See Figure 6.14 for an illustration.)
Note that the RNA secondary structure in Figure 6.13 satisfies properties (i) through (iv). From a structural point of view, condition (i) arises simply because the RNA molecule cannot bend too sharply; and conditions (ii) and (iii) are the fundamental Watson-Crick rules of base-pairing. Condition (iv) is the striking one, since it’s not obvious why it should hold in nature. But while there are sporadic exceptions to it in real molecules (via so-called pseudo- knotting), it does turn out to be a good approximation to the spatial constraints on real RNA secondary structures.
Now, out of all the secondary structures that are possible for a single RNA molecule, which are the ones that are likely to arise under physiological conditions? The usual hypothesis is that a single-stranded RNA molecule will form the secondary structure with the optimum total free energy. The correct model for the free energy of a secondary structure is a subject of much debate; but a first approximation here is to assume that the free energy of a secondary structure is proportional simply to the number of base pairs that it contains.
Thus, having said all this, we can state the basic RNA secondary structure prediction problem very simply: We want an efficient algorithm that takes
3 Note that the symbol T from the alphabet of DNA has been replaced by a U , but this is not important for us here.
G GU
CA CG AU UA GC UA
ACAUGAUGGCCAUGU
6.5 RNA Secondary Structure: Dynamic Programming over Intervals
275
(a) (b)
Figure 6.14 Two views of an RNA secondary structure. In the second view, (b), the string has been “stretched” lengthwise, and edges connecting matched pairs appear as noncrossing “bubbles” over the string.
a single-stranded RNA molecule B = b1b2 . . . bn and determines a secondary structure S with the maximum possible number of base pairs.
Designing and Analyzing the Algorithm
A First Attempt at Dynamic Programming The natural first attempt to apply dynamic programming would presumably be based on the following subproblems: We say that OPT(j) is the maximum number of base pairs in a secondary structure on b1b2 . . . bj. By the no-sharp-turns condition above, we know that OPT(j) = 0 for j ≤ 5; and we know that OPT(n) is the solution we’re looking for.
The trouble comes when we try writing down a recurrence that expresses OPT(j) in terms of the solutions to smaller subproblems. We can get partway there: in the optimal secondary structure on b1b2 . . . bj, it’s the case that either
. j is not involved in a pair; or
. j pairs with t for some t < j − 4.
In the first case, we just need to consult our solution for OPT(j − 1). The second case is depicted in Figure 6.15(a); because of the noncrossing condition, we now know that no pair can have one end between 1 and t − 1 and the other end between t + 1 and j − 1. We’ve therefore effectively isolated two new subproblems: one on the bases b1b2 . . . bt−1, and the other on the bases bt+1 . . . bj−1. The first is solved by OPT(t − 1), but the second is not on our list of subproblems, because it does not begin with b1.
276
Chapter 6 Dynamic Programming
1 2
i
Including the pair (t, j) results in two independent subproblems.
t– 1 t t + 1 (a)
t– 1 t t + 1 (b)
j– 1 j
j– 1 j
Figure 6.15 Schematic views of the dynamic programming recurrence using (a) one variable, and (b) two variables.
This is the insight that makes us realize we need to add a variable. We need to be able to work with subproblems that do not begin with b1; in other words, we need to consider subproblems on bibi+1 . . . bj for all choices of i ≤ j.
Dynamic Programming over Intervals Once we make this decision, our previous reasoning leads straight to a successful recurrence. Let OPT(i, j) denote the maximum number of base pairs in a secondary structure on bibi+1 . . . bj. The no-sharp-turns condition lets us initialize OPT(i, j) = 0 whenever i ≥ j − 4. (For notational convenience, we will also allow ourselves to refer to OPT(i, j) even when i > j; in this case, its value is 0.)

Now, in the optimal secondary structure on bibi+1 . . . bj, we have the same alternatives as before:

. j is not involved in a pair; or

. j pairs with t for some t < j − 4.
In the first case, we have OPT(i, j) = OPT(i, j − 1). In the second case, depicted in Figure 6.15(b), we recur on the two subproblems OPT(i, t − 1) and OPT(t + 1, j − 1); as argued above, the noncrossing condition has isolated these two subproblems from each other.
We have therefore justified the following recurrence.
(6.13) OPT(i, j) = max(OPT(i, j − 1), max(1 + OPT(i, t − 1) + OPT(t + 1, j − 1))), where the max is taken over t such that bt and bj are an allowable base pair (under conditions (i) and (ii) from the definition of a secondary structure).
Now we just have to make sure we understand the proper order in which to build up the solutions to the subproblems. The form of (6.13) reveals that we’re always invoking the solution to subproblems on shorter intervals: those
6.5 RNA Secondary Structure: Dynamic Programming over Intervals
277
RNA sequence ACCGGUAGU
444 333 222
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
1
0
0
0
0
0
0
1
1
0
0
1
1
1
i = 1
j= 6 7 8 9
Initial values
i = 1
j= 6 7 8 9
Filling in the values for k = 5
i= 1
j= 6 7 8 9
Filling in the values for k = 6
0
0
0
0
0
0
1
1
0
0
1
1
1
1
1
0
0
0
0
0
0
1
1
0
0
1
1
1
1
1
2
44 33 22
i =1
j= 6 7 8 9
Filling in the values for k = 7
i = 1
j= 6 7 8 9
Filling in the values for k = 8
Figure 6.16 The iterations of the algorithm on a sample instance of the RNA Secondary Structure Prediction Problem.
for which k = j − i is smaller. Thus things will work without any trouble if we build up the solutions in order of increasing interval length.
Initialize OPT(i,j)=0 whenever i≥j−4 For k=5, 6,...,n−1
For i=1,2,...n−k
Set j=i+k
Compute OPT(i, j) using the recurrence in (6.13)
Endfor
Endfor
Return OPT(1, n)
As an example of this algorithm executing, we consider the input ACCGGUAGU, a subsequence of the sequence in Figure 6.14. As with the Knapsack Problem, we need two dimensions to depict the array M: one for the left endpoint of the interval being considered, and one for the right end- point. In the figure, we only show entries corresponding to [i, j] pairs with i < j − 4, since these are the only ones that can possibly be nonzero.
It is easy to bound the running time: there are O(n2) subproblems to solve, and evaluating the recurrence in (6.13) takes time O(n) for each. Thus the running time is O(n3).
278
Chapter 6 Dynamic Programming
As always, we can recover the secondary structure itself (not just its value) by recording how the minima in (6.13) are achieved and tracing back through the computation.
6.6 Sequence Alignment
For the remainder of this chapter, we consider two further dynamic program- ming algorithms that each have a wide range of applications. In the next two sections we discuss sequence alignment, a fundamental problem that arises in comparing strings. Following this, we turn to the problem of computing shortest paths in graphs when edges have costs that may be negative.
The Problem
Dictionaries on the Web seem to get more and more useful: often it seems easier to pull up a bookmarked online dictionary than to get a physical dictionary down from the bookshelf. And many online dictionaries offer functions that you can’t get from a printed one: if you’re looking for a definition and type in a word it doesn’t contain—say, ocurrance—it will come back and ask, “Perhaps you mean occurrence?” How does it do this? Did it truly know what you had in mind?
Let’s defer the second question to a different book and think a little about the first one. To decide what you probably meant, it would be natural to search the dictionary for the word most “similar” to the one you typed in. To do this, we have to answer the question: How should we define similarity between two words or strings?
Intuitively, we’d like to say that ocurrance and occurrence are similar because we can make the two words identical if we add a c to the first word and change the a to an e. Since neither of these changes seems so large, we conclude that the words are quite similar. To put it another way, we can nearly line up the two words letter by letter:
o-currance
occurrence
The hyphen (-) indicates a gap where we had to add a letter to the second word to get it to line up with the first. Moreover, our lining up is not perfect in that an e is lined up with an a.
We want a model in which similarity is determined roughly by the number of gaps and mismatches we incur when we line up the two words. Of course, there are many possible ways to line up the two words; for example, we could have written
6.6 Sequence Alignment
279
o-curr-ance
occurre-nce
which involves three gaps and no mismatches. Which is better: one gap and one mismatch, or three gaps and no mismatches?
This discussion has been made easier because we know roughly what the correspondence ought to look like. When the two strings don’t look like English words—for example, abbbaabbbbaab and ababaaabbbbbab—it may take a little work to decide whether they can be lined up nicely or not:
abbbaa--bbbbaab
ababaaabbbbba-b
Dictionary interfaces and spell-checkers are not the most computationally intensive application for this type of problem. In fact, determining similarities among strings is one of the central computational problems facing molecular biologists today.
Strings arise very naturally in biology: an organism’s genome—its full set of genetic material—is divided up into giant linear DNA molecules known as chromosomes, each of which serves conceptually as a one-dimensional chemical storage device. Indeed, it does not obscure reality very much to think of it as an enormous linear tape, containing a string over the alphabet {A,C,G,T}. The string of symbols encodes the instructions for building protein molecules; using a chemical mechanism for reading portions of the chromosome, a cell can construct proteins that in turn control its metabolism.
Why is similarity important in this picture? To a first approximation, the sequence of symbols in an organism’s genome can be viewed as determining the properties of the organism. So suppose we have two strains of bacteria, X and Y, which are closely related evolutionarily. Suppose further that we’ve determined that a certain substring in the DNA of X codes for a certain kind of toxin. Then, if we discover a very “similar” substring in the DNA of Y, we might be able to hypothesize, before performing any experiments at all, that this portion of the DNA in Y codes for a similar kind of toxin. This use of computation to guide decisions about biological experiments is one of the hallmarks of the field of computational biology.
All this leaves us with the same question we asked initially, while typing badly spelled words into our online dictionary: How should we define the notion of similarity between two strings?
In the early 1970s, the two molecular biologists Needleman and Wunsch proposed a definition of similarity, which, basically unchanged, has become
280
Chapter 6 Dynamic Programming
the standard definition in use today. Its position as a standard was reinforced by its simplicity and intuitive appeal, as well as through its independent discovery by several other researchers around the same time. Moreover, this definition of similarity came with an efficient dynamic programming algorithm to compute it. In this way, the paradigm of dynamic programming was independently discovered by biologists some twenty years after mathematicians and computer scientists first articulated it.
The definition is motivated by the considerations we discussed above, and in particular by the notion of “lining up” two strings. Suppose we are given two strings X and Y , where X consists of the sequence of symbols x1x2 . . . xm and Y consists of the sequence of symbols y1y2 . . . yn. Consider the sets {1, 2, . . . , m} and {1, 2, . . . , n} as representing the different positions in the strings X and Y, and consider a matching of these sets; recall that a matching is a set of ordered pairs with the property that each item occurs in at most one pair. We say that a matching M of these two sets is an alignment if there are no “crossing” pairs: if (i, j), (i′, j′) ∈ M and i < i′, then j < j′. Intuitively, an alignment gives a way of lining up the two strings, by telling us which pairs of positions will be lined up with one another. Thus, for example,
stop- -tops
corresponds to the alignment {(2, 1), (3, 2), (4, 3)}.
Our definition of similarity will be based on finding the optimal alignment between X and Y, according to the following criteria. Suppose M is a given alignment between X and Y.
. First, there is a parameter δ > 0 that defines a gap penalty. For each position of X or Y that is not matched in M—it is a gap—we incur a cost of δ.

. Second, for each pair of letters p, q in our alphabet, there is a mismatch cost of αpq for lining up p with q. Thus, for each (i, j) ∈ M, we pay the appropriate mismatch cost αxiyj for lining up xi with yj. One generally assumes that αpp = 0 for each letter p—there is no mismatch cost to line up a letter with another copy of itself—although this will not be necessary in anything that follows.

. The cost of M is the sum of its gap and mismatch costs, and we seek an alignment of minimum cost.

The process of minimizing this cost is often referred to as sequence alignment in the biology literature. The quantities δ and {αpq} are external parameters that must be plugged into software for sequence alignment; indeed, a lot of work goes into choosing the settings for these parameters. From our point of

view, in designing an algorithm for sequence alignment, we will take them as given. To go back to our first example, notice how these parameters determine which alignment of ocurrance and occurrence we should prefer: the first is strictly better if and only if δ + αae < 3δ.
Designing the Algorithm
We now have a concrete numerical definition for the similarity between strings X and Y: it is the minimum cost of an alignment between X and Y. The lower this cost, the more similar we declare the strings to be. We now turn to the problem of computing this minimum cost, and an optimal alignment that yields it, for a given pair of strings X and Y.
One of the approaches we could try for this problem is dynamic program- ming, and we are motivated by the following basic dichotomy.
. In the optimal alignment M, either (m, n) ∈ M or (m, n) ̸∈ M. (That is, either the last symbols in the two strings are matched to each other, or they aren’t.)
By itself, this fact would be too weak to provide us with a dynamic program- ming solution. Suppose, however, that we compound it with the following basic fact.
(6.14) Let M be any alignment of X and Y. If (m, n) ̸∈ M, then either the mth position of X or the nth position of Y is not matched in M.
Proof. Suppose by way of contradiction that (m, n) ̸∈ M, and there are num- bersi

There is an equivalent way to write (6.14) that exposes three alternative possibilities, and leads directly to the formulation of a recurrence.

(6.15) In an optimal alignment M, at least one of the following is true:

(i) (m,n)∈M; or

(ii) the mth position of X is not matched; or

(iii) the nth position of Y is not matched.

Now, let OPT(i, j) denote the minimum cost of an alignment between x1x2 …xi and y1y2 …yj. If case (i) of (6.15) holds, we pay αxmyn and then align x1x2 . . . xm−1 as well as possible with y1y2 . . . yn−1; we get OPT(m, n) = αxmyn +OPT(m−1,n−1). If case (ii) holds, we pay a gap cost of δ since the mth position of X is not matched, and then we align x1x2 . . . xm−1 as well as

6.6 Sequence Alignment

281

282

Chapter 6 Dynamic Programming

possible with y1y2 …yn. In this way, we get OPT(m,n)=δ+OPT(m−1,n). Similarly, if case (iii) holds, we get OPT(m, n) = δ + OPT(m, n − 1).

Using the same argument for the subproblem of finding the minimum-cost alignment between x1x2 . . . xi and y1y2 . . . yj, we get the following fact.

(6.16) The minimum alignment costs satisfy the following recurrence for i ≥ 1 and j ≥ 1:

OPT(i,j)=min[αxiyj +OPT(i−1,j−1),δ+OPT(i−1,j),δ+OPT(i,j−1)]. Moreover, (i, j) is in an optimal alignment M for this subproblem if and only

if the minimum is achieved by the first of these values.

We have maneuvered ourselves into a position where the dynamic pro- gramming algorithm has become clear: We build up the values of OPT(i, j) using the recurrence in (6.16). There are only O(mn) subproblems, and OPT(m, n) is the value we are seeking.

We now specify the algorithm to compute the value of the optimal align- ment. For purposes of initialization, we note that OPT(i, 0) = OPT(0, i) = iδ for all i, since the only way to line up an i-letter word with a 0-letter word is to use i gaps.

Alignment(X ,Y )

Array A[0…m,0…n] Initialize A[i, 0]= iδ for each i Initialize A[0,j]=jδ for each j For j=1,…,n

For i=1,…,m

Use the recurrence (6.16) to compute A[i, j]

Endfor

Endfor

Return A[m, n]

As in previous dynamic programming algorithms, we can trace back through the array A, using the second part of fact (6.16), to construct the alignment itself.

Analyzing the Algorithm

The correctness of the algorithm follows directly from (6.16). The running time is O(mn), since the array A has O(mn) entries, and at worst we spend constant time on each.

6.6 Sequence Alignment

283

x3 x2 x1

y1 y2 y3 y4

Figure 6.17 A graph-based picture of sequence alignment.

There is an appealing pictorial way in which people think about this sequence alignment algorithm. Suppose we build a two-dimensional m × n grid graph GXY, with the rows labeled by symbols in the string X, the columns labeled by symbols in Y, and directed edges as in Figure 6.17.

We number the rows from 0 to m and the columns from 0 to n; we denote the node in the ith row and the jth column by the label (i, j). We put costs on the edges of GXY: the cost of each horizontal and vertical edge is δ, and the cost of the diagonal edge from (i − 1, j − 1) to (i, j) is αxiyj.

The purpose of this picture now emerges: the recurrence in (6.16) for OPT(i, j) is precisely the recurrence one gets for the minimum-cost path in GXY from (0, 0) to (i, j). Thus we can show

(6.17) Let f (i, j) denote the minimum cost of a path from (0, 0) to (i, j) in GXY. Then for all i, j, we have f(i, j) = OPT(i, j).

Proof. We can easily prove this by induction on i + j. When i + j = 0, we have i=j=0, and indeed f(i,j)=OPT(i,j)=0.

Now consider arbitrary values of i and j, and suppose the statement is true for all pairs (i′, j′) with i′ + j′ < i + j. The last edge on the shortest path to (i, j) is either from (i − 1, j − 1), (i − 1, j), or (i, j − 1). Thus we have
f(i,j)=min[αxiyj +f(i−1,j−1),δ+f(i−1,j),δ+f(i,j−1)] =min[αxiyj +OPT(i−1,j−1),δ+OPT(i−1,j),δ+OPT(i,j−1)] = OPT(i, j),
where we pass from the first line to the second using the induction hypothesis, and we pass from the second to the third using (6.16).
284
Chapter 6 Dynamic Programming
8
6
6
n
a
e
m
—
—name
Figure 6.18 The OPT values for the problem of aligning the words mean to name.
Thus the value of the optimal alignment is the length of the shortest path in GXY from (0, 0) to (m, n). (We’ll call any path in GXY from (0, 0) to (m, n) a corner-to-corner path.) Moreover, the diagonal edges used in a shortest path correspond precisely to the pairs used in a minimum-cost alignment. These connections to the Shortest-Path Problem in the graph GXY do not directly yield an improvement in the running time for the sequence alignment problem; however, they do help one’s intuition for the problem and have been useful in suggesting algorithms for more complex variations on sequence alignment.
For an example, Figure 6.18 shows the value of the shortest path from (0, 0) to each node (i, j) for the problem of aligning the words mean and name. For the purpose of this example, we assume that δ = 2; matching a vowel with a different vowel, or a consonant with a different consonant, costs 1; while matching a vowel and a consonant with each other costs 3. For each cell in the table (representing the corresponding node), the arrow indicates the last step of the shortest path leading to that node—in other words, the way that the minimum is achieved in (6.16). Thus, by following arrows backward from node (4, 4), we can trace back to construct the alignment.
6.7 Sequence Alignment in Linear Space via Divide and Conquer
In the previous section, we showed how to compute the optimal alignment between two strings X and Y of lengths m and n, respectively. Building up the two-dimensional m-by-n array of optimal solutions to subproblems, OPT(·, ·), turned out to be equivalent to constructing a graph GXY with mn nodes laid out in a grid and looking for the cheapest path between opposite corners. In either of these ways of formulating the dynamic programming algorithm, the running time is O(mn), because it takes constant time to determine the value in each of the mn cells of the array OPT; and the space requirement is O(mn) as well, since it was dominated by the cost of storing the array (or the graph GXY ).
The Problem
The question we ask in this section is: Should we be happy with O(mn) as a space bound? If our application is to compare English words, or even English sentences, it is quite reasonable. In biological applications of sequence alignment, however, one often compares very long strings against one another; and in these cases, the (mn) space requirement can potentially be a more severe problem than the (mn) time requirement. Suppose, for example, that we are comparing two strings of 100,000 symbols each. Depending on the underlying processor, the prospect of performing roughly 10 billion primitive
5
5
3
4
5
6
5
4
3
2
4
4
2
1
3
4
6
0
2
4
6
8
6.7 Sequence Alignment in Linear Space via Divide and Conquer
285
operations might be less cause for worry than the prospect of working with a single 10-gigabyte array.
Fortunately, this is not the end of the story. In this section we describe a very clever enhancement of the sequence alignment algorithm that makes it work in O(mn) time using only O(m + n) space. In other words, we can bring the space requirement down to linear while blowing up the running time by at most an additional constant factor. For ease of presentation, we’ll describe various steps in terms of paths in the graph GXY , with the natural equivalence back to the sequence alignment problem. Thus, when we seek the pairs in an optimal alignment, we can equivalently ask for the edges in a shortest corner-to-corner path in GXY.
The algorithm itself will be a nice application of divide-and-conquer ideas. The crux of the technique is the observation that, if we divide the problem into several recursive calls, then the space needed for the computation can be reused from one call to the next. The way in which this idea is used, however, is fairly subtle.
Designing the Algorithm
We first show that if we only care about the value of the optimal alignment, and not the alignment itself, it is easy to get away with linear space. The crucial observation is that to fill in an entry of the array A, the recurrence in (6.16) only needs information from the current column of A and the previous column of A. Thus we will “collapse” the array A to an m × 2 array B: as the algorithm iterates through values of j, entries of the form B[i, 0] will hold the “previous” column’s value A[i, j − 1], while entries of the form B[i, 1]will hold the “current” column’s value A[i, j].
Space-Efficient-Alignment(X ,Y )
Array B[0...m,0...1]
Initialize B[i,0]=iδ for each i (just as in column 0 of A) For j=1,...,n
B[0,1]=jδ (since this corresponds to entry A[0,j]) For i=1,...,m
B[i, 1]= min[αxiyj + B[i − 1, 0],
δ+B[i−1,1], δ+B[i,0]]
Endfor
Move column 1 of B to column 0 to make room for next iteration:
Update B[i, 0]= B[i, 1] for each i Endfor
286
Chapter 6 Dynamic Programming
It is easy to verify that when this algorithm completes, the array entry B[i, 1]holds the value of OPT(i, n) for i = 0, 1, . . . , m. Moreover, it uses O(mn) time and O(m) space. The problem is: where is the alignment itself? We haven’t left enough information around to be able to run a procedure like Find-Alignment. Since B at the end of the algorithm only contains the last two columns of the original dynamic programming array A, if we were to try tracing back to get the path, we’d run out of information after just these two columns. We could imagine getting around this difficulty by trying to “predict” what the alignment is going to be in the process of running our space-efficient procedure. In particular, as we compute the values in the jth column of the (now implicit) array A, we could try hypothesizing that a certain entry has a very small value, and hence that the alignment that passes through this entry is a promising candidate to be the optimal one. But this promising alignment might run into big problems later on, and a different alignment that currently looks much less attractive could turn out to be the optimal one.
There is, in fact, a solution to this problem—we will be able to recover the alignment itself using O(m + n) space—but it requires a genuinely new idea. The insight is based on employing the divide-and-conquer technique that we’ve seen earlier in the book. We begin with a simple alternative way to implement the basic dynamic programming solution.
A Backward Formulation of the Dynamic Program Recall that we use f (i, j) to denote the length of the shortest path from (0, 0) to (i, j) in the graph GXY . (As we showed in the initial sequence alignment algorithm, f(i,j) has the same value as OPT(i, j).) Now let’s define g(i, j) to be the length of the shortest path from (i, j) to (m, n) in GXY . The function g provides an equally natural dynamic programming approach to sequence alignment, except that we build it up in reverse: we start with g(m, n) = 0, and the answer we want is g(0, 0). By strict analogy with (6.16), we have the following recurrence for g.
(6.18) For i

OPT(i, v) = min(OPT(i − 1, v), min(OPT(i − 1, w) + cvw)). w∈V

Using this recurrence, we get the following dynamic programming algo- rithm to compute the value OPT(n − 1, s).

w

t

P

Figure 6.22 The minimum-cost path P from v to t using at most i edges.

294

Chapter 6 Dynamic Programming

–4

6 –3

–1 d 4

a

b–2 t

e2 –3

Shortest-Path(G, s, t)

n= number of nodes in G

Array M[0…n−1,V]

Define M[0,t]=0 and M[0,v]=∞ for all other v∈V For i=1,…,n−1

For v∈V in any order

Compute M[i, v] using the recurrence (6.23)

Endfor

Endfor

Return M[n − 1, s]

The correctness of the method follows directly by induction from (6.23). We can bound the running time as follows. The table M has n2 entries; and each entry can take O(n) time to compute, as there are at most n nodes w ∈ V we have to consider.

(6.24) The Shortest-Path method correctly computes the minimum cost of an s-t path in any graph that has no negative cycles, and runs in O(n3) time.

Given the table M containing the optimal values of the subproblems, the shortest path using at most i edges can be obtained in O(in) time, by tracing back through smaller subproblems.

As an example, consider the graph in Figure 6.23(a), where the goal is to find a shortest path from each node to t. The table in Figure 6.23(b) shows the array M, with entries corresponding to the values M[i, v] from the algorithm. Thus a single row in the table corresponds to the shortest path from a particular node to t, as we allow the path to use an increasing number of edges. For example, the shortest path from node d to t is updated four times, as it changes from d-t, to d-a-t, to d-a-b-e-t, and finally to d-a-b-e-c-t.

Extensions: Some Basic Improvements to the Algorithm

An Improved Running-Time Analysis We can actually provide a better running-time analysis for the case in which the graph G does not have too many edges. A directed graph with n nodes can have close to n2 edges, since there could potentially be an edge between each pair of nodes, but many graphs are much sparser than this. When we work with a graph for which the number of edges m is significantly less than n2, we’ve already seen in a number of cases earlier in the book that it can be useful to write the running- time in terms of both m and n; this way, we can quantify our speed-up on graphs with relatively fewer edges.

8

c3

(a)

012345

0

0

0

0

0

0

∞

–3

–3

–4

–6

–6

∞

∞

0

–2

–2

–2

∞

3

3

3

3

3

∞

4

3

3

2

0

∞

2

0

0

0

0

t a b c d e

(b)

Figure 6.23 For the directed graph in (a), the Shortest- Path Algorithm constructs the dynamic programming table in (b).

If we are a little more careful in the analysis of the method above, we can improve the running-time bound to O(mn) without significantly changing the algorithm itself.

(6.25) The Shortest-Path method can be implemented in O(mn) time. Proof. Consider the computation of the array entry M[i, v] according to the

recurrence (6.23); we have

M[i, v]= min(M[i − 1, v], min(M[i − 1, w]+ cvw)).

w∈V

We assumed it could take up to O(n) time to compute this minimum, since there are n possible nodes w. But, of course, we need only compute this minimum over all nodes w for which v has an edge to w; let us use nv to denote this number. Then it takes time O(nv) to compute the array entry M[i, v]. We have to compute an entry for every node v and every index 0≤i≤n−1, so this gives a running-time bound of

O n nv . v∈V

In Chapter 3, we performed exactly this kind of analysis for other graph

nodes in V, and so each edge is counted exactly once by this expression. Thus

we have n = m. Plugging this into our expression v∈V v

On nv v∈V

for the running time, we get a running-time bound of O(mn).

Improving the Memory Requirements We can also significantly improve the memory requirements with only a small change to the implementation. A common problem with many dynamic programming algorithms is the large space usage, arising from the M array that needs to be stored. In the Bellman- Ford Algorithm as written, this array has size n2; however, we now show how to reduce this to O(n). Rather than recording M[i, v] for each value i, we will use and update a single value M[v] for each node v, the length of the shortest path from v to t that we have found so far. We still run the algorithm for

6.8 Shortest Paths in a Graph

295

algorithms, and used (3.9) from that chapter to bound the expression

v∈V nv

for undirected graphs. Here we are dealing with directed graphs, and nv denotes

the number of edges leaving v. In a sense, it is even easier to work out the

value of v∈V nv for the directed case: each edge leaves exactly one of the

296

Chapter 6 Dynamic Programming

iterations i = 1, 2, . . . , n − 1, but the role of i will now simply be as a counter; in each iteration, and for each node v, we perform the update

M[v]= min(M[v], min(cvw + M[w])). w∈V

We now observe the following fact.

(6.26) Throughout the algorithm M[v] is the length of some path from v to t, and after i rounds of updates the value M[v] is no larger than the length of the shortest path from v to t using at most i edges.

Given (6.26), we can then use (6.22) as before to show that we are done after n − 1 iterations. Since we are only storing an M array that indexes over the nodes, this requires only O(n) working memory.

Finding the Shortest Paths One issue to be concerned about is whether this space-efficient version of the algorithm saves enough information to recover the shortest paths themselves. In the case of the Sequence Alignment Problem in the previous section, we had to resort to a tricky divide-and-conquer method to recover the solution from a similar space-efficient implementation. Here, however, we will be able to recover the shortest paths much more easily.

To help with recovering the shortest paths, we will enhance the code by

having each node v maintain the first node (after itself) on its path to the

destination t; we will denote this first node by first[v]. To maintain first[v],

we update its value whenever the distance M[v] is updated. In other words,

whenever the value of M[v] is reset to the minimum min(cvw + M[w]), we set

w∈V

Now let P denote the directed “pointer graph” whose nodes are V, and

first[v] to the node w that attains this minimum.

whose edges are {(v, first[v])}. The main observation is the following.

(6.27) If the pointer graph P contains a cycle C, then this cycle must have negative cost.

Proof. Notice that if first[v]=w at any time, then we must have M[v]≥ cvw + M[w]. Indeed, the left- and right-hand sides are equal after the update that sets first[v] equal to w; and since M[w] may decrease, this equation may turn into an inequality.

Let v1, v2, . . . , vk be the nodes along the cycle C in the pointer graph, and assume that (vk , v1) is the last edge to have been added. Now, consider the values right before this last update. At this time we have M[vi]≥ cvivi+1 + M[vi+1]for all i=1,…,k−1, and we also have M[vk]>cvkv1 +M[v1]since we are about to update M[vk] and change first[vk] to v1. Adding all these

inequalities, the M[vi] values cancel, and we get 0 > k−1 cv v + cv v : a i=1ii+1 k1

negative cycle, as claimed.

6.9 Shortest Paths and Distance Vector Protocols

297

Now note that if G has no negative cycles, then (6.27) implies that the pointer graph P will never have a cycle. For a node v, consider the path we get by following the edges in P, from v to first[v] = v1, to first[v1] = v2, and so forth. Since the pointer graph has no cycles, and the sink t is the only node that has no outgoing edge, this path must lead to t. We claim that when the algorithm terminates, this is in fact a shortest path in G from v to t.

(6.28) Suppose G has no negative cycles, and consider the pointer graph P at the termination of the algorithm. For each node v, the path in P from v to t is a shortest v-t path in G.

Proof. Consider a node v and let w = first[v]. Since the algorithm terminated, we must have M[v]=cvw +M[w]. The value M[t]=0, and hence the length of the path traced out by the pointer graph is exactly M[v], which we know is the shortest-path distance.

Note that in the more space-efficient version of Bellman-Ford, the path whose length is M[v] after i iterations can have substantially more edges than i. For example, if the graph is a single path from s to t, and we perform updates in the reverse of the order the edges appear on the path, then we get the final shortest-path values in just one iteration. This does not always happen, so we cannot claim a worst-case running-time improvement, but it would be nice to be able to use this fact opportunistically to speed up the algorithm on instances where it does happen. In order to do this, we need a stopping signal in the algorithm—something that tells us it’s safe to terminate before iteration n − 1 is reached.

Such a stopping signal is a simple consequence of the following observa- tion: If we ever execute a complete iteration i in which no M[v]value changes, then no M[v] value will ever change again, since future iterations will begin with exactly the same set of array entries. Thus it is safe to stop the algorithm. Note that it is not enough for a particular M[v] value to remain the same; in order to safely terminate, we need for all these values to remain the same for a single iteration.

6.9 Shortest Paths and Distance Vector Protocols

One important application of the Shortest-Path Problem is for routers in a communication network to determine the most efficient path to a destination. We represent the network using a graph in which the nodes correspond to routers, and there is an edge between v and w if the two routers are connected by a direct communication link. We define a cost cvw representing the delay on the link (v, w); the Shortest-Path Problem with these costs is to determine the path with minimum delay from a source node s to a destination t. Delays are

298

Chapter 6 Dynamic Programming

naturally nonnegative, so one could use Dijkstra’s Algorithm to compute the shortest path. However, Dijkstra’s shortest-path computation requires global knowledge of the network: it needs to maintain a set S of nodes for which shortest paths have been determined, and make a global decision about which node to add next to S. While routers can be made to run a protocol in the background that gathers enough global information to implement such an algorithm, it is often cleaner and more flexible to use algorithms that require only local knowledge of neighboring nodes.

If we think about it, the Bellman-Ford Algorithm discussed in the previous section has just such a “local” property. Suppose we let each node v maintain its value M[v]; then to update this value, v needs only obtain the value M[w] from each neighbor w, and compute

min(cvw + M[w]) w∈V

based on the information obtained.

We now discuss an improvement to the Bellman-Ford Algorithm that makes it better suited for routers and, at the same time, a faster algorithm in practice. Our current implementation of the Bellman-Ford Algorithm can be thought of as a pull-based algorithm. In each iteration i, each node v has to contact each neighbor w, and “pull” the new value M[w] from it. If a node w has not changed its value, then there is no need for v to get the value again; however, v has no way of knowing this fact, and so it must execute the pull anyway.

This wastefulness suggests a symmetric push-based implementation, where values are only transmitted when they change. Specifically, each node w whose distance value M[w]changes in an iteration informs all its neighbors of the new value in the next iteration; this allows them to update their values accordingly. If M[w] has not changed, then the neighbors of w already have the current value, and there is no need to “push” it to them again. This leads to savings in the running time, as not all values need to be pushed in each iter- ation. We also may terminate the algorithm early, if no value changes during an iteration. Here is a concrete description of the push-based implementation.

Push-Based-Shortest-Path(G, s, t)

n= number of nodes in G

Array M[V]

Initialize M[t]=0 and M[v]=∞ for all other v∈V For i=1,…,n−1

For w∈V in any order

If M[w] has been updated in the previous iteration then

6.9 Shortest Paths and Distance Vector Protocols

299

For all edges (v, w) in any order

M[v]= min(M[v], cvw + M[w])

If this changes the value of M[v], then first[v]=w

Endfor

Endfor

If no value changed in this iteration, then end the algorithm

Endfor

Return M[s]

In this algorithm, nodes are sent updates of their neighbors’ distance values in rounds, and each node sends out an update in each iteration in which it has changed. However, if the nodes correspond to routers in a network, then we do not expect everything to run in lockstep like this; some routers may report updates much more quickly than others, and a router with an update to report may sometimes experience a delay before contacting its neighbors. Thus the routers will end up executing an asynchronous version of the algorithm: each time a node w experiences an update to its M[w] value, it becomes “active” and eventually notifies its neighbors of the new value. If we were to watch the behavior of all routers interleaved, it would look as follows.

Asynchronous-Shortest-Path(G, s, t)

n= number of nodes in G

Array M[V]

Initialize M[t]=0 and M[v]=∞ for all other v∈V Declare t to be active and all other nodes inactive While there exists an active node

Choose an active node w

For all edges (v, w) in any order

M[v]= min(M[v], cvw + M[w])

If this changes the value of M[v], then

first[v]=w

v becomes active Endfor

w becomes inactive EndWhile

One can show that even this version of the algorithm, with essentially no coordination in the ordering of updates, will converge to the correct values of the shortest-path distances to t, assuming only that each time a node becomes active, it eventually contacts its neighbors.

The algorithm we have developed here uses a single destination t, and all nodes v ∈ V compute their shortest path to t. More generally, we are

300

Chapter 6 Dynamic Programming

presumably interested in finding distances and shortest paths between all pairs of nodes in a graph. To obtain such distances, we effectively use n separate computations, one for each destination. Such an algorithm is referred to as a distance vector protocol, since each node maintains a vector of distances to every other node in the network.

Problems with the Distance Vector Protocol

One of the major problems with the distributed implementation of Bellman- Ford on routers (the protocol we have been discussing above) is that it’s derived from an initial dynamic programming algorithm that assumes edge costs will remain constant during the execution of the algorithm. Thus far we’ve been designing algorithms with the tacit understanding that a program executing the algorithm will be running on a single computer (or a centrally managed set of computers), processing some specified input. In this context, it’s a rather benign assumption to require that the input not change while the program is actually running. Once we start thinking about routers in a network, however, this assumption becomes troublesome. Edge costs may change for all sorts of reasons: links can become congested and experience slow-downs; or a link (v, w) may even fail, in which case the cost cvw effectively increases to ∞.

Here’s an indication of what can go wrong with our shortest-path algo- rithm when this happens. If an edge (v, w) is deleted (say the link goes down), it is natural for node v to react as follows: it should check whether its shortest path to some node t used the edge (v, w), and, if so, it should increase the distance using other neighbors. Notice that this increase in distance from v can now trigger increases at v’s neighbors, if they were relying on a path through v, and these changes can cascade through the network. Consider the extremely simple example in Figure 6.24, in which the original graph has three edges (s, v), (v, s) and (v, t), each of cost 1.

Now suppose the edge (v, t) in Figure 6.24 is deleted. How does node v react? Unfortunately, it does not have a global map of the network; it only knows the shortest-path distances of each of its neighbors to t. Thus it does

The deleted edge causes an unbounded sequence of updates by s and v.

1

svt

1

1

Deleted

When the edge (v, t) is deleted, the distributed Bellman-Ford Algorithm will begin “counting to infinity.”

Figure 6.24

not know that the deletion of (v, t) has eliminated all paths from s to t. Instead, it sees that M[s]= 2, and so it updates M[v]= cvs + M[s]= 3, assuming that it will use its cost-1 edge to s, followed by the supposed cost-2 path from s to t. Seeing this change, node s will update M[s]= csv + M[v]= 4, based on its cost-1 edge to v, followed by the supposed cost-3 path from v to t. Nodes s and v will continue updating their distance to t until one of them finds an alternate route; in the case, as here, that the network is truly disconnected, these updates will continue indefinitely—a behavior known as the problem of counting to infinity.

To avoid this problem and related difficulties arising from the limited amount of information available to nodes in the Bellman-Ford Algorithm, the designers of network routing schemes have tended to move from distance vector protocols to more expressive path vector protocols, in which each node stores not just the distance and first hop of their path to a destination, but some representation of the entire path. Given knowledge of the paths, nodes can avoid updating their paths to use edges they know to be deleted; at the same time, they require significantly more storage to keep track of the full paths. In the history of the Internet, there has been a shift from distance vector protocols to path vector protocols; currently, the path vector approach is used in the Border Gateway Protocol (BGP) in the Internet core.

* 6.10 Negative Cycles in a Graph

So far in our consideration of the Bellman-Ford Algorithm, we have assumed that the underlying graph has negative edge costs but no negative cycles. We now consider the more general case of a graph that may contain negative cycles.

The Problem

There are two natural questions we will consider.

. How do we decide if a graph contains a negative cycle?

. How do we actually find a negative cycle in a graph that contains one?

The algorithm developed for finding negative cycles will also lead to an improved practical implementation of the Bellman-Ford Algorithm from the previous sections.

It turns out that the ideas we’ve seen so far will allow us to find negative cycles that have a path reaching a sink t. Before we develop the details of this, let’s compare the problem of finding a negative cycle that can reach a given t with the seemingly more natural problem of finding a negative cycle anywhere in the graph, regardless of its position related to a sink. It turns out that if we

6.10 Negative Cycles in a Graph

301

302

Chapter 6

Dynamic Programming

Any negative cycle in G will be able to reach t.

G

t

Figure 6.25 The augmented graph.

develop a solution to the first problem, we’ll be able to obtain a solution to the second problem as well, in the following way. Suppose we start with a graph G, add a new node t to it, and connect each other node v in the graph to node t via an edge of cost 0, as shown in Figure 6.25. Let us call the new “augmented graph” G′.

(6.29) The augmented graph G′ has a negative cycle C such that there is a path from C to the sink t if and only if the original graph has a negative cycle.

Proof. Assume G has a negative cycle. Then this cycle C clearly has an edge to t in G′, since all nodes have an edge to t.

Now suppose G′ has a negative cycle with a path to t. Since no edge leaves t in G′, this cycle cannot contain t. Since G′ is the same as G aside from the node t, it follows that this cycle is also a negative cycle of G.

So it is really enough to solve the problem of deciding whether G has a negative cycle that has a path to a given sink node t, and we do this now.

Designing and Analyzing the Algorithm

To get started thinking about the algorithm, we begin by adopting the original version of the Bellman-Ford Algorithm, which was less efficient in its use of space. We first extend the definitions of OPT(i, v) from the Bellman-Ford Algorithm, defining them for values i ≥ n. With the presence of a negative cycle in the graph, (6.22) no longer applies, and indeed the shortest path may

get shorter and shorter as we go around a negative cycle. In fact, for any node v on a negative cycle that has a path to t, we have the following.

(6.30) If node v can reach node t and is contained in a negative cycle, then lim OPT(i, v) = −∞.

i→∞

If the graph has no negative cycles, then (6.22) implies following statement.

(6.31) If there are no negative cycles in G, then OPT(i, v) = OPT(n − 1, v) for all nodes v and all i≥n.

But for how large an i do we have to compute the values OPT(i, v) before concluding that the graph has no negative cycles? For example, a node v may satisfy the equation OPT(n, v) = OPT(n − 1, v), and yet still lie on a negative cycle. (Do you see why?) However, it turns out that we will be in good shape if this equation holds for all nodes.

(6.32) There is no negative cycle with a path to t if and only if OPT(n, v) = OPT(n − 1, v) for all nodes v.

Proof. Statement(6.31)hasalreadyprovedtheforwarddirection.Fortheother direction, we use an argument employed earlier for reasoning about when it’s safe to stop the Bellman-Ford Algorithm early. Specifically, suppose OPT(n, v) = OPT(n − 1, v) for all nodes v. The values of OPT(n + 1, v) can be computed from OPT(n, v); but all these values are the same as the corresponding OPT(n − 1, v). It follows that we will have OPT(n + 1, v) = OPT(n − 1, v). Extending this reasoning to future iterations, we see that none of the values will ever change again, that is, OPT(i, v) = OPT(n − 1, v) for all nodes v and all i ≥ n. Thus there cannot be a negative cycle C that has a path to t; for any node w on this cycle C, (6.30) implies that the values OPT(i, w) would have to become arbitrarily negative as i increased.

Statement (6.32) gives an O(mn) method to decide if G has a negative cycle that can reach t. We compute values of OPT(i, v) for nodes of G and for values of i up to n. By (6.32), there is no negative cycle if and only if there is some value of i≤n at which OPT(i,v)=OPT(i−1,v) for all nodes v.

So far we have determined whether or not the graph has a negative cycle with a path from the cycle to t, but we have not actually found the cycle. To find a negative cycle, we consider a node v such that OPT(n, v) ̸= OPT(n − 1, v): for this node, a path P from v to t of cost OPT(n, v) must use exactly n edges. We find this minimum-cost path P from v to t by tracing back through the subproblems. As in our proof of (6.22), a simple path can only have n − 1

6.10 Negative Cycles in a Graph

303

304

Chapter 6 Dynamic Programming

edges, so P must contain a cycle C. We claim that this cycle C has negative cost.

(6.33) If G has n nodes and OPT(n,v)̸=OPT(n−1,v), then a path P from v to t of cost OPT(n, v) contains a cycle C, and C has negative cost.

Proof. FirstobservethatthepathPmusthavenedges,asOPT(n,v)̸=OPT(n− 1, v), and so every path using n − 1 edges has cost greater than that of the path P. In a graph with n nodes, a path consisting of n edges must repeat a node somewhere; let w be a node that occurs on P more than once. Let C be the cycle on P between two consecutive occurrences of node w. If C were not a negative cycle, then deleting C from P would give us a v-t path with fewer than n edges and no greater cost. This contradicts our assumption that OPT(n, v) ̸= OPT(n − 1, v), and hence C must be a negative cycle.

(6.34) The algorithm above finds a negative cycle in G, if such a cycle exists, and runs in O(mn) time.

Extensions: Improved Shortest Paths and Negative Cycle Detection Algorithms

At the end of Section 6.8 we discussed a space-efficient implementation of the Bellman-Ford algorithm for graphs with no negative cycles. Here we implement the detection of negative cycles in a comparably space-efficient way. In addition to the savings in space, this will also lead to a considerable speedup in practice even for graphs with no negative cycles. The implementation will be based on the same pointer graph P derived from the “first edges” (v, first[v]) that we used for the space-efficient implementation in Section 6.8. By (6.27), we know that if the pointer graph ever has a cycle, then the cycle has negative cost, and we are done. But if G has a negative cycle, does this guarantee that the pointer graph will ever have a cycle? Furthermore, how much extra computation time do we need for periodically checking whether P has a cycle?

Ideally, we would like to determine whether a cycle is created in the pointer graph P every time we add a new edge (v, w) with first[v]= w. An additional advantage of such “instant” cycle detection will be that we will not have to wait for n iterations to see that the graph has a negative cycle: We can terminate as soon as a negative cycle is found. Earlier we saw that if a graph G has no negative cycles, the algorithm can be stopped early if in some iteration the shortest path values M[v] remain the same for all nodes v. Instant negative cycle detection will be an analogous early termination rule for graphs that have negative cycles.

Consider a new edge (v, w), with first[v]= w, that is added to the pointer graph P. Before we add (v, w) the pointer graph has no cycles, so it consists of paths from each node v to the sink t. The most natural way to check whether adding edge (v, w) creates a cycle in P is to follow the current path from w to the terminal t in time proportional to the length of this path. If we encounter v along this path, then a cycle has been formed, and hence, by (6.27), the graph has a negative cycle. Consider Figure 6.26, for example, where in both (a) and (b) the pointer first[v] is being updated from u to w; in (a), this does not result in a (negative) cycle, but in (b) it does. However, if we trace out the sequence of pointers from v like this, then we could spend as much as O(n) time following the path to t and still not find a cycle. We now discuss a method that does not require an O(n) blow-up in the running time.

We know that before the new edge (v, w) was added, the pointer graph was a directed tree. Another way to test whether the addition of (v, w) creates a cycle is to consider all nodes in the subtree directed toward v. If w is in this subtree, then (v, w) forms a cycle; otherwise it does not. (Again, consider the two sample cases in Figure 6.26.) To be able to find all nodes in the subtree directed toward v, we need to have each node v maintain a list of all other nodes whose selected edges point to v. Given these pointers, we can find the subtree in time proportional to the size of the subtree pointing to v, at most O(n) as before. However, here we will be able to make additional use of the work done. Notice that the current distance value M[x] for all nodes x in the subtree was derived from node v’s old value. We have just updated v’s distance, and hence we know that the distance values of all these nodes will be updated again. We’ll mark each of these nodes x as “dormant,” delete the

w

tt

w

vu vu

6.10 Negative Cycles in a Graph

305

Update to first[v] = w

(a)

Update to first[v] = w

(b)

Figure 6.26 Changing the pointer graph P when first[v] is updated from u to w. In (b), this creates a (negative) cycle, whereas in (a) it does not.

306

Chapter 6 Dynamic Programming

edge (x, first[x]) from the pointer graph, and not use x for future updates until its distance value changes.

This can save a lot of future work in updates, but what is the effect on the worst-case running time? We can spend as much as O(n) extra time marking nodes dormant after every update in distances. However, a node can be marked dormant only if a pointer had been defined for it at some point in the past, so the time spent on marking nodes dormant is at most as much as the time the algorithm spends updating distances.

Now consider the time the algorithm spends on operations other than marking nodes dormant. Recall that the algorithm is divided into iterations, where iteration i + 1 processes nodes whose distance has been updated in iteration i. For the original version of the algorithm, we showed in (6.26) that after i iterations, the value M[v]is no larger than the value of the shortest path from v to t using at most i edges. However, with many nodes dormant in each iteration, this may not be true anymore. For example, if the shortest path from v to t using at most i edges starts on edge e=(v,w), and w is dormant in this iteration, then we may not update the distance value M[v], and so it stays at a value higher than the length of the path through the edge (v, w). This seems like a problem—however, in this case, the path through edge (v, w) is not actually the shortest path, so M[v] will have a chance to get updated later to an even smaller value.

So instead of the simpler property that held for M[v]in the original versions of the algorithm, we now have the the following claim.

(6.35) Throughout the algorithm M[v]is the length of some simple path from v to t; the path has at least i edges if the distance value M[v] is updated in iteration i; and after i iterations, the value M[v] is the length of the shortest path for all nodes v where there is a shortest v-t path using at most i edges.

Proof. The first pointers maintain a tree of paths to t, which implies that all paths used to update the distance values are simple. The fact that updates in iteration i are caused by paths with at least i edges is easy to show by induction on i. Similarly, we use induction to show that after iteration i the value M[v] is the distance on all nodes v where the shortest path from v to t uses at most i edges. Note that nodes v where M[v] is the actual shortest-path distance cannot be dormant, as the value M[v] will be updated in the next iteration for all dormant nodes.

Using this claim, we can see that the worst-case running time of the algorithm is still bounded by O(mn): Ignoring the time spent on marking nodes dormant, each iteration is implemented in O(m) time, and there can be at most n − 1 iterations that update values in the array M without finding

a negative cycle, as simple paths can have at most n − 1 edges. Finally, the time spent marking nodes dormant is bounded by the time spent on updates. We summarize the discussion with the following claim about the worst-case performance of the algorithm. In fact, as mentioned above, this new version is in practice the fastest implementation of the algorithm even for graphs that do not have negative cycles, or even negative-cost edges.

(6.36) The improved algorithm outlined above finds a negative cycle in G if such a cycle exists. It terminates immediately if the pointer graph P of first[v] pointers contains a cycle C, or if there is an iteration in which no update occurs to any distance value M[v]. The algorithm uses O(n) space, has at most n iterations, and runs in O(mn) time in the worst case.

Solved Exercises

Solved Exercise 1

Suppose you are managing the construction of billboards on the Stephen Daedalus Memorial Highway, a heavily traveled stretch of road that runs west-east for M miles. The possible sites for billboards are given by numbers x1, x2, . . . , xn, each in the interval [0, M] (specifying their position along the highway, measured in miles from its western end). If you place a billboard at location xi, you receive a revenue of ri > 0.

Regulations imposed by the county’s Highway Department require that no two of the billboards be within less than or equal to 5 miles of each other. You’d like to place billboards at a subset of the sites so as to maximize your total revenue, subject to this restriction.

Example. Suppose M = 20, n = 4,

{x1, x2, x3, x4} = {6, 7, 12, 14},

and

{r1,r2,r3,r4}={5,6,5,1}.

Then the optimal solution would be to place billboards at x1 and x3, for a total

revenue of 10.

Give an algorithm that takes an instance of this problem as input and returns the maximum total revenue that can be obtained from any valid subset of sites. The running time of the algorithm should be polynomial in n.

Solution We can naturally apply dynamic programming to this problem if we reason as follows. Consider an optimal solution for a given input instance; in this solution, we either place a billboard at site xn or not. If we don’t, the optimal solution on sites x1, . . . , xn is really the same as the optimal solution

Solved Exercises

307

308

Chapter 6 Dynamic Programming

on sites x1, . . . , xn−1; if we do, then we should eliminate xn and all other sites that are within 5 miles of it, and find an optimal solution on what’s left. The same reasoning applies when we’re looking at the problem defined by just the first j sites, x1, . . . , xj: we either include xj in the optimal solution or we don’t, with the same consequences.

Let’s define some notation to help express this. For a site xj, we let e(j) denote the easternmost site xi that is more than 5 miles from xj. Since sites are numbered west to east, this means that the sites x1, x2, . . . , xe(j) are still valid options once we’ve chosen to place a billboard at xj, but the sites xe(j)+1, . . . , xj−1 are not.

Now, our reasoning above justifies the following recurrence. If we let OPT(j) denote the revenue from the optimal subset of sites among x1, . . . , xj, then we have

OPT(j) = max(rj + OPT(e(j)), OPT(j − 1)).

We now have most of the ingredients we need for a dynamic programming algorithm. First, we have a set of n subproblems, consisting of the first j sites for j=0,1,2,…,n. Second, we have a recurrence that lets us build up the solutions to subproblems, given by OPT(j) = max(rj + OPT(e(j)), OPT(j − 1)).

To turn this into an algorithm, we just need to define an array M that will store the OPT values and throw a loop around the recurrence that builds up the values M[j] in order of increasing j.

Initialize M[0]=0 and M[1]=r1 For j=2,3,…,n:

Compute M[j] using the recurrence Endfor

Return M[n]

As with all the dynamic programming algorithms we’ve seen in this chapter, an optimal set of billboards can be found by tracing back through the values in array M.

Given the values e(j) for all j, the running time of the algorithm is O(n), since each iteration of the loop takes constant time. We can also compute all e(j) values in O(n) time as follows. For each site location xi, we define xi′ = xi − 5. Wethenmergethesortedlistx1,…,xn withthesortedlistx1′,…,xn′ inlinear time, as we saw how to do in Chapter 2. We now scan through this merged list; when we get to the entry xj′, we know that anything from this point onward to xj cannot be chosen together with xj (since it’s within 5 miles), and so we

simply define e(j) to be the largest value of i for which we’ve seen xi in our scan.

Here’s a final observation on this problem. Clearly, the solution looks very much like that of the Weighted Interval Scheduling Problem, and there’s a fundamental reason for that. In fact, our billboard placement problem can be directly encoded as an instance of Weighted Interval Scheduling, as follows. Suppose that for each site xi, we define an interval with endpoints [xi − 5, xi] and weight ri. Then, given any nonoverlapping set of intervals, the corresponding set of sites has the property that no two lie within 5 miles of each other. Conversely, given any such set of sites (no two within 5 miles), the intervals associated with them will be nonoverlapping. Thus the collections of nonoverlapping intervals correspond precisely to the set of valid billboard placements, and so dropping the set of intervals we’ve just defined (with their weights) into an algorithm for Weighted Interval Scheduling will yield the desired solution.

Solved Exercise 2

Through some friends of friends, you end up on a consulting visit to the cutting-edge biotech firm Clones ‘R’ Us (CRU). At first you’re not sure how your algorithmic background will be of any help to them, but you soon find yourself called upon to help two identical-looking software engineers tackle a perplexing problem.

The problem they are currently working on is based on the concatenation of sequences of genetic material. If X and Y are each strings over a fixed alphabet S, then XY denotes the string obtained by concatenating them— writing X followed by Y. CRU has identified a target sequence A of genetic material, consisting of m symbols, and they want to produce a sequence that is as similar to A as possible. For this purpose, they have a library L consisting of k (shorter) sequences, each of length at most n. They can cheaply produce any sequence consisting of copies of the strings in L concatenated together (with repetitions allowed).

Thus we say that a concatenation over L is any sequence of the form B1B2 . . . Bl, where each Bi belongs the set L. (Again, repetitions are allowed, so Bi and Bj could be the same string in L, for different values of i and j.) The problem is to find a concatenation over {Bi} for which the sequence alignment cost is as small as possible. (For the purpose of computing the sequence alignment cost, you may assume that you are given a gap cost δ and a mismatch cost αpq for each pair p, q ∈ S.)

Give a polynomial-time algorithm for this problem.

Solved Exercises

309

310

Chapter 6 Dynamic Programming

Solution This problem is vaguely reminiscent of Segmented Least Squares: we have a long sequence of “data” (the string A) that we want to “fit” with shorter segments (the strings in L).

If we wanted to pursue this analogy, we could search for a solution as follows. Let B = B1B2 . . . Bl denote a concatenation over L that aligns as well as possible with the given string A. (That is, B is an optimal solution to the input instance.) Consider an optimal alignment M of A with B, let t be the first position in A that is matched with some symbol in Bl, and let Al denote the substring of A from position t to the end. (See Figure 6.27 for an illustration of this with l = 3.) Now, the point is that in this optimal alignment M, the substring Al is optimally aligned with Bl; indeed, if there were a way to better align Al with Bl, we could substitute it for the portion of M that aligns Al with Bl and obtain a better overall alignment of A with B.

This tells us that we can look at the optimal solution as follows. There’s some final piece of Al that is aligned with one of the strings in L, and for this piece all we’re doing is finding the string in L that aligns with it as well as possible. Having found this optimal alignment for Al, we can break it off and continue to find the optimal solution for the remainder of A.

Thinking about the problem this way doesn’t tell us exactly how to proceed—we don’t know how long Al is supposed to be, or which string in L it should be aligned with. But this is the kind of thing we can search over in a dynamic programming algorithm. Essentially, we’re in about the same spot we were in with the Segmented Least Squares Problem: there we knew that we had to break off some final subsequence of the input points, fit them as well as possible with one line, and then iterate on the remaining input points.

So let’s set up things to make the search for Al possible. First, let A[x : y] denote the substring of A consisting of its symbols from position x to position y, inclusive. Let c(x, y) denote the cost of the optimal alignment of A[x : y] with any string in L. (That is, we search over each string in L and find the one that

t

A

Figure 6.27 In the optimal concatentation of strings to align with A, there is a final string (B3 in the figure) that aligns with a substring of A (A3 in the figure) that extends from some position t to the end.

A3

B2

B3

B1

aligns best with A[x : y].) Let OPT(j) denote the alignment cost of the optimal solution on the string A[1 : j].

The argument above says that an optimal solution on A[1 : j] consists of identifying a final “segment boundary” t < j, finding the optimal alignment of A[t:j]with a single string in L, and iterating on A[1:t−1]. The cost of this alignment of A[t : j] is just c(t , j), and the cost of aligning with what’s left is just OPT(t − 1). This suggests that our subproblems fit together very nicely, and it justifies the following recurrence.
(6.37) OPT(j) = mint

So, given a sequence of n weeks, a plan is specified by a choice of “low-stress,” “high-stress,” or “none” for each of the n weeks, with the property that if “high-stress” is chosen for week i > 1, then “none” has to be chosen for week i − 1. (It’s okay to choose a high-stress job in week 1.) The value of the plan is determined in the natural way: for each i, you add li to the value if you choose “low-stress” in week i, and you add hi to the value if you choose “high-stress” in week i. (You add 0 if you choose “none” in week i.)

Theproblem.Givensetsofvaluesl1,l2,…,ln andh1,h2,…,hn,finda plan of maximum value. (Such a plan will be called optimal.)

Example. Suppose n = 4, and the values of li and hi are given by the following table. Then the plan of maximum value would be to choose “none” in week 1, a high-stress job in week 2, and low-stress jobs in weeks 3 and 4. The value of this plan would be 0 + 50 + 10 + 10 = 70.

Week 1 Week 2 Week 3 Week 4

l 10 1 10 10 h 5 50 5 1

Exercises

313

314

Chapter 6

Dynamic Programming

(a)

Show that the following algorithm does not correctly solve this problem, by giving an instance on which it does not return the correct answer.

For iterations i = 1 to n If hi+1 > li + li+1 then

Output “Choose no job in week i”

Output “Choose a high-stress job in week i + 1” Continue with iteration i + 2

Else

Output “Choose a low-stress job in week i” Continue with iteration i + 1

Endif End

To avoid problems with overflowing array bounds, we define hi = li = 0 when i > n.

In your example, say what the correct answer is and also what the above algorithm finds.

Give an efficient algorithm that takes values for l1, l2, . . . , ln and h1, h2, . . . , hn and returns the value of an optimal plan.

(b)

3. LetG=(V,E)beadirectedgraphwithnodesv1,…,vn.WesaythatGis an ordered graph if it has the following properties.

(i) Each edge goes from a node with a lower index to a node with a higher index. That is, every directed edge has the form (vi, vj) with i < j.
(ii) Eachnodeexceptvnhasatleastoneedgeleavingit.Thatis,forevery node vi, i = 1, 2, . . . , n − 1, there is at least one edge of the form (vi, vj).
The length of a path is the number of edges in it. The goal in this
question is to solve the following problem (see Figure 6.29 for an exam- ple).
Given an ordered graph G, find the length of the longest path that begins at v1 and ends at vn.
(a) Show that the following algorithm does not correctly solve this problem, by giving an example of an ordered graph on which it does not return the correct answer.
Set w=v1 Set L=0
v1 v2 v3 v4 v5
Figure 6.29 The correct answer for this ordered graph is 3: The longest path from v1 to vn uses the three edges (v1, v2),(v2, v4), and (v4, v5).
While there is an edge out of the node w Choose the edge (w, vj)
for which j is as small as possible Set w=vj
Increase L by 1
end while
Return L as the length of the longest path
In your example, say what the correct answer is and also what the algorithm above finds.
(b) Give an efficient algorithm that takes an ordered graph G and returns the length of the longest path that begins at v1 and ends at vn. (Again, the length of a path is the number of edges in the path.)
4. Supposeyou’rerunningalightweightconsultingbusiness—justyou,two associates, and some rented equipment. Your clients are distributed between the East Coast and the West Coast, and this leads to the following question.
Each month, you can either run your business from an office in New York (NY) or from an office in San Francisco (SF). In month i, you’ll incur an operating cost of Ni if you run the business out of NY; you’ll incur an operating cost of Si if you run the business out of SF. (It depends on the distribution of client demands for that month.)
However, if you run the business out of one city in month i, and then out of the other city in month i + 1, then you incur a fixed moving cost of M to switch base offices.
Given a sequence of n months, a plan is a sequence of n locations— each one equal to either NY or SF—such that the ith location indicates the city in which you will be based in the ith month. The cost of a plan is the sum of the operating costs for each of the n months, plus a moving cost of M for each time you switch cities. The plan can begin in either city.
Exercises
315
316
Chapter 6 Dynamic Programming
The problem. Given a value for the moving cost M, and sequences of operating costs N1,...,Nn and S1,...,Sn, find a plan of minimum cost. (Such a plan will be called optimal.)
Example. Suppose n = 4, M = 10, and the operating costs are given by the following table.
Month 1 Month 2 Month 3 Month 4
NY 1 3 20 30 SF 50 20 2 4
Then the plan of minimum cost would be the sequence of locations
[NY,NY,SF,SF],
with a total cost of 1 + 3 + 2 + 4 + 10 = 20, where the final term of 10 arises
because you change locations once.
(a) Show that the following algorithm does not correctly solve this problem, by giving an instance on which it does not return the correct answer.
Fori = 1ton
If Ni < Si then
Output "NY in Month i" Else
Output "SF in Month i" End
In your example, say what the correct answer is and also what the algorithm above finds.
(b) Give an example of an instance in which every optimal plan must move (i.e., change locations) at least three times.
Provide a brief explanation, saying why your example has this property.
(c) Give an efficient algorithm that takes values for n, M, and sequences ofoperatingcostsN1,...,Nn andS1,...,Sn,andreturnsthecost of an optimal plan.
5. Assomeofyouknowwell,andothersofyoumaybeinterestedtolearn, a number of languages (including Chinese and Japanese) are written without spaces between the words. Consequently, software that works with text written in these languages must address the word segmentation problem—inferring likely boundaries between consecutive words in the
text. If English were written without spaces, the analogous problem would consist of taking a string like “meetateight” and deciding that the best segmentation is “meet at eight” (and not “me et at eight,” or “meet ate ight,” or any of a huge number of even less plausible alternatives). How could we automate this process?
A simple approach that is at least reasonably effective is to find a segmentation that simply maximizes the cumulative “quality” of its indi- vidual constituent words. Thus, suppose you are given a black box that, for any string of letters x = x1x2 . . . xk, will return a number quality(x). This number can be either positive or negative; larger numbers correspond to more plausible English words. (So quality(“me”) would be positive, while quality(“ght”) would be negative.)
Given a long string of letters y = y1y2 . . . yn, a segmentation of y is a partition of its letters into contiguous blocks of letters; each block corre- sponds to a word in the segmentation. The total quality of a segmentation is determined by adding up the qualities of each of its blocks. (So we’d get the right answer above provided that quality(“meet”) + quality(“at”) + quality(“eight”) was greater than the total quality of any other segmenta- tion of the string.)
Give an efficient algorithm that takes a string y and computes a segmentation of maximum total quality. (You can treat a single call to the black box computing quality(x) as a single computational step.)
(A final note, not necessary for solving the problem: To achieve better performance, word segmentation software in practice works with a more complex formulation of the problem—for example, incorporating the notion that solutions should not only be reasonable at the word level, but also form coherent phrases and sentences. If we consider the example “theyouthevent,” there are at least three valid ways to segment this into common English words, but one constitutes a much more coherent phrase than the other two. If we think of this in the terminology of formal languages, this broader problem is like searching for a segmentation that also can be parsed well according to a grammar for the underlying language. But even with these additional criteria and constraints, dynamic programming approaches lie at the heart of a number of successful segmentation systems.)
6. In a word processor, the goal of “pretty-printing” is to take text with a ragged right margin, like this,
Call me Ishmael.
Some years ago,
never mind how long precisely,
Exercises
317
318
Chapter 6
Dynamic Programming
having little or no money in my purse,
and nothing particular to interest me on shore,
I thought I would sail about a little
and see the watery part of the world.
and turn it into text whose right margin is as “even” as possible, like this.
Call me Ishmael. Some years ago, never
mind how long precisely, having little
or no money in my purse, and nothing
particular to interest me on shore, I
thought I would sail about a little
and see the watery part of the world.
To make this precise enough for us to start thinking about how to write a pretty-printer for text, we need to figure out what it means for the right margins to be “even.” So suppose our text consists of a sequence of words, W = {w1, w2, . . . , wn}, where wi consists of ci characters. We have a maximum line length of L. We will assume we have a fixed-width font and ignore issues of punctuation or hyphenation.
A formatting of W consists of a partition of the words in W into lines. In the words assigned to a single line, there should be a space after each word except the last; and so if wj , wj+1, . . . , wk are assigned to one line, then we should have
⎡⎤
k−1
⎣(ci +1)⎦+ck ≤L.
i=j
We will call an assignment of words to a line valid if it satisfies this inequality. The difference between the left-hand side and the right-hand side will be called the slack of the line—that is, the number of spaces left at the right margin.
Give an efficient algorithm to find a partition of a set of words W into valid lines, so that the sum of the squares of the slacks of all lines (including the last line) is minimized.
7. As a solved exercise in Chapter 5, we gave an algorithm with O(n log n) running time for the following problem. We’re looking at the price of a given stock over n consecutive days, numbered i = 1, 2, . . . , n. For each day i, we have a price p(i) per share for the stock on that day. (We’ll assume for simplicity that the price was fixed during each day.) We’d like to know: How should we choose a day i on which to buy the stock and a later day j > i on which to sell it, if we want to maximize the profit per

share, p(j) − p(i)? (If there is no way to make money during the n days, we should conclude this instead.)

In the solved exercise, we showed how to find the optimal pair of days i and j in time O(n log n). But, in fact, it’s possible to do better than this. Show how to find the optimal numbers i and j in time O(n).

8. TheresidentsoftheundergroundcityofZiondefendthemselvesthrough a combination of kung fu, heavy artillery, and efficient algorithms. Re- cently they have become interested in automated methods that can help fend off attacks by swarms of robots.

Here’s what one of these robot attacks looks like.

. A swarm of robots arrives over the course of n seconds; in the ith second, xi robots arrive. Based on remote sensing data, you know this sequence x1, x2, . . . , xn in advance.

. You have at your disposal an electromagnetic pulse (EMP), which can destroy some of the robots as they arrive; the EMP’s power depends on how long it’s been allowed to charge up. To make this precise, there is a function f (·) so that if j seconds have passed since the EMP was last used, then it is capable of destroying up to f (j) robots.

. So specifically, if it is used in the kth second, and it has been j seconds since it was previously used, then it will destroy min(xk , f (j)) robots. (After this use, it will be completely drained.)

. We will also assume that the EMP starts off completely drained, so if it is used for the first time in the jth second, then it is capable of destroying up to f (j) robots.

The problem. Given the data on robot arrivals x1, x2, . . . , xn, and given the recharging function f (·), choose the points in time at which you’re going to activate the EMP so as to destroy as many robots as possible.

Example. Suppose n = 4, and the values of xi and f (i) are given by the following table.

i1234

xi 1 10 10 1

f(i) 1 2 4 8

The best solution would be to activate the EMP in the 3rd and the 4th seconds. In the 3rd second, the EMP has gotten to charge for 3 seconds, and so it destroys min(10, 4) = 4 robots; In the 4th second, the EMP has only gotten to charge for 1 second since its last use, and it destroys min(1, 1) = 1 robot. This is a total of 5.

Exercises

319

320

Chapter 6

Dynamic Programming

(a)

Show that the following algorithm does not correctly solve this problem, by giving an instance on which it does not return the correct answer.

Schedule-EMP(x1, . . . , xn)

Let j be the smallest number for which f(j) ≥ xn

(If no such j exists then set j = n) Activate the EMP in the nth second

If n−j≥1 then

Continue recursively on the input x1, . . . , xn−j (i.e., invoke Schedule-EMP(x1, . . . , xn−j))

In your example, say what the correct answer is and also what the algorithm above finds.

Give an efficient algorithm that takes the data on robot arrivals x1, x2, . . . , xn, and the recharging function f (·), and returns the maxi- mum number of robots that can be destroyed by a sequence of EMP activations.

(b)

9. You’re helping to run a high-performance computing system capable of processing several terabytes of data per day. For each of n days, you’re presented with a quantity of data; on day i, you’re presented with xi terabytes. For each terabyte you process, you receive a fixed revenue, but any unprocessed data becomes unavailable at the end of the day (i.e., you can’t work on it in any future day).

You can’t always process everything each day because you’re con- strained by the capabilities of your computing system, which can only process a fixed number of terabytes in a given day. In fact, it’s running some one-of-a-kind software that, while very sophisticated, is not totally reliable, and so the amount of data you can process goes down with each day that passes since the most recent reboot of the system. On the first day after a reboot, you can process s1 terabytes, on the second day after a reboot, you can process s2 terabytes, and so on, up to sn; we assume s1 > s2 > s3 > . . . > sn > 0. (Of course, on day i you can only process up to xi terabytes, regardless of how fast your system is.) To get the system back to peak performance, you can choose to reboot it; but on any day you choose to reboot the system, you can’t process any data at all.

The problem. Given the amounts of available data x1, x2, . . . , xn for the next n days, and given the profile of your system as expressed by s1, s2, . . . , sn (and starting from a freshly rebooted system on day 1), choose

the days on which you’re going to reboot so as to maximize the total amount of data you process.

Example. Suppose n = 4, and the values of xi and si are given by the following table.

Day 1 Day 2 Day 3 Day 4

x 10 1 7 7 s8421

The best solution would be to reboot on day 2 only; this way, you process 8 terabytes on day 1, then 0 on day 2, then 7 on day 3, then 4 on day 4, for a total of 19. (Note that if you didn’t reboot at all, you’d process 8 + 1 + 2 + 1 = 12; and other rebooting strategies give you less than 19 as well.)

(a) Give an example of an instance with the following properties.

– There is a “surplus” of data in the sense that xi > s1 for every i. – The optimal solution reboots the system at least twice.

In addition to the example, you should say what the optimal solution is. You do not need to provide a proof that it is optimal.

(b) Give an efficient algorithm that takes values for x1, x2, . . . , xn and s1, s2, . . . , sn and returns the total number of terabytes processed by an optimal solution.

10. You’retryingtorunalargecomputingjobinwhichyouneedtosimulate a physical system for as many discrete steps as you can. The lab you’re working in has two large supercomputers (which we’ll call A and B) which are capable of processing this job. However, you’re not one of the high- priority users of these supercomputers, so at any given point in time, you’re only able to use as many spare cycles as these machines have available.

Here’s the problem you face. Your job can only run on one of the machines in any given minute. Over each of the next n minutes, you have a “profile” of how much processing power is available on each machine. In minute i, you would be able to run ai > 0 steps of the simulation if your job is on machine A, and bi > 0 steps of the simulation if your job is on machine B. You also have the ability to move your job from one machine to the other; but doing this costs you a minute of time in which no processing is done on your job.

So, given a sequence of n minutes, a plan is specified by a choice of A, B, or “move” for each minute, with the property that choices A and

Exercises

321

322

Chapter 6 Dynamic Programming

B cannot appear in consecutive minutes. For example, if your job is on machine A in minute i, and you want to switch to machine B, then your choice for minute i + 1 must be move, and then your choice for minute i + 2 can be B. The value of a plan is the total number of steps that you manage to execute over the n minutes: so it’s the sum of ai over all minutes in which the job is on A, plus the sum of bi over all minutes in which the job is on B.

Theproblem.Givenvaluesa1,a2,…,an andb1,b2,…,bn,findaplanof maximum value. (Such a strategy will be called optimal.) Note that your plan can start with either of the machines A or B in minute 1.

Example. Suppose n = 4, and the values of ai and bi are given by the following table.

Minute 1 Minute 2 Minute 3 Minute 4

A 10 1 1 10 B 5 1 20 20

Then the plan of maximum value would be to choose A for minute 1, then move for minute 2, and then B for minutes 3 and 4. The value of this plan would be 10 + 0 + 20 + 20 = 50.

(a) Show that the following algorithm does not correctly solve this problem, by giving an instance on which it does not return the correct answer.

In minute 1, choose the machine achieving the larger of a1, b1 Set i=2

While i≤n

What was the choice in minute i − 1? If A:

If bi+1 > ai + ai+1 then

Choose move in minute i and B in minute i + 1 Proceed to iteration i + 2

Else

Choose A in minute i Proceed to iteration i + 1

Endif

If B: behave as above with roles of A and B reversed

EndWhile

In your example, say what the correct answer is and also what the algorithm above finds.

(b) Give an efficient algorithm that takes values for a1, a2, . . . , an and b1, b2, . . . , bn and returns the value of an optimal plan.

11. Suppose you’re consulting for a company that manufactures PC equip- ment and ships it to distributors all over the country. For each of the next n weeks, they have a projected supply si of equipment (measured in pounds), which has to be shipped by an air freight carrier.

Each week’s supply can be carried by one of two air freight companies, A or B.

. Company A charges a fixed rate r per pound (so it costs r · si to ship a week’s supply si).

. Company B makes contracts for a fixed amount c per week, indepen- dent of the weight. However, contracts with company B must be made in blocks of four consecutive weeks at a time.

A schedule, for the PC company, is a choice of air freight company (A or B) for each of the n weeks, with the restriction that company B, whenever it is chosen, must be chosen for blocks of four contiguous weeks at a time. The cost of the schedule is the total amount paid to company A and B, according to the description above.

Give a polynomial-time algorithm that takes a sequence of supply values s1, s2, . . . , sn and returns a schedule of minimum cost.

Example. Suppose r = 1, c = 10, and the sequence of values is 11,9,9,12,12,12,12,9,9,11.

Then the optimal schedule would be to choose company A for the first three weeks, then company B for a block of four consecutive weeks, and then company A for the final three weeks.

12. Supposewewanttoreplicateafileoveracollectionofnservers,labeled S1, S2, . . . , Sn. To place a copy of the file at server Si results in a placement cost of ci, for an integer ci > 0.

Now, if a user requests the file from server Si, and no copy of the file is present at Si, then the servers Si+1, Si+2, Si+3 . . . are searched in order until a copy of the file is finally found, say at server Sj, where j > i. This results in an access cost of j − i. (Note that the lower-indexed servers Si−1, Si−2, . . . are not consulted in this search.) The access cost is 0 if Si holds a copy of the file. We will require that a copy of the file be placed at server Sn, so that all such searches will terminate, at the latest, at Sn.

Exercises

323

324

Chapter 6 Dynamic Programming

We’d like to place copies of the files at the servers so as to minimize the sum of placement and access costs. Formally, we say that a configu- rationisachoice,foreachserverSi withi=1,2,…,n−1,ofwhetherto place a copy of the file at Si or not. (Recall that a copy is always placed at Sn.) The total cost of a configuration is the sum of all placement costs for servers with a copy of the file, plus the sum of all access costs associated with all n servers.

Give a polynomial-time algorithm to find a configuration of minimum total cost.

13. Theproblemofsearchingforcyclesingraphsarisesnaturallyinfinancial

trading applications. Consider a firm that trades shares in n different

companies. For each pair i ̸= j, they maintain a trade ratio rij, meaning

that one share of i trades for rij shares of j. Here we allow the rate r to be

fractional; that is, r = 2 means that you can trade three shares of i to get ij 3

two shares of j.

A trading cycle for a sequence of shares i1, i2, . . . , ik consists of successively trading shares in company i1 for shares in company i2, then shares in company i2 for shares i3, and so on, finally trading shares in ik back to shares in company i1. After such a sequence of trades, one ends up with shares in the same company i1 that one starts with. Trading around a cycle is usually a bad idea, as you tend to end up with fewer shares than you started with. But occasionally, for short periods of time, there are opportunities to increase shares. We will call such a cycle an opportunity cycle, if trading along the cycle increases the number of shares. This happens exactly if the product of the ratios along the cycle is above 1. In analyzing the state of the market, a firm engaged in trading would like to know if there are any opportunity cycles.

Give a polynomial-time algorithm that finds such an opportunity cycle, if one exists.

14. Alargecollectionofmobilewirelessdevicescannaturallyformanetwork in which the devices are the nodes, and two devices x and y are connected by an edge if they are able to directly communicate with each other (e.g., by a short-range radio link). Such a network of wireless devices is a highly dynamic object, in which edges can appear and disappear over time as the devices move around. For instance, an edge (x, y) might disappear as x and y move far apart from each other and lose the ability to communicate directly.

In a network that changes over time, it is natural to look for efficient ways of maintaining a path between certain designated nodes. There are

two opposing concerns in maintaining such a path: we want paths that are short, but we also do not want to have to change the path frequently as the network structure changes. (That is, we’d like a single path to continue working, if possible, even as the network gains and loses edges.) Here is a way we might model this problem.

Suppose we have a set of mobile nodes V, and at a particular point in time there is a set E0 of edges among these nodes. As the nodes move, the set of edges changes from E0 to E1, then to E2, then to E3, and so on, to an edgesetEb.Fori=0,1,2,…,b,letGi denotethegraph(V,Ei).Soifwewere to watch the structure of the network on the nodes V as a “time lapse,” it would look precisely like the sequence of graphs G0, G1, G2, . . . , Gb−1, Gb. We will assume that each of these graphs Gi is connected.

Now consider two particular nodes s, t ∈ V. For an s-t path P in one of the graphs Gi, we define the length of P to be simply the number of edges in P, and we denote this l(P). Our goal is to produce a sequence of pathsP0,P1,…,Pb sothatforeachi,Pi isans-tpathinGi.Wewantthe paths to be relatively short. We also do not want there to be too many changes—points at which the identity of the path switches. Formally, we define changes(P0,P1,…,Pb) to be the number of indices i (0≤i≤b−1) for which Pi ̸= Pi+1.

Fix a constant K > 0. We define the cost of the sequence of paths P0,P1,…,Pb tobe

b i=0

(a) Suppose it is possible to choose a single path P that is an s-t path in each of the graphs G0, G1, . . . , Gb. Give a polynomial-time algorithm to find the shortest such path.

(b) Give a polynomial-time algorithm to find a sequence of paths P0,P1,…,Pb of minimum cost, where Pi is an s-t path in Gi for i = 0, 1, . . . , b.

15. Onmostcleardays,agroupofyourfriendsintheAstronomyDepartment gets together to plan out the astronomical events they’re going to try observing that night. We’ll make the following assumptions about the events.

. There are n events, which for simplicity we’ll assume occur in se- quence separated by exactly one minute each. Thus event j occurs at minute j; if they don’t observe this event at exactly minute j, then they miss out on it.

cost(P0,P1,…,Pb)=

l(Pi)+K ·changes(P0,P1,…,Pb).

Exercises

325

326

Chapter 6

Dynamic Programming

.

.

The sky is mapped according to a one-dimensional coordinate system (measured in degrees from some central baseline); event j will be taking place at coordinate dj, for some integer value dj. The telescope starts at coordinate 0 at minute 0.

The last event, n, is much more important than the others; so it is required that they observe event n.

The Astronomy Department operates a large telescope that can be used for viewing these events. Because it is such a complex instrument, it can only move at a rate of one degree per minute. Thus they do not expect to be able to observe all n events; they just want to observe as many as possible, limited by the operation of the telescope and the requirement that event n must be observed.

We say that a subset S of the events is viewable if it is possible to observe each event j ∈ S at its appointed time j, and the telescope has adequate time (moving at its maximum of one degree per minute) to move between consecutive events in S.

The problem. Given the coordinates of each of the n events, find a viewable subset of maximum size, subject to the requirement that it should contain event n. Such a solution will be called optimal.

Example. Suppose the one-dimensional coordinates of the events are as shown here.

Event 123456789 Coordinate 1 –4 –1 4 5 –4 6 7 –2

Then the optimal solution is to observe events 1, 3, 6, 9. Note that the telescope has time to move from one event in this set to the next, even moving at one degree per minute.

(a) Show that the following algorithm does not correctly solve this problem, by giving an instance on which it does not return the correct answer.

Mark all events j with |dn−dj|>n−j as illegal (as observing them would prevent you from observing event n)

Mark all other events as legal

Initialize current position to coordinate 0 at minute 0 While not at end of event sequence

Find the earliest legal event j that can be reached without exceeding the maximum movement rate of the telescope

Add j to the set S

Update current position to be coord.~dj at minute j Endwhile

Output the set S

In your example, say what the correct answer is and also what

the algorithm above finds.

(b) Give an efficient algorithm that takes values for the coordinates

d1, d2, . . . , dn of the events and returns the size of an optimal solution.

16. There are many sunny days in Ithaca, New York; but this year, as it happens, the spring ROTC picnic at Cornell has fallen on a rainy day. The ranking officer decides to postpone the picnic and must notify everyone by phone. Here is the mechanism she uses to do this.

Each ROTC person on campus except the ranking officer reports to a unique superior officer. Thus the reporting hierarchy can be described by a tree T, rooted at the ranking officer, in which each other node v has a parent node u equal to his or her superior officer. Conversely, we will call v a direct subordinate of u. See Figure 6.30, in which A is the ranking officer, B and D are the direct subordinates of A, and C is the direct subordinate of B.

To notify everyone of the postponement, the ranking officer first calls each of her direct subordinates, one at a time. As soon as each subordinate gets the phone call, he or she must notify each of his or her direct subordinates, one at a time. The process continues this way until everyone has been notified. Note that each person in this process can only call direct subordinates on the phone; for example, in Figure 6.30, A would not be allowed to call C.

We can picture this process as being divided into rounds. In one round, each person who has already learned of the postponement can call one of his or her direct subordinates on the phone. The number of rounds it takes for everyone to be notified depends on the sequence in which each person calls their direct subordinates. For example, in Figure 6.30, it will take only two rounds if A starts by calling B, but it will take three rounds if A starts by calling D.

Give an efficient algorithm that determines the minimum number of rounds needed for everyone to be notified, and outputs a sequence of phone calls that achieves this minimum number of rounds.

17. Yourfriendshavebeenstudyingtheclosingpricesoftechstocks,looking for interesting patterns. They’ve defined something called a rising trend, as follows.

A should call B before D. A

BD

C

Figure 6.30 A hierarchy with four people. The fastest broadcast scheme is for A to call B in the first round. In the second round, A calls D and B calls C. If A were to call D first, then C could not learn the news until the third round.

Exercises

327

328

Chapter 6 Dynamic Programming

They have the closing price for a given stock recorded for n days in succession; let these prices be denoted P[1], P[2], . . . , P[n]. A rising trend in these prices is a subsequence of the prices P[i1], P[i2], . . . , P[ik], for days i1 < i2 < . . . < ik, so that
. i1 = 1, and
. P[ij]< P[ij+1] for each j = 1, 2, . . . , k − 1.
Thus a rising trend is a subsequence of the days—beginning on the first day and not necessarily contiguous—so that the price strictly increases over the days in this subsequence.
They are interested in finding the longest rising trend in a given sequence of prices.
Example. Suppose n = 7, and the sequence of prices is 10,1,2,11,3,4,12.
Then the longest rising trend is given by the prices on days 1, 4, and 7. Note that days 2, 3, 5, and 6 consist of increasing prices; but because this subsequence does not begin on day 1, it does not fit the definition of a rising trend.
(a) Show that the following algorithm does not correctly return the length of the longest rising trend, by giving an instance on which it fails to return the correct answer.
Define i=1 L=1
For j=2 to n
If P[j]>P[i] then

Set i=j.

Add 1 to L Endif

Endfor

In your example, give the actual length of the longest rising trend, and say what the algorithm above returns.

(b) Give an efficient algorithm that takes a sequence of prices P[1], P[2], . . . , P[n] and returns the length of the longest rising trend.

18. Consider the sequence alignment problem over a four-letter alphabet {z1, z2, z3, z4}, with a given gap cost and given mismatch costs. Assume that each of these parameters is a positive integer.

Suppose you are given two strings A=a1a2 …am and B=b1b2 …bn and a proposed alignment between them. Give an O(mn) algorithm to decide whether this alignment is the unique minimum-cost alignment between A and B.

19. You’re consulting for a group of people (who would prefer not to be mentioned here by name) whose jobs consist of monitoring and analyzing electronic signals coming from ships in coastal Atlantic waters. They want a fast algorithm for a basic primitive that arises frequently: “untangling” a superposition of two known signals. Specifically, they’re picturing a situation in which each of two ships is emitting a short sequence of 0s and 1s over and over, and they want to make sure that the signal they’re hearing is simply an interleaving of these two emissions, with nothing extra added in.

This describes the whole problem; we can make it a little more explicit as follows. Given a string x consisting of 0s and 1s, we write xk to denote k copies of x concatenated together. We say that a string x′ is a repetition of x if it is a prefix of xk for some number k. So x′ = 10110110110 is a repetition of x = 101.

We say that a string s is an interleaving of x and y if its symbols can be partitioned into two (not necessarily contiguous) subsequences s′ and s′′, so that s′ is a repetition of x and s′′ is a repetition of y. (So each symbol in s must belong to exactly one of s′ or s′′.) For example, if x = 101 and y = 00, then s = 100010101 is an interleaving of x and y, since characters 1,2,5,7,8,9 form 101101—a repetition of x—and the remaining characters 3,4,6 form 000—a repetition of y.

In terms of our application, x and y are the repeating sequences from the two ships, and s is the signal we’re listening to: We want to make sure s “unravels” into simple repetitions of x and y. Give an efficient algorithm that takes strings s, x, and y and decides if s is an interleaving of x and y.

20. Supposeit’snearingtheendofthesemesterandyou’retakingncourses, each with a final project that still has to be done. Each project will be graded on the following scale: It will be assigned an integer number on a scale of 1 to g > 1, higher numbers being better grades. Your goal, of course, is to maximize your average grade on the n projects.

You have a total of H > n hours in which to work on the n projects cumulatively, and you want to decide how to divide up this time. For simplicity, assume H is a positive integer, and you’ll spend an integer number of hours on each project. To figure out how best to divide up yourtime,you’vecomeupwithasetoffunctions{fi :i=1,2,…,n}(rough

Exercises

329

330

Chapter 6 Dynamic Programming

estimates, of course) for each of your n courses; if you spend h ≤ H hours on the project for course i, you’ll get a grade of fi(h). (You may assume that the functions fi are nondecreasing: if h < h′, then fi(h) ≤ fi(h′).)
So the problem is: Given these functions {fi}, decide how many hours to spend on each project (in integer values only) so that your average grade, as computed according to the fi, is as large as possible. In order to be efficient, the running time of your algorithm should be polynomial in n, g, and H; none of these quantities should appear as an exponent in your running time.
21. Some time back, you helped a group of friends who were doing sim- ulations for a computation-intensive investment company, and they’ve come back to you with a new problem. They’re looking at n consecutive days of a given stock, at some point in the past. The days are numbered i = 1, 2, . . . , n; for each day i, they have a price p(i) per share for the stock on that day.
For certain (possibly large) values of k, they want to study what they call k-shot strategies. A k-shot strategy is a collection of m pairs of days (b1, s1), . . . , (bm, sm), where 0 ≤ m ≤ k and
1 ≤ b1 < s1 < b2 < s2 . . . < bm < sm ≤ n.
We view these as a set of up to k nonoverlapping intervals, during each of which the investors buy 1,000 shares of the stock (on day bi) and then sell it (on day si). The return of a given k-shot strategy is simply the profit obtained from the m buy-sell transactions, namely,
m i=1
The investors want to assess the value of k-shot strategies by running simulations on their n-day trace of the stock price. Your goal is to design an efficient algorithm that determines, given the sequence of prices, the k- shot strategy with the maximum possible return. Since k may be relatively large in these simulations, your running time should be polynomial in both n and k; it should not contain k in the exponent.
22. To assess how “well-connected” two nodes in a directed graph are, one can not only look at the length of the shortest path between them, but can also count the number of shortest paths.
This turns out to be a problem that can be solved efficiently, subject to some restrictions on the edge costs. Suppose we are given a directed graph G = (V , E), with costs on the edges; the costs may be positive or
1,000
p(si) − p(bi).
negative, but every cycle in the graph has strictly positive cost. We are also given two nodes v, w ∈ V. Give an efficient algorithm that computes the number of shortest v-w paths in G. (The algorithm should not list all the paths; just the number suffices.)
23. SupposeyouaregivenadirectedgraphG=(V,E)withcostsontheedges ce for e ∈ E and a sink t (costs may be negative). Assume that you also have finite values d(v) for v ∈ V. Someone claims that, for each node v ∈ V, the quantity d(v) is the cost of the minimum-cost path from node v to the sink t.
(a) Give a linear-time algorithm (time O(m) if the graph has m edges) that verifies whether this claim is correct.
(b) Assume that the distances are correct, and d(v) is finite for all v ∈ V. Now you need to compute distances to a different sink t′. Give an O(m log n) algorithm for computing distances d′(v) for all nodes v ∈ V to the sink node t′. (Hint: It is useful to consider a new cost function defined as follows: for edge e = (v, w), let ce′ = ce − d(v) + d(w). Is there a relation between costs of paths for the two different costs c and c′?)
24. Gerrymandering is the practice of carving up electoral districts in very careful ways so as to lead to outcomes that favor a particular political party. Recent court challenges to the practice have argued that through this calculated redistricting, large numbers of voters are being effectively (and intentionally) disenfranchised.
Computers, it turns out, have been implicated as the source of some of the “villainy” in the news coverage on this topic: Thanks to powerful software, gerrymandering has changed from an activity carried out by a bunch of people with maps, pencil, and paper into the industrial-strength process that it is today. Why is gerrymandering a computational problem? There are database issues involved in tracking voter demographics down to the level of individual streets and houses; and there are algorithmic issues involved in grouping voters into districts. Let’s think a bit about what these latter issues look like.
Suppose we have a set of n precincts P1, P2, . . . , Pn, each containing m registered voters. We’re supposed to divide these precincts into two districts, each consisting of n/2 of the precincts. Now, for each precinct, we have information on how many voters are registered to each of two political parties. (Suppose, for simplicity, that every voter is registered to one of these two.) We’ll say that the set of precincts is susceptible to gerrymandering if it is possible to perform the division into two districts in such a way that the same party holds a majority in both districts.
Exercises
331
332
Chapter 6 Dynamic Programming
Give an algorithm to determine whether a given set of precincts is susceptible to gerrymandering; the running time of your algorithm should be polynomial in n and m.
Example. Suppose we have n = 4 precincts, and the following information on registered voters.
Precinct
Number registered for party A Number registered for party B
1 2
55 43 45 57
3 4
60 47 40 53
This set of precincts is susceptible since, if we grouped precincts 1 and 4 into one district, and precincts 2 and 3 into the other, then party A would have a majority in both districts. (Presumably, the “we” who are doing the grouping here are members of party A.) This example is a quick illustration of the basic unfairness in gerrymandering: Although party A holds only a slim majority in the overall population (205 to 195), it ends up with a majority in not one but both districts.
25. Considertheproblemfacedbyastockbrokertryingtosellalargenumber of shares of stock in a company whose stock price has been steadily falling in value. It is always hard to predict the right moment to sell stock, but owning a lot of shares in a single company adds an extra complication: the mere act of selling many shares in a single day will have an adverse effect on the price.
Since future market prices, and the effect of large sales on these prices, are very hard to predict, brokerage firms use models of the market to help them make such decisions. In this problem, we will consider the following simple model. Suppose we need to sell x shares of stock in a company, and suppose that we have an accurate model of the market: it predicts that the stock price will take the values p1, p2, . . . , pn over the next n days. Moreover, there is a function f(·) that predicts the effect of large sales: if we sell y shares on a single day, it will permanently decrease the price by f (y) from that day onward. So, if we sell y1 shares on day 1, we obtain a price per share of p1 − f (y1), for a total income of y1 · (p1 − f (y1)). Having sold y1 shares on day 1, we can then sell y2 shares on day 2 for a price per share of p2 − f (y1) − f (y2); this yields an additional income of y2 · (p2 − f (y1) − f (y2)). This process continues over all n days. (Note, as in our calculation for day 2, that the decreases from earlier days are absorbed into the prices for all later days.)
Design an efficient algorithm that takes the prices p1, . . . , pn and the function f (·) (written as a list of values f (1), f (2), . . . , f (x)) and determines
the best way to sell x shares by day n. In other words, find natural numbers y1,y2,...,yn so that x=y1+...+yn, and selling yi shares on day i for i = 1, 2, . . . , n maximizes the total income achievable. You should assume that the share value pi is monotone decreasing, and f(·) is monotone increasing; that is, selling a larger number of shares causes a larger drop in the price. Your algorithm’s running time can have a polynomial dependence on n (the number of days), x (the number of shares), and p1 (the peak price of the stock).
Example Consider the case when n = 3; the prices for the three days are 90, 80, 40; and f (y) = 1 for y ≤ 40,000 and f (y) = 20 for y > 40, 000. Assume you start with x = 100, 000 shares. Selling all of them on day 1 would yield a price of 70 per share, for a total income of 7,000,000. On the other hand, selling 40,000 shares on day 1 yields a price of 89 per share, and selling the remaining 60,000 shares on day 2 results in a price of 59 per share, for a total income of 7,100,000.

26. Consider the following inventory problem. You are running a company that sells some large product (let’s assume you sell trucks), and predic- tions tell you the quantity of sales to expect over the next n months. Let di denote the number of sales you expect in month i. We’ll assume that all sales happen at the beginning of the month, and trucks that are not sold are stored until the beginning of the next month. You can store at most S trucks, and it costs C to store a single truck for a month. You receive shipments of trucks by placing orders for them, and there is a fixed ordering fee of K each time you place an order (regardless of the number of trucks you order). You start out with no trucks. The problem is to design an algorithm that decides how to place orders so that you satisfy all the demands {di}, and minimize the costs. In summary:

. There are two parts to the cost: (1) storage—it costs C for every truck on hand that is not needed that month; (2) ordering fees—it costs K for every order placed.

. In each month you need enough trucks to satisfy the demand di, but the number left over after satisfying the demand for the month should not exceed the inventory limit S.

Give an algorithm that solves this problem in time that is polynomial in n and S.

27. The owners of an independently operated gas station are faced with the following situation. They have a large underground tank in which they store gas; the tank can hold up to L gallons at one time. Ordering gas is quite expensive, so they want to order relatively rarely. For each order,

Exercises

333

334

Chapter 6 Dynamic Programming

they need to pay a fixed price P for delivery in addition to the cost of the gas ordered. However, it costs c to store a gallon of gas for an extra day, so ordering too much ahead increases the storage cost.

They are planning to close for a week in the winter, and they want their tank to be empty by the time they close. Luckily, based on years of experience, they have accurate projections for how much gas they will need each day until this point in time. Assume that there are n days left until they close, and they need gi gallons of gas for each of the days i=1,…,n. Assume that the tank is empty at the end of day 0. Give an algorithm to decide on which days they should place orders, and how much to order so as to minimize their total cost.

28. Recall the scheduling problem from Section 4.2 in which we sought to minimize the maximum lateness. There are n jobs, each with a deadline di and a required processing time ti, and all jobs are available to be scheduled starting at time s. For a job i to be done, it needs to be assigned a period from si ≥ s to fi = si + ti, and different jobs should be assigned nonoverlapping intervals. As usual, such an assignment of times will be called a schedule.

In this problem, we consider the same setup, but want to optimize a different objective. In particular, we consider the case in which each job must either be done by its deadline or not at all. We’ll say that a subset J of the jobs is schedulable if there is a schedule for the jobs in J so that each of them finishes by its deadline. Your problem is to select a schedulable subset of maximum possible size and give a schedule for this subset that allows each job to finish by its deadline.

(a) Prove that there is an optimal solution J (i.e., a schedulable set of maximum size) in which the jobs in J are scheduled in increasing order of their deadlines.

(b) Assume that all deadlines di and required times ti are integers. Give an algorithm to find an optimal solution. Your algorithm should run in time polynomial in the number of jobs n, and the maximum deadline D = maxi di.

29. Let G = (V, E) be a graph with n nodes in which each pair of nodes is joined by an edge. There is a positive weight wij on each edge (i, j); and we will assume these weights satisfy the triangle inequality wik ≤ wij + wjk . For a subset V′ ⊆ V, we will use G[V′] to denote the subgraph (with edge weights) induced on the nodes in V′.

We are given a set X ⊆ V of k terminals that must be connected by edges. We say that a Steiner tree on X is a set Z so that X ⊆ Z ⊆ V, together

with a spanning subtree T of G[Z]. The weight of the Steiner tree is the weight of the tree T.

Show that there is function f (·) and a polynomial function p(·) so that the problem of finding a minimum-weight Steiner tree on X can be solved in time O(f (k) · p(n)).

Notes and Further Reading

Richard Bellman is credited with pioneering the systematic study of dynamic programming (Bellman 1957); the algorithm in this chapter for segmented least squares is based on Bellman’s work from this early period (Bellman 1961). Dynamic programming has since grown into a technique that is widely used across computer science, operations research, control theory, and a number of other areas. Much of the recent work on this topic has been concerned with stochastic dynamic programming: Whereas our problem formulations tended to tacitly assume that all input is known at the outset, many problems in scheduling, production and inventory planning, and other domains involve uncertainty, and dynamic programming algorithms for these problems encode this uncertainty using a probabilistic formulation. The book by Ross (1983) provides an introduction to stochastic dynamic programming.

Many extensions and variations of the Knapsack Problem have been studied in the area of combinatorial optimization. As we discussed in the chapter, the pseudo-polynomial bound arising from dynamic programming can become prohibitive when the input numbers get large; in these cases, dynamic programming is often combined with other heuristics to solve large instances of Knapsack Problems in practice. The book by Martello and Toth (1990) is devoted to computational approaches to versions of the Knapsack Problem.

Dynamic programming emerged as a basic technique in computational bi- ology in the early 1970s, in a flurry of activity on the problem of sequence comparison. Sankoff (2000) gives an interesting historical account of the early work in this period. The books by Waterman (1995) and Gusfield (1997) pro- vide extensive coverage of sequence alignment algorithms (as well as many related algorithms in computational biology); Mathews and Zuker (2004) dis- cuss further approaches to the problem of RNA secondary structure prediction. The space-efficient algorithm for sequence alignment is due to Hirschberg (1975).

The algorithm for the Shortest-Path Problem described in this chapter is based originally on the work of Bellman (1958) and Ford (1956). Many op- timizations, motivated both by theoretical and experimental considerations,

Notes and Further Reading

335

336

Chapter 6 Dynamic Programming

have been added to this basic approach to shortest paths; a Web site main- tained by Andrew Goldberg contains state-of-the-art code that he has de- veloped for this problem (among a number of others), based on work by Cherkassky, Goldberg and Radzik (1994). The applications of shortest-path methods to Internet routing, and the trade-offs among the different algorithms for networking applications, are covered in books by Bertsekas and Gallager (1992), Keshav (1997), and Stewart (1998).

Notes on the Exercises Exercise 5 is based on discussions with Lillian Lee; Exercise 6 is based on a result of Donald Knuth; Exercise 25 is based on results of Dimitris Bertsimas and Andrew Lo; and Exercise 29 is based on a result of S. Dreyfus and R. Wagner.

Chapter 7 Network Flow

In this chapter, we focus on a rich set of algorithmic problems that grow, in a sense, out of one of the original problems we formulated at the beginning of the course: Bipartite Matching.

Recall the set-up of the Bipartite Matching Problem. A bipartite graph G=(V,E) is an undirected graph whose node set can be partitioned as V=X∪Y, with the property that every edge e∈E has one end in X and the other end in Y. We often draw bipartite graphs as in Figure 7.1, with the nodes in X in a column on the left, the nodes in Y in a column on the right, and each edge crossing from the left column to the right column.

Now, we’ve already seen the notion of a matching at several points in the course: We’ve used the term to describe collections of pairs over a set, with the property that no element of the set appears in more than one pair. (Think of men (X) matched to women (Y) in the Stable Matching Problem, or characters in the Sequence Alignment Problem.) In the case of a graph, the edges constitute pairs of nodes, and we consequently say that a matching in a graph G = (V, E) is a set of edges M ⊆ E with the property that each node appears in at most one edge of M. A set of edges M is a perfect matching if every node appears in exactly one edge of M.

Matchings in bipartite graphs can model situations in which objects are being assigned to other objects. We have seen a number of such situations in our earlier discussions of graphs and bipartite graphs. One natural example arises when the nodes in X represent jobs, the nodes in Y represent machines, and an edge (xi, yj) indicates that machine yj is capable of processing job xi. A perfect matching is, then, a way of assigning each job to a machine that can process it, with the property that each machine is assigned exactly one job. Bipartite graphs can represent many other relations that arise between two

x1 y1 x2 y2 x3 y3 x4 y4 x5 y5

Figure 7.1 A bipartite graph.

338

Chapter 7 Network Flow

distinct sets of objects, such as the relation between customers and stores; or houses and nearby fire stations; and so forth.

One of the oldest problems in combinatorial algorithms is that of deter- mining the size of the largest matching in a bipartite graph G. (As a special case, note that G has a perfect matching if and only if |X| = |Y| and it has a matching of size |X|.) This problem turns out to be solvable by an algorithm that runs in polynomial time, but the development of this algorithm needs ideas fundamentally different from the techniques that we’ve seen so far.

Rather than developing the algorithm directly, we begin by formulating a general class of problems—network flow problems—that includes the Bipartite Matching Problem as a special case. We then develop a polynomial-time algorithm for a general problem, the Maximum-Flow Problem, and show how this provides an efficient algorithm for Bipartite Matching as well. While the initial motivation for network flow problems comes from the issue of traffic in a network, we will see that they have applications in a surprisingly diverse set of areas and lead to efficient algorithms not just for Bipartite Matching, but for a host of other problems as well.

7.1 The Maximum-Flow Problem and the Ford-Fulkerson Algorithm

The Problem

One often uses graphs to model transportation networks—networks whose edges carry some sort of traffic and whose nodes act as “switches” passing traffic between different edges. Consider, for example, a highway system in which the edges are highways and the nodes are interchanges; or a computer network in which the edges are links that can carry packets and the nodes are switches; or a fluid network in which edges are pipes that carry liquid, and the nodes are junctures where pipes are plugged together. Network models of this type have several ingredients: capacities on the edges, indicating how much they can carry; source nodes in the graph, which generate traffic; sink (or destination) nodes in the graph, which can “absorb” traffic as it arrives; and finally, the traffic itself, which is transmitted across the edges.

Flow Networks We’ll be considering graphs of this form, and we refer to the traffic as flow—an abstract entity that is generated at source nodes, transmitted across edges, and absorbed at sink nodes. Formally, we’ll say that a flow network is a directed graph G = (V , E) with the following features.

. Associated with each edge e is a capacity, which is a nonnegative number that we denote ce.

7.1 The Maximum-Flow Problem and the Ford-Fulkerson Algorithm

339

. There is a single source node s ∈ V.

. There is a single sink node t ∈ V.

Nodes other than s and t will be called internal nodes.

u

20

s 30 t

20

v

Figure 7.2 A flow network, with source s and sink t. The numbers next to the edges are the capacities.

We will make two assumptions about the flow networks we deal with: first,

that no edge enters the source s and no edge leaves the sink t; second, that

there is at least one edge incident to each node; and third, that all capacities

are integers. These assumptions make things cleaner to think about, and while 10 they eliminate a few pathologies, they preserve essentially all the issues we

want to think about.

10

Figure 7.2 illustrates a flow network with four nodes and five edges, and capacity values given next to each edge.

Defining Flow Next we define what it means for our network to carry traffic, or flow. We say that an s-t flow is a function f that maps each edge e to a nonnegative real number, f : E → R+; the value f (e) intuitively represents the amount of flow carried by edge e. A flow f must satisfy the following two properties.1

(i) (Capacityconditions)Foreache∈E,wehave0≤f(e)≤ce.

(ii) (Conservation conditions) For each node v other than s and t, we have

f(e)= f(e). e into v e out of v

f(e) sums the flow value f(e) over all edges entering node v,

f(e) is the sum of flow values over all edges leaving node v.

Thus the flow on an edge cannot exceed the capacity of the edge. For every node other than the source and the sink, the amount of flow entering must equal the amount of flow leaving. The source has no entering edges (by our assumption), but it is allowed to have flow going out; in other words, it can generate flow. Symmetrically, the sink is allowed to have flow coming in, even though it has no edges leaving it. The value of a flow f, denoted ν(f), is defined to be the amount of flow generated at the source:

ν(f)= f(e). e out of s

and f

1 Our notion of flow models traffic as it goes through the network at a steady rate. We have a single variable f (e) to denote the amount of flow on edge e. We do not model bursty traffic, where the flow fluctuates over time.

Here

e into v

while

e out of v

To make the notation more compact, we define fout(v)= f(e)

in e out of v (v)= eintov f(e). We can extend this to sets of vertices; if S⊆V, we

340

Chapter 7 Network Flow

define f out(S) = f (e) and f in(S) = f (e). In this terminology, e out of S e into S

the conservation condition for nodes v ̸= s, t becomes f in(v) = f out(v); and we can write ν(f ) = f out(s).

The Maximum-Flow Problem Given a flow network, a natural goal is to arrange the traffic so as to make as efficient use as possible of the available capacity. Thus the basic algorithmic problem we will consider is the following: Given a flow network, find a flow of maximum possible value.

As we think about designing algorithms for this problem, it’s useful to consider how the structure of the flow network places upper bounds on the maximum value of an s-t flow. Here is a basic “obstacle” to the existence of large flows: Suppose we divide the nodes of the graph into two sets, A and B, so that s∈A and t∈B. Then, intuitively, any flow that goes from s to t must cross from A into B at some point, and thereby use up some of the edge capacity from A to B. This suggests that each such “cut” of the graph puts a bound on the maximum possible flow value. The maximum-flow algorithm that we develop here will be intertwined with a proof that the maximum-flow value equals the minimum capacity of any such division, called the minimum cut. As a bonus, our algorithm will also compute the minimum cut. We will see that the problem of finding cuts of minimum capacity in a flow network turns out to be as valuable, from the point of view of applications, as that of finding a maximum flow.

Designing the Algorithm

Suppose we wanted to find a maximum flow in a network. How should we go about doing this? It takes some testing out to decide that an approach such as dynamic programming doesn’t seem to work—at least, there is no algorithm known for the Maximum-Flow Problem that could really be viewed as naturally belonging to the dynamic programming paradigm. In the absence of other ideas, we could go back and think about simple greedy approaches, to see where they break down.

Suppose we start with zero flow: f (e) = 0 for all e. Clearly this respects the capacity and conservation conditions; the problem is that its value is 0. We now try to increase the value of f by “pushing” flow along a path from s to t, up to the limits imposed by the edge capacities. Thus, in Figure 7.3, we might choose the path consisting of the edges {(s, u), (u, v), (v, t)} and increase the flow on each of these edges to 20, and leave f (e) = 0 for the other two. In this way, we still respect the capacity conditions—since we only set the flow as high as the edge capacities would allow—and the conservation conditions— since when we increase flow on an edge entering an internal node, we also increase it on an edge leaving the node. Now, the value of our flow is 20, and we can ask: Is this the maximum possible for the graph in the figure? If we

7.1 The Maximum-Flow Problem and the Ford-Fulkerson Algorithm

341

uuu

20 10

30 t s

10 20

20 10

30 t s

10 20

20

10

30 t

s

10 20

vvv

(a) (b) (c) Figure 7.3 (a) The network of Figure 7.2. (b) Pushing 20 units of flow along the path

s, u, v, t. (c) The new kind of augmenting path using the edge (u, v) backward.

think about it, we see that the answer is no, since it is possible to construct a flow of value 30. The problem is that we’re now stuck—there is no s-t path on which we can directly push flow without exceeding some capacity—and yet we do not have a maximum flow. What we need is a more general way of pushing flow from s to t, so that in a situation such as this, we have a way to increase the value of the current flow.

Essentially, we’d like to perform the following operation denoted by a dotted line in Figure 7.3(c). We push 10 units of flow along (s, v); this now results in too much flow coming into v. So we “undo” 10 units of flow on (u, v); this restores the conservation condition at v but results in too little flow leaving u. So, finally, we push 10 units of flow along (u, t), restoring the conservation condition at u. We now have a valid flow, and its value is 30. See Figure 7.3, where the dark edges are carrying flow before the operation, and the dashed edges form the new kind of augmentation.

This is a more general way of pushing flow: We can push forward on edges with leftover capacity, and we can push backward on edges that are already carrying flow, to divert it in a different direction. We now define the residual graph, which provides a systematic way to search for forward- backward operations such as this.

The Residual Graph Given a flow network G, and a flow f on G, we define the residual graph Gf of G with respect to f as follows. (See Figure 7.4 for the residual graph of the flow on Figure 7.3 after pushing 20 units of flow along the path s, u, v, t.)

. ThenodesetofGf isthesameasthatofG.

. For each edge e=(u,v) of G on which f(e)

This completes the definition of the residual graph Gf . Note that each edge e in G can give rise to one or two edges in Gf: If 0

7.1 The Maximum-Flow Problem and the Ford-Fulkerson Algorithm

345

Proof. The first edge e of P must be an edge out of s in the residual graph Gf; and since the path is simple, it does not visit s again. Since G has no edges entering s, the edge e must be a forward edge. We increase the flow on this edge by bottleneck(P,f), and we do not change the flow on any other edge incident to s. Therefore the value of f′ exceeds the value of f by bottleneck(P , f ).

We need one more observation to prove termination: We need to be able

to bound the maximum possible flow value. Here’s one upper bound: If all the

edges out of s could be completely saturated with flow, the value of the flow

would be c . Let C denote this sum. Thus we have ν(f)≤C for all eoutofs e

s-t flows f. (C may be a huge overestimate of the maximum value of a flow in G, but it’s handy for us as a finite, simply stated bound.) Using statement (7.3), we can now prove termination.

(7.4) Suppose, as above, that all capacities in the flow network G are integers. Then the Ford-Fulkerson Algorithm terminates in at most C iterations of the While loop.

Proof. We noted above that no flow in G can have value greater than C, due to the capacity condition on the edges leaving s. Now, by (7.3), the value of the flow maintained by the Ford-Fulkerson Algorithm increases in each iteration; so by (7.2), it increases by at least 1 in each iteration. Since it starts with the value 0, and cannot go higher than C, the While loop in the Ford-Fulkerson Algorithm can run for at most C iterations.

Next we consider the running time of the Ford-Fulkerson Algorithm. Let n denote the number of nodes in G, and m denote the number of edges in G. We have assumed that all nodes have at least one incident edge, hence m ≥ n/2, and so we can use O(m + n) = O(m) to simplify the bounds.

(7.5) Suppose, as above, that all capacities in the flow network G are integers. Then the Ford-Fulkerson Algorithm can be implemented to run in O(mC) time.

Proof. We know from (7.4) that the algorithm terminates in at most C itera- tions of the While loop. We therefore consider the amount of work involved in one iteration when the current flow is f .

The residual graph Gf has at most 2m edges, since each edge of G gives rise to at most two edges in the residual graph. We will maintain Gf using an adjacency list representation; we will have two linked lists for each node v, one containing the edges entering v, and one containing the edges leaving v. To find an s-t path in Gf , we can use breadth-first search or depth-first search,

346

Chapter 7 Network Flow

which run in O(m + n) time; by our assumption that m ≥ n/2, O(m + n) is the same as O(m). The procedure augment(f , P) takes time O(n), as the path P has at most n − 1 edges. Given the new flow f ′, we can build the new residual graph in O(m) time: For each edge e of G, we construct the correct forward and backward edges in Gf′.

A somewhat more efficient version of the algorithm would maintain the linked lists of edges in the residual graph Gf as part of the augment procedure that changes the flow f via augmentation.

7.2 Maximum Flows and Minimum Cuts in a Network

We now continue with the analysis of the Ford-Fulkerson Algorithm, an activity that will occupy this whole section. In the process, we will not only learn a lot about the algorithm, but also find that analyzing the algorithm provides us with considerable insight into the Maximum-Flow Problem itself.

Analyzing the Algorithm: Flows and Cuts

Our next goal is to show that the flow that is returned by the Ford-Fulkerson

Algorithm has the maximum possible value of any flow in G. To make progress

toward this goal, we return to an issue that we raised in Section 7.1: the way in

which the structure of the flow network places upper bounds on the maximum

value of an s-t flow. We have already seen one upper bound: the value ν(f) of

any s-t-flow f is at most C = c . Sometimes this bound is useful, but eoutofs e

sometimes it is very weak. We now use the notion of a cut to develop a much more general means of placing upper bounds on the maximum-flow value.

Consider dividing the nodes of the graph into two sets, A and B, so that

s ∈ A and t ∈ B. As in our discussion in Section 7.1, any such division places

an upper bound on the maximum possible flow value, since all the flow must

cross from A to B somewhere. Formally, we say that an s-t cut is a partition

(A, B) of the vertex set V, so that s ∈ A and t ∈ B. The capacity of a cut (A, B),

which we will denote c(A, B), is simply the sum of the capacities of all edges

out of A: c(A,B)= c . eoutofA e

Cuts turn out to provide very natural upper bounds on the values of flows, as expressed by our intuition above. We make this precise via a sequence of facts.

(7.6) Let f be any s-t flow, and (A, B) any s-t cut. Then ν(f) = fout(A) − fin(A).

7.2 Maximum Flows and Minimum Cuts in a Network

347

This statement is actually much stronger than a simple upper bound. It says that by watching the amount of flow f sends across a cut, we can exactly measure the flow value: It is the total amount that leaves A, minus the amount that “swirls back” into A. This makes sense intuitively, although the proof requires a little manipulation of sums.

Proof. By definition ν(f ) = f out(s). By assumption we have f in(s) = 0, as the source s has no entering edges, so we can write ν(f ) = f out(s) − f in(s). Since every node v in A other than s is internal, we know that f out(v) − f in(v) = 0 for all such nodes. Thus

ν(f ) = (f out(v) − f in(v)), v∈A

since the only term in this sum that is nonzero is the one in which v is set to s.

Let’s try to rewrite the sum on the right as follows. If an edge e has both ends in A, then f (e) appears once in the sum with a “+” and once with a “−”, and hence these two terms cancel out. If e has only its tail in A, then f(e) appears just once in the sum, with a “+”. If e has only its head in A, then f (e) also appears just once in the sum, with a “−”. Finally, if e has neither end in A, then f(e) doesn’t appear in the sum at all. In view of this, we have

out in out in

f (v)−f (v)=

Putting together these two equations, we have the statement of (7.6).

v∈A

e out of A

f(e)− f(e)=f (A)−f (A). e into A

If A = {s}, then fout(A) = fout(s), and fin(A) = 0 as there are no edges entering the source by assumption. So the statement for this set A = {s} is exactly the definition of the flow value ν(f).

Note that if (A, B) is a cut, then the edges into B are precisely the edges out of A. Similarly, the edges out of B are precisely the edges into A. Thus we have f out(A) = f in(B) and f in(A) = f out(B), just by comparing the definitions for these two expressions. So we can rephrase (7.6) in the following way.

(7.7) Letf beanys-tflow,and(A,B)anys-tcut.Thenν(f)=fin(B)−fout(B).

If we set A = V − {t} and B = {t} in (7.7), we have ν(f ) = f in(B) − f out(B) = f in(t) − f out(t). By our assumption the sink t has no leaving edges, so we have f out(t) = 0. This says that we could have originally defined the value of a flow equally well in terms of the sink t: It is fin(t), the amount of flow arriving at the sink.

A very useful consequence of (7.6) is the following upper bound. (7.8) Let f be any s-t flow, and (A, B) any s-t cut. Then ν(f ) ≤ c(A, B).

348

Chapter 7

Network Flow

Proof.

ν(f) = fout(A) − fin(A)

≤fout(A)

= f(e)

e out of A

≤ ce

e out of A = c(A, B).

Here the first line is simply (7.6); we pass from the first to the second since f in(A) ≥ 0, and we pass from the third to the fourth by applying the capacity conditions to each term of the sum.

In a sense, (7.8) looks weaker than (7.6), since it is only an inequality rather than an equality. However, it will be extremely useful for us, since its right-hand side is independent of any particular flow f . What (7.8) says is that the value of every flow is upper-bounded by the capacity of every cut. In other words, if we exhibit any s-t cut in G of some value c∗, we know immediately by (7.8) that there cannot be an s-t flow in G of value greater than c∗. Conversely, if we exhibit any s-t flow in G of some value ν∗, we know immediately by (7.8) that there cannot be an s-t cut in G of value less than ν∗.

Analyzing the Algorithm: Max-Flow Equals Min-Cut

Let f denote the flow that is returned by the Ford-Fulkerson Algorithm. We want to show that f has the maximum possible value of any flow in G, and we do this by the method discussed above: We exhibit an s-t cut (A∗, B∗) for which ν(f) = c(A∗, B∗). This immediately establishes that f has the maximum value of any flow, and that (A∗, B∗) has the minimum capacity of any s-t cut.

The Ford-Fulkerson Algorithm terminates when the flow f has no s-t path in the residual graph Gf . This turns out to be the only property needed for proving its maximality.

(7.9) If f is an s-t-flow such that there is no s-t path in the residual graph Gf , then there is an s-t cut (A∗, B∗) in G for which ν(f) = c(A∗, B∗). Consequently, f has the maximum value of any flow in G, and (A∗, B∗) has the minimum capacity of any s-t cut in G.

Proof. Thestatementclaimstheexistenceofacutsatisfyingacertaindesirable property; thus we must now identify such a cut. To this end, let A∗ denote the set of all nodes v in G for which there is an s-v path in Gf. Let B∗ denote the set of all other nodes: B∗ = V − A∗.

Residual graph

(u, v) is saturated with flow.

v

t

u

(u, v) carries

no flow.

7.2 Maximum Flows and Minimum Cuts in a Network

349

s

u v

A*

Figure 7.5 The (A∗, B∗) cut in the proof of (7.9).

First we establish that (A∗, B∗) is indeed an s-t cut. It is clearly a partition of V. The source s belongs to A∗ since there is always a path from s to s. Moreover, t ̸∈ A∗ by the assumption that there is no s-t path in the residual graph; hence t ∈ B∗ as desired.

Next,supposethate=(u,v)isanedgeinGforwhichu∈A∗ andv∈B∗,as shown in Figure 7.5. We claim that f (e) = ce. For if not, e would be a forward edge in the residual graph Gf , and since u ∈ A∗, there is an s-u path in Gf ; appending e to this path, we would obtain an s-v path in Gf , contradicting our assumption that v ∈ B∗.

Now suppose that e′ = (u′, v′) is an edge in G for which u′ ∈ B∗ and v′ ∈ A∗. We claim that f(e′)=0. For if not, e′ would give rise to a backward edge e′′ = (v′ , u′) in the residual graph Gf , and since v′ ∈ A∗, there is an s-v′ path in Gf ; appending e′′ to this path, we would obtain an s-u′ path in Gf , contradicting our assumption that u′ ∈ B∗.

So all edges out of A∗ are completely saturated with flow, while all edges into A∗ are completely unused. We can now use (7.6) to reach the desired conclusion:

ν(f)=fout(A∗)−fin(A∗)

= f(e)− f(e) e out of A∗ e into A∗

ce − 0 =c(A∗,B∗).

=

e out of A∗

B*

350

Chapter 7 Network Flow

Note how, in retrospect, we can see why the two types of residual edges— forward and backward—are crucial in analyzing the two terms in the expres- sion from (7.6).

Given that the Ford-Fulkerson Algorithm terminates when there is no s-t in the residual graph, (7.6) immediately implies its optimality.

(7.10) The flow f returned by the Ford-Fulkerson Algorithm is a maximum flow.

We also observe that our algorithm can easily be extended to compute a minimum s-t cut (A∗, B∗), as follows.

(7.11) Given a flow f of maximum value, we can compute an s-t cut of minimum capacity in O(m) time.

Proof. We simply follow the construction in the proof of (7.9). We construct the residual graph Gf , and perform breadth-first search or depth-first search to determine the set A∗ of all nodes that s can reach. We then define B∗ = V − A∗, and return the cut (A∗, B∗).

Note that there can be many minimum-capacity cuts in a graph G; the procedure in the proof of (7.11) is simply finding a particular one of these cuts, starting from a maximum flow f .

As a bonus, we have obtained the following striking fact through the analysis of the algorithm.

(7.12) In every flow network, there is a flow f and a cut (A, B) so that ν(f)=c(A,B).

The point is that f in (7.12) must be a maximum s-t flow; for if there were a flow f ′ of greater value, the value of f ′ would exceed the capacity of (A, B), and this would contradict (7.8). Similarly, it follows that (A, B) in (7.12) is a minimum cut—no other cut can have smaller capacity—for if there were a cut (A′ , B′) of smaller capacity, it would be less than the value of f , and this again would contradict (7.8). Due to these implications, (7.12) is often called the Max-Flow Min-Cut Theorem, and is phrased as follows.

(7.13) In every flow network, the maximum value of an s-t flow is equal to the minimum capacity of an s-t cut.

7.2 Maximum Flows and Minimum Cuts in a Network

351

Further Analysis: Integer-Valued Flows

Among the many corollaries emerging from our analysis of the Ford-Fulkerson Algorithm, here is another extremely important one. By (7.2), we maintain an integer-valued flow at all times, and by (7.9), we conclude with a maximum flow. Thus we have

(7.14) If all capacities in the flow network are integers, then there is a maximum flow f for which every flow value f(e) is an integer.

Note that (7.14) does not claim that every maximum flow is integer-valued, only that some maximum flow has this property. Curiously, although (7.14) makes no reference to the Ford-Fulkerson Algorithm, our algorithmic approach here provides what is probably the easiest way to prove it.

Real Numbers as Capacities? Finally, before moving on, we can ask how crucial our assumption of integer capacities was (ignoring (7.4), (7.5) and (7.14), which clearly needed it). First we notice that allowing capacities to be rational numbers does not make the situation any more general, since we can determine the least common multiple of all capacities, and multiply them all by this value to obtain an equivalent problem with integer capacities.

But what if we have real numbers as capacities? Where in the proof did we rely on the capacities being integers? In fact, we relied on it quite crucially: We used (7.2) to establish, in (7.4), that the value of the flow increased by at least 1 in every step. With real numbers as capacities, we should be concerned that the value of our flow keeps increasing, but in increments that become arbitrarily smaller and smaller; and hence we have no guarantee that the number of iterations of the loop is finite. And this turns out to be an extremely real worry, for the following reason: With pathological choices for the augmenting path, the Ford-Fulkerson Algorithm with real-valued capacities can run forever.

However, one can still prove that the Max-Flow Min-Cut Theorem (7.12) is true even if the capacities may be real numbers. Note that (7.9) assumed only that the flow f has no s-t path in its residual graph Gf , in order to conclude that there is an s-t cut of equal value. Clearly, for any flow f of maximum value, the residual graph has no s-t-path; otherwise there would be a way to increase the value of the flow. So one can prove (7.12) in the case of real-valued capacities by simply establishing that for every flow network, there exists a maximum flow.

Of course, the capacities in any practical application of network flow would be integers or rational numbers. However, the problem of pathological choices for the augmenting paths can manifest itself even with integer capacities: It can make the Ford-Fulkerson Algorithm take a gigantic number of iterations.

352

Chapter 7 Network Flow

In the next section, we discuss how to select augmenting paths so as to avoid the potential bad behavior of the algorithm.

7.3 Choosing Good Augmenting Paths

In the previous section, we saw that any way of choosing an augmenting path

increases the value of the flow, and this led to a bound of C on the number of

augmentations, where C = c . When C is not very large, this can be eoutofs e

a reasonable bound; however, it is very weak when C is large.

To get a sense for how bad this bound can be, consider the example graph in Figure 7.2; but this time assume the capacities are as follows: The edges (s, v), (s, u), (v, t) and (u, t) have capacity 100, and the edge (u, v) has capacity 1, as shown in Figure 7.6. It is easy to see that the maximum flow has value 200, andhasf(e)=100fortheedges(s,v),(s,u),(v,t)and(u,t)andvalue0onthe edge (u, v). This flow can be obtained by a sequence of two augmentations, using the paths of nodes s, u, t and path s, v, t. But consider how bad the Ford-Fulkerson Algorithm can be with pathological choices for the augmenting paths. Suppose we start with augmenting path P1 of nodes s, u, v, t in this order (as shown in Figure 7.6). This path has bottleneck(P1, f ) = 1. After this augmentation, we have f (e) = 1 on the edge e = (u, v), so the reverse edge is in the residual graph. For the next augmenting path, we choose the path P2 of the nodes s, v, u, t in this order. In this second augmentation, we get bottleneck(P2 , f ) = 1 as well. After this second augmentation, we have f (e) = 0 for the edge e = (u, v), so the edge is again in the residual graph. Suppose we alternate between choosing P1 and P2 for augmentation. In this case, each augmentation will have 1 as the bottleneck capacity, and it will take 200 augmentations to get the desired flow of value 200. This is exactly the bound we proved in (7.4), since C = 200 in this example.

Designing a Faster Flow Algorithm

The goal of this section is to show that with a better choice of paths, we can improve this bound significantly. A large amount of work has been devoted to finding good ways of choosing augmenting paths in the Maximum-Flow Problem so as to minimize the number of iterations. We focus here on one of the most natural approaches and will mention other approaches at the end of the section. Recall that augmentation increases the value of the maximum flow by the bottleneck capacity of the selected path; so if we choose paths with large bottleneck capacity, we will be making a lot of progress. A natural idea is to select the path that has the largest bottleneck capacity. Having to find such paths can slow down each individual iteration by quite a bit. We will avoid this slowdown by not worrying about selecting the path that has exactly

uu

7.3 Choosing Good Augmenting Paths

353

100 99 100

100

P1 1

s1ts1t P2

100 100

100 1

vv

(a) (b)

uu

99 99 98 99 121

P1 1 s1ts1t

1 99 P2 1

99 1 99 2

vv

(c) (d)

Figure 7.6 Parts (a) through (d) depict four iterations of the Ford-Fulkerson Algorithm using a bad choice of augmenting paths: The augmentations alternate between the path P1 through the nodes s, u, v, t in order and the path P2 through the nodes s, v, u, t in order.

the largest bottleneck capacity. Instead, we will maintain a so-called scaling parameter , and we will look for paths that have bottleneck capacity of at least .

Let Gf () be the subset of the residual graph consisting only of edges with residual capacity of at least . We will work with values of that are powers of 2. The algorithm is as follows.

Scaling Max-Flow

Initially f(e) = 0 for all e in G

Initially set to be the largest power of 2 that is no larger

than the maximum capacity out of s: ≤ maxe out of s ce While ≥1

While there is an s-t path in the graph Gf() Let P be a simple s-t path in Gf()

98

354

Chapter 7

Network Flow

f ′ = augment(f , P)

Update f to be f ′ and update Gf () Endwhile

=/2 Endwhile

Return f

Analyzing the Algorithm

First observe that the new Scaling Max-Flow Algorithm is really just an implementation of the original Ford-Fulkerson Algorithm. The new loops, the value , and the restricted residual graph Gf () are only used to guide the selection of residual path—with the goal of using edges with large residual capacity for as long as possible. Hence all the properties that we proved about the original Max-Flow Algorithm are also true for this new version: the flow remains integer-valued throughout the algorithm, and hence all residual capacities are integer-valued.

(7.15) If the capacities are integer-valued, then throughout the Scaling Max- Flow Algorithm the flow and the residual capacities remain integer-valued. This implies that when = 1, Gf () is the same as Gf , and hence when the algorithm terminates the flow, f is of maximum value.

Next we consider the running time. We call an iteration of the outside

While loop—with a fixed value of —the -scaling phase. It is easy to give

an upper bound on the number of different -scaling phases, in terms of the

value C = c that we also used in the previous section. The initial eoutofs e

value of is at most C, it drops by factors of 2, and it never gets below 1. Thus,

(7.16) The number of iterations of the outer While loop is at most 1+ ⌈log2 C⌉.

The harder part is to bound the number of augmentations done in each scaling phase. The idea here is that we are using paths that augment the flow by a lot, and so there should be relatively few augmentations. During the - scaling phase, we only use edges with residual capacity of at least . Using (7.3), we have

(7.17) During the -scaling phase, each augmentation increases the flow value by at least .

The key insight is that at the end of the -scaling phase, the flow f cannot be too far from the maximum possible value.

(7.18) Let f be the flow at the end of the -scaling phase. There is an s-t cut (A, B) in G for which c(A, B) ≤ ν(f) + m, where m is the number of edges in the graph G. Consequently, the maximum flow in the network has value at most ν(f ) + m.

Proof. This proof is analogous to our proof of (7.9), which established that the flow returned by the original Max-Flow Algorithm is of maximum value.

As in that proof, we must identify a cut (A, B) with the desired property. Let A denote the set of all nodes v in G for which there is an s-v path in Gf (). Let B denote the set of all other nodes: B=V −A. We can see that (A,B) is indeed an s-t cut as otherwise the phase would not have ended.

Now consider an edge e = (u, v) in G for which u ∈ A and v ∈ B. We claim that ce < f (e) + . For if this were not the case, then e would be a forward edge in the graph Gf(), and since u∈A, there is an s-u path in Gf(); appending e to this path, we would obtain an s-v path in Gf (), contradicting our assumption that v ∈ B. Similarly, we claim that for any edge e′ = (u′, v′) in G for which u′ ∈B and v′ ∈A, we have f(e′)<. Indeed, if f(e′)≥, then e′ would give rise to a backward edge e′′ = (v′ , u′) in the graph Gf (), and since v′ ∈ A, there is an s-v′ path in Gf (); appending e′′ to this path, we would obtain an s-u′ path in Gf (), contradicting our assumption that u′ ∈ B.
So all edges e out of A are almost saturated—they satisfy ce < f (e) + — and all edges into A are almost empty—they satisfy f (e) < . We can now use (7.6) to reach the desired conclusion:
ν(f)= f(e)− f(e)
e out of A e into A
≥ (ce − ) −
e out of A e into A
= ce− − e out of A e out of A e into A
≥c(A,B)−m.
Here the first inequality follows from our bounds on the flow values of edges across the cut, and the second inequality follows from the simple fact that the graph only contains m edges total.
The maximum-flow value is bounded by the capacity of any cut by (7.8). We use the cut (A, B) to obtain the bound claimed in the second statement.
7.3 Choosing Good Augmenting Paths
355
356
Chapter 7 Network Flow
(7.19) The number of augmentations in a scaling phase is at most 2m.
Proof. The statement is clearly true in the first scaling phase: we can use each of the edges out of s only for at most one augmentation in that phase. Now consider a later scaling phase , and let fp be the flow at the end of the previous scaling phase. In that phase, we used ′ = 2 as our parameter. By (7.18), the maximum flow f ∗ is at most ν(f ∗) ≤ ν(fp) + m′ = ν(fp) + 2m. In the -scaling phase, each augmentation increases the flow by at least , and hence there can be at most 2m augmentations.
An augmentation takes O(m) time, including the time required to set up the graph and find the appropriate path. We have at most 1 + ⌈log2 C⌉ scaling phases and at most 2m augmentations in each scaling phase. Thus we have the following result.
(7.20) The Scaling Max-Flow Algorithm in a graph with m edges and integer capacities finds a maximum flow in at most 2m(1 + ⌈log2 C⌉) augmentations. It can be implemented to run in at most O(m2 log2 C) time.
When C is large, this time bound is much better than the O(mC) bound that applied to an arbitrary implementation of the Ford-Fulkerson Algorithm. In our example at the beginning of this section, we had capacities of size 100, but we could just as well have used capacities of size 2100; in this case, the generic Ford-Fulkerson Algorithm could take time proportional to 2100, while the scaling algorithm will take time proportional to log2(2100) = 100. One way to view this distinction is as follows: The generic Ford-Fulkerson Algorithm requires time proportional to the magnitude of the capacities, while the scaling algorithm only requires time proportional to the number of bits needed to specify the capacities in the input to the problem. As a result, the scaling algorithm is running in time polynomial in the size of the input (i.e., the number of edges and the numerical representation of the capacities), and so it meets our traditional goal of achieving a polynomial-time algorithm. Bad implementations of the Ford-Fulkerson Algorithm, which can require close to C iterations, do not meet this standard of polynomiality. (Recall that in Section 6.4 we used the term pseudo-polynomial to describe such algorithms, which are polynomial in the magnitudes of the input numbers but not in the number of bits needed to represent them.)
Extensions: Strongly Polynomial Algorithms
Could we ask for something qualitatively better than what the scaling algo- rithm guarantees? Here is one thing we could hope for: Our example graph (Figure 7.6) had four nodes and five edges; so it would be nice to use a
7.4 The Preflow-Push Maximum-Flow Algorithm
357
number of iterations that is polynomial in the numbers 4 and 5, completely independently of the values of the capacities. Such an algorithm, which is polynomial in |V| and |E| only, and works with numbers having a polyno- mial number of bits, is called a strongly polynomial algorithm. In fact, there is a simple and natural implementation of the Ford-Fulkerson Algorithm that leads to such a strongly polynomial bound: each iteration chooses the aug- menting path with the fewest number of edges. Dinitz, and independently Edmonds and Karp, proved that with this choice the algorithm terminates in at most O(mn) iterations. In fact, these were the first polynomial algorithms for the Maximum-Flow Problem. There has since been a huge amount of work devoted to improving the running times of maximum-flow algorithms. There are currently algorithms that achieve running times of O(mn log n), O(n3), and O(min(n2/3, m1/2)m log n log U), where the last bound assumes that all capac- ities are integral and at most U. In the next section, we’ll discuss a strongly polynomial maximum-flow algorithm based on a different principle.
* 7.4 The Preflow-Push Maximum-Flow Algorithm
From the very beginning, our discussion of the Maximum-Flow Problem has been centered around the idea of an augmenting path in the residual graph. However, there are some very powerful techniques for maximum flow that are not explicitly based on augmenting paths. In this section we study one such technique, the Preflow-Push Algorithm.
Designing the Algorithm
Algorithms based on augmenting paths maintain a flow f , and use the augment procedure to increase the value of the flow. By way of contrast, the Preflow- Push Algorithm will, in essence, increase the flow on an edge-by-edge basis. Changing the flow on a single edge will typically violate the conservation con- dition, and so the algorithm will have to maintain something less well behaved than a flow—something that does not obey conservation—as it operates.
Preflows We say that an s-t preflow (preflow, for short) is a function f that maps each edge e to a nonnegative real number, f : E → R+. A preflow f must satisfy the capacity conditions:
(i) Foreache∈E,wehave0≤f(e)≤ce.
In place of the conservation conditions, we require only inequalities: Each
node other than s must have at least as much flow entering as leaving.
(ii) For each node v other than the source s, we have
f(e)≥ f(e). e into v e out of v
358
Chapter 7 Network Flow
We will call the difference
ef(v)= f(e)− f(e) e into v e out of v
the excess of the preflow at node v. Notice that a preflow where all nodes other than s and t have zero excess is a flow, and the value of the flow is exactly ef (t) = −ef (s). We can still define the concept of a residual graph Gf for a preflow f , just as we did for a flow. The algorithm will “push” flow along edges of the residual graph (using both forward and backward edges).
Preflows and Labelings The Preflow-Push Algorithm will maintain a preflow and work on converting the preflow into a flow. The algorithm is based on the physical intuition that flow naturally finds its way “downhill.” The “heights” for this intuition will be labels h(v) for each node v that the algorithm will define and maintain, as shown in Figure 7.7. We will push flow from nodes with higher labels to those with lower labels, following the intuition that fluid flows downhill. To make this precise, a labeling is a function h : V → Z≥0 from the nodes to the nonnegative integers. We will also refer to the labels as heights of the nodes. We will say that a labeling h and an s-t preflow f are compatible if
(i) (Sourceandsinkconditions)h(t)=0andh(s)=n,
(ii) (Steepness conditions) For all edges (v, w) ∈ Ef in the residual graph, we
have h(v) ≤ h(w) + 1. Heights
4
3
2
1
Edges in the residual graph may not be too steep.
0t
Nodes
Figure 7.7 A residual graph and a compatible labeling. No edge in the residual graph can be too “steep”—its tail can be at most one unit above its head in height. The source node s must have h(s) = n and is not drawn in the figure.
7.4 The Preflow-Push Maximum-Flow Algorithm
359
Intuitively, the height difference n between the source and the sink is meant to ensure that the flow starts high enough to flow from s toward the sink t, while the steepness condition will help by making the descent of the flow gradual enough to make it to the sink.
The key property of a compatible preflow and labeling is that there can be no s-t path in the residual graph.
(7.21) If s-t preflow f is compatible with a labeling h, then there is no s-t path in the residual graph Gf .
Proof. We prove the statement by contradiction. Let P be a simple s-t path in the residual graph G. Assume that the nodes along P are s,v1,...,vk =t. By definition of a labeling compatible with preflow f , we have that h(s) = n. The edge (s, v1) is in the residual graph, and hence h(v1) ≥ h(s) − 1 = n − 1. Using induction on i and the steepness condition for the edge (vi−1, vi), we get that for all nodes vi in path P the height is at least h(vi) ≥ n − i. Notice that the last node of the path is vk =t; hence we get that h(t)≥n−k. However, h(t)=0 by definition; and k < n as the path P is simple. This contradiction proves the claim.
Recall from (7.9) that if there is no s-t path in the residual graph Gf of a flow f , then the flow has maximum value. This implies the following corollary.
(7.22) If s-t flow f is compatible with a labeling h, then f is a flow of maximum value.
Note that (7.21) applies to preflows, while (7.22) is more restrictive in that it applies only to flows. Thus the Preflow-Push Algorithm will maintain a preflow f and a labeling h compatible with f , and it will work on modifying f and h so as to move f toward being a flow. Once f actually becomes a flow, we can invoke (7.22) to conclude that it is a maximum flow. In light of this, we can view the Preflow-Push Algorithm as being in a way orthogonal to the Ford- Fulkerson Algorithm. The Ford-Fulkerson Algorithm maintains a feasible flow while changing it gradually toward optimality. The Preflow-Push Algorithm, on the other hand, maintains a condition that would imply the optimality of a preflow f , if it were to be a feasible flow, and the algorithm gradually transforms the preflow f into a flow.
To start the algorithm, we will need to define an initial preflow f and labeling h that are compatible. We will use h(v) = 0 for all v ̸= s, and h(s) = n, as our initial labeling. To make a preflow f compatible with this labeling, we need to make sure that no edges leaving s are in the residual graph (as these edges do not satisfy the steepness condition). To this end, we define the initial
360
Chapter 7 Network Flow
preflow as f(e) = ce for all edges e = (s, v) leaving the source, and f(e) = 0 for all other edges.
(7.23) The initial preflow f and labeling h are compatible.
Pushing and Relabeling Next we will discuss the steps the algorithm makes toward turning the preflow f into a feasible flow, while keeping it compatible with some labeling h. Consider any node v that has excess—that is, ef (v) > 0. If there is any edge e in the residual graph Gf that leaves v and goes to a node w at a lower height (note that h(w) is at most 1 less than h(v) due to the steepness condition), then we can modify f by pushing some of the excess flow from v to w. We will call this a push operation.

push(f , h, v, w)

Applicable if ef(v)>0, h(w)

for all edges (v, w) ∈ Ef Increase h(v) by 1 Return(f , h)

The Full Preflow-Push Algorithm

rithm is as follows.

we have h(w) ≥ h(v)

So, in summary, the Preflow-Push Algo-

Preflow-Push

Initially h(v)=0 for all v̸=s and h(s)=n and

f(e)=ce for all e=(s,v) and f(e)=0 for all other edges While there is a node v̸= t with excess ef(v)>0

Let v be a node with excess

If there is w such that push(f , h, v, w) can be applied then

push(f , h, v, w)

7.4 The Preflow-Push Maximum-Flow Algorithm

361

Else

relabel(f , h, v)

Endwhile

Return(f)

Analyzing the Algorithm

As usual, this algorithm is somewhat underspecified. For an implementation of the algorithm, we will have to specify which node with excess to choose, and how to efficiently select an edge on which to push. However, it is clear that each iteration of this algorithm can be implemented in polynomial time. (We’ll discuss later how to implement it reasonably efficiently.) Further, it is not hard to see that the preflow f and the labeling h are compatible throughout the algorithm. If the algorithm terminates—something that is far from obvious based on its description—then there are no nodes other than t with positive excess, and hence the preflow f is in fact a flow. It then follows from (7.22) that f would be a maximum flow at termination.

We summarize a few simple observations about the algorithm.

(7.24) Throughout the Preflow-Push Algorithm: (i) the labels are nonnegative integers;

(ii) f is a preflow, and if the capacities are integral, then the preflow f is integral; and

(iii) the preflow f and labeling h are compatible.

If the algorithm returns a preflow f , then f is a flow of maximum value.

Proof. By (7.23) the initial preflow f and labeling h are compatible. We will show using induction on the number of push and relabel operations that f and h satisfy the properties of the statement. The push operation modifies the preflow f, but the bounds on δ guarantee that the f returned satisfies the capacity constraints, and that excesses all remain nonnegative, so f is a preflow. To see that the preflow f and the labeling h are compatible, note that push(f , h, v, w) can add one edge to the residual graph, the reverse edge (v, w), and this edge does satisfy the steepness condition. The relabel operation increases the label of v, and hence increases the steepness of all edges leaving v. However, it only applies when no edge leaving v in the residual graph is going downward, and hence the preflow f and the labeling h are compatible after relabeling.

The algorithm terminates if no node other than s or t has excess. In this case, f is a flow by definition; and since the preflow f and the labeling h

362

Chapter 7 Network Flow

remain compatible throughout the algorithm, (7.22) implies that f is a flow of maximum value.

Next we will consider the number of push and relabel operations. First we will prove a limit on the relabel operations, and this will help prove a limit on the maximum number of push operations possible. The algorithm never changes the label of s (as the source never has positive excess). Each other node v starts with h(v) = 0, and its label increases by 1 every time it changes. So we simply need to give a limit on how high a label can get. We only consider a node v for relabel when v has excess. The only source of flow in the network is the source s; hence, intuitively, the excess at v must have originated at s. The following consequence of this fact will be key to bounding the labels.

(7.25) Let f be a preflow. If the node v has excess, then there is a path in Gf from v to the source s.

Proof. Let A denote all the nodes w such that there is a path from w to s in the residual graph Gf , and let B = V − A. We need to show that all nodes with excess are in A.

Notice that s ∈ A. Further, no edges e = (x, y) leaving A can have positive flow, as an edge with f (e) > 0 would give rise to a reverse edge (y, x) in the residual graph, and then y would have been in A. Now consider the sum of excesses in the set B, and recall that each node in B has nonnegative excess, as s ̸∈ B.

in out 0≤ ef(v)= (f (v)−f (v))

v∈B v∈B

Let’s rewrite the sum on the right as follows. If an edge e has both ends in B, then f (e) appears once in the sum with a “+” and once with a “−”, and hence these two terms cancel out. If e has only its head in B, then e leaves A, and we saw above that all edges leaving A have f (e) = 0. If e has only its tail in B, then f(e) appears just once in the sum, with a “−”. So we get

0≤ef(v)=−fout(B). v∈B

Since flows are nonnegative, we see that the sum of the excesses in B is zero; since each individual excess in B is nonnegative, they must therefore all be 0.

Now we are ready to prove that the labels do not change too much. Recall that n denotes the number of nodes in V.

7.4 The Preflow-Push Maximum-Flow Algorithm

363

(7.26) Throughout the algorithm, all nodes have h(v) ≤ 2n − 1.

Proof. The initial labels h(t) = 0 and h(s) = n do not change during the algorithm. Consider some other node v ̸= s, t. The algorithm changes v’s label only when applying the relabel operation, so let f and h be the preflow and labeling returned by a relabel(f , h, v) operation. By (7.25) there is a path P in the residual graph Gf from v to s. Let |P| denote the number of edges in P, and note that |P| ≤ n − 1. The steepness condition implies that heights of the nodes can decrease by at most 1 along each edge in P, and hence h(v) − h(s) ≤ |P|, which proves the statement.

Labels are monotone increasing throughout the algorithm, so this state- ment immediately implies a limit on the number of relabeling operations.

(7.27) Throughout the algorithm, each node is relabeled at most 2n − 1 times, and the total number of relabeling operations is less than 2n2.

Next we will bound the number of push operations. We will distinguish two kinds of push operations. A push(f , h, v, w) operation is saturating if either e=(v,w) is a forward edge in Ef and δ=ce −f(e), or (v,w) is a backward edge with e = (w, v) and δ = f (e). In other words, the push is saturating if, after the push, the edge (v, w) is no longer in the residual graph. All other push operations will be referred to as nonsaturating.

(7.28) Throughout the algorithm, the number of saturating push operations is at most 2nm.

Proof. Consider an edge (v,w) in the residual graph. After a saturating push(f , h, v, w) operation, we have h(v) = h(w) + 1, and the edge (v, w) is no longer in the residual graph Gf , as shown in Figure 7.8. Before we can push again along this edge, first we have to push from w to v to make the edge (v, w) appear in the residual graph. However, in order to push from w to v, we first need for w’s label to increase by at least 2 (so that w is above v). The label of w can increase by 2 at most n − 1 times, so a saturating push from v to w can occur at most n times. Each edge e ∈ E can give rise to two edges in the residual graph, so overall we can have at most 2nm saturating pushes.

The hardest part of the analysis is proving a bound on the number of nonsaturating pushes, and this also will be the bottleneck for the theoretical bound on the running time.

(7.29) Throughout the algorithm, the number of nonsaturating push opera- tions is at most 2n2m.

364

Chapter 7

Network Flow

Heights

4

3

2

1

0

The height of node w has to increase by 2 before it can push flow back to node v.

v

w

Figure 7.8 After a saturating push(f , h, v, w), the height of v exceeds the height of w by 1.

Proof. For this proof, we will use a so-called potential function method. For a preflow f and a compatible labeling h, we define

v:ef (v)>0

to be the sum of the heights of all nodes with positive excess. ( is often called a potential since it resembles the “potential energy” of all nodes with positive excess.)

In the initial preflow and labeling, all nodes with positive excess are at height 0, so (f , h) = 0. (f , h) remains nonnegative throughout the algo- rithm. A nonsaturating push(f , h, v, w) operation decreases (f , h) by at least 1, since after the push the node v will have no excess, and w, the only node that gets new excess from the operation, is at a height 1 less than v. How- ever, each saturating push and each relabel operation can increase (f , h). A relabel operation increases (f , h) by exactly 1. There are at most 2n2 relabel operations, so the total increase in (f , h) due to relabel opera- tions is 2n2. A saturating push(f , h, v, w) operation does not change labels, but it can increase (f , h), since the node w may suddenly acquire positive excess after the push. This would increase (f , h) by the height of w, which is at most 2n − 1. There are at most 2nm saturating push operations, so the total increase in (f , h) due to push operations is at most 2mn(2n − 1). So, between the two causes, (f , h) can increase by at most 4mn2 during the algorithm.

(f , h) =

h(v)

t

Nodes

7.4 The Preflow-Push Maximum-Flow Algorithm

365

But since remains nonnegative throughout, and it decreases by at least 1 on each nonsaturating push operation, it follows that there can be at most 4mn2 nonsaturating push operations.

Extensions: An Improved Version of the Algorithm

There has been a lot of work devoted to choosing node selection rules for the Preflow-Push Algorithm to improve the worst-case running time. Here we consider a simple rule that leads to an improved O(n3) bound on the number of nonsaturating push operations.

(7.30) If at each step we choose the node with excess at maximum height, then the number of nonsaturating push operations throughout the algorithm is at most 4n3.

Proof. Consider the maximum height H = maxv:ef (v)>0 h(v) of any node with excess as the algorithm proceeds. The analysis will use this maximum height H in place of the potential function in the previous O(n2m) bound.

This maximum height H can only increase due to relabeling (as flow is always pushed to nodes at lower height), and so the total increase in H throughout the algorithm is at most 2n2 by (7.26). H starts out 0 and remains nonnegative, so the number of times H changes is at most 4n2.

Now consider the behavior of the algorithm over a phase of time in which H remains constant. We claim that each node can have at most one nonsaturating push operation during this phase. Indeed, during this phase, flow is being pushed from nodes at height H to nodes at height H − 1; and after a nonsaturating push operation from v, it must receive flow from a node at height H + 1 before we can push from it again.

Since there are at most n nonsaturating push operations between each change to H, and H changes at most 4n2 times, the total number of nonsatu- rating push operations is at most 4n3.

As a follow-up to (7.30), it is interesting to note that experimentally the computational bottleneck of the method is the number of relabeling operations, and a better experimental running time is obtained by variants that work on increasing labels faster than one by one. This is a point that we pursue further in some of the exercises.

Implementing the Preflow-Push Algorithm

Finally, we need to briefly discuss how to implement this algorithm efficiently. Maintaining a few simple data structures will allow us to effectively implement

366

Chapter 7 Network Flow

the operations of the algorithm in constant time each, and overall to imple- ment the algorithm in time O(mn) plus the number of nonsaturating push operations. Hence the generic algorithm will run in O(mn2) time, while the version that always selects the node at maximum height will run in O(n3) time.

We can maintain all nodes with excess on a simple list, and so we will be able to select a node with excess in constant time. One has to be a bit more careful to be able to select a node with maximum height H in constant time. In order to do this, we will maintain a linked list of all nodes with excess at every possible height. Note that whenever a node v gets relabeled, or continues to have positive excess after a push, it remains a node with maximum height H. Thus we only have to select a new node after a push when the current node v no longer has positive excess. If node v was at height H, then the new node at maximum height will also be at height H or, if no node at height H has excess, then the maximum height will be H − 1, since the previous push operation out of v pushed flow to a node at height H − 1.

Now assume we have selected a node v, and we need to select an edge (v, w) on which to apply push(f , h, v, w) (or relabel(f , h, v) if no such w exists). To be able to select an edge quickly, we will use the adjacency list representation of the graph. More precisely, we will maintain, for each node v, all possible edges leaving v in the residual graph (both forward and backward edges) in a linked list, and with each edge we keep its capacity and flow value. Note that this way we have two copies of each edge in our data structure: a forward and a backward copy. These two copies will have pointers to each other, so that updates done at one copy can be carried over to the other one in O(1) time. We will select edges leaving a node v for push operations in the order they appear on node v’s list. To facilitate this selection, we will maintain a pointer current(v) for each node v to the last edge on the list that has been considered for a push operation. So, if node v no longer has excess after a nonsaturating push operation out of node v, the pointer current(v) will stay at this edge, and we will use the same edge for the next push operation out of v. After a saturating push operation out of node v, we advance current(v) to the next edge on the list.

The key observation is that, after advancing the pointer current(v) from an edge (v, w), we will not want to apply push to this edge again until we relabel v.

(7.31) After the current(v) pointer is advanced from an edge (v, w), we cannot apply push to this edge until v gets relabeled.

Proof. At the moment current(v) is advanced from the edge (v, w), there is some reason push cannot be applied to this edge. Either h(w) ≥ h(v), or the

7.5 A First Application: The Bipartite Matching Problem

367

edge is not in the residual graph. In the first case, we clearly need to relabel v before applying a push on this edge. In the latter case, one needs to apply push to the reverse edge (w, v) to make (v, w) reenter the residual graph. However, when we apply push to edge (w, v), then w is above v, and so v needs to be relabeled before one can push flow from v to w again.

Since edges do not have to be considered again for push before relabeling, we get the following.

(7.32) When the current(v) pointer reaches the end of the edge list for v, the relabel operation can be applied to node v.

After relabeling node v, we reset current(v) to the first edge on the list and start considering edges again in the order they appear on v’s list.

(7.33) The running time of the Preflow-Push Algorithm, implemented using the above data structures, is O(mn) plus O(1) for each nonsaturating push operation. In particular, the generic Preflow-Push Algorithm runs in O(n2m) time, while the version where we always select the node at maximum height runs in O(n3) time.

Proof. The initial flow and relabeling is set up in O(m) time. Both push and

relabel operations can be implemented in O(1) time, once the operation

has been selected. Consider a node v. We know that v can be relabeled at

most 2n times throughout the algorithm. We will consider the total time the

algorithm spends on finding the right edge on which to push flow out of node v,

between two times that node v gets relabeled. If node v has dv adjacent edges,

then by (7.32) we spend O(dv) time on advancing the current(v) pointer

between consecutive relabelings of v. Thus the total time spent on advancing

the current pointers throughout the algorithm is O( nd ) = O(mn), as v∈V v

claimed.

7.5 A First Application: The Bipartite Matching Problem

Having developed a set of powerful algorithms for the Maximum-Flow Prob- lem, we now turn to the task of developing applications of maximum flows and minimum cuts in graphs. We begin with two very basic applications. First, in this section, we discuss the Bipartite Matching Problem mentioned at the beginning of this chapter. In the next section, we discuss the more general Disjoint Paths Problem.

368

Chapter 7 Network Flow

The Problem

One of our original goals in developing the Maximum-Flow Problem was to be able to solve the Bipartite Matching Problem, and we now show how to do this. Recall that a bipartite graph G = (V , E) is an undirected graph whose node set can be partitioned as V = X ∪ Y, with the property that every edge e ∈ E has one end in X and the other end in Y. A matching M in G is a subset of the edges M ⊆ E such that each node appears in at most one edge in M. The Bipartite Matching Problem is that of finding a matching in G of largest possible size.

Designing the Algorithm

The graph defining a matching problem is undirected, while flow networks are directed; but it is actually not difficult to use an algorithm for the Maximum- Flow Problem to find a maximum matching.

Beginning with the graph G in an instance of the Bipartite Matching Problem, we construct a flow network G′ as shown in Figure 7.9. First we direct all edges in G from X to Y. We then add a node s, and an edge (s,x) from s to each node in X. We add a node t, and an edge (y,t) from each node in Y to t. Finally, we give each edge in G′ a capacity of 1.

We now compute a maximum s-t flow in this network G′. We will discover that the value of this maximum is equal to the size of the maximum matching in G. Moreover, our analysis will show how one can use the flow itself to recover the matching.

st

(a) (b)

Figure 7.9 (a) A bipartite graph. (b) The corresponding flow network, with all capacities equal to 1.

7.5 A First Application: The Bipartite Matching Problem

369

Analyzing the Algorithm

The analysis is based on showing that integer-valued flows in G′ encode matchings in G in a fairly transparent fashion. First, suppose there is a matching in G consisting of k edges (xi1, yi1), . . . , (xik , yik). Then consider the flow f that sends one unit along each path of the form s, xij , yij , t—that is, f (e) = 1 for each edge on one of these paths. One can verify easily that the capacity and conservation conditions are indeed met and that f is an s-t flow of value k.

Conversely, suppose there is a flow f′ in G′ of value k. By the integrality theorem for maximum flows (7.14), we know there is an integer-valued flow f of value k; and since all capacities are 1, this means that f (e) is equal to either 0 or 1 for each edge e. Now, consider the set M′ of edges of the form (x, y) on which the flow value is 1.

Here are three simple facts about the set M′.

(7.34) M′ contains k edges.

Proof. To prove this, consider the cut (A, B) in G′ with A = {s} ∪ X. The value of the flow is the total flow leaving A, minus the total flow entering A. The first of these terms is simply the cardinality of M′, since these are the edges leaving A that carry flow, and each carries exactly one unit of flow. The second of these terms is 0, since there are no edges entering A. Thus, M′ contains k edges.

(7.35) Each node in X is the tail of at most one edge in M′.

Proof. To prove this, suppose x ∈ X were the tail of at least two edges in M′. Since our flow is integer-valued, this means that at least two units of flow leave from x. By conservation of flow, at least two units of flow would have to come into x—but this is not possible, since only a single edge of capacity 1 enters x. Thus x is the tail of at most one edge in M′.

By the same reasoning, we can show

(7.36) Each node in Y is the head of at most one edge in M′.

Combining these facts, we see that if we view M′ as a set of edges in the original bipartite graph G, we get a matching of size k. In summary, we have proved the following fact.

(7.37) The size of the maximum matching in G is equal to the value of the maximum flow in G′; and the edges in such a matching in G are the edges that carry flow from X to Y in G′.

370

Chapter 7 Network Flow

Note the crucial way in which the integrality theorem (7.14) figured in this construction: we needed to know if there is a maximum flow in G′ that takes only the values 0 and 1.

Bounding the Running Time Now let’s consider how quickly we can com-

pute a maximum matching in G. Let n = |X| = |Y|, and let m be the number

of edges of G. We’ll tacitly assume that there is at least one edge incident to

each node in the original problem, and hence m ≥ n/2. The time to compute

a maximum matching is dominated by the time to compute an integer-valued

maximum flow in G′, since converting this to a matching in G is simple. For

this flow problem, we have that C = c =|X|=n, as s has an edge eoutofs e

of capacity 1 to each node of X. Thus, by using the O(mC) bound in (7.5), we get the following.

(7.38) The Ford-Fulkerson Algorithm can be used to find a maximum match- ing in a bipartite graph in O(mn) time.

It’s interesting that if we were to use the “better” bounds of O(m2 log2 C) or O(n3) that we developed in the previous sections, we’d get the inferior running times of O(m2 log n) or O(n3) for this problem. There is nothing contradictory in this. These bounds were designed to be good for all instances, even when C is very large relative to m and n. But C = n for the Bipartite Matching Problem, and so the cost of this extra sophistication is not needed.

It is worthwhile to consider what the augmenting paths mean in the network G′. Consider the matching M consisting of edges (x2, y2), (x3, y3), and (x5, y5) in the bipartite graph in Figure 7.1; see also Figure 7.10. Let f be the corresponding flow in G′. This matching is not maximum, so f is not a maximum s-t flow, and hence there is an augmenting path in the residual graph Gf′ . One such augmenting path is marked in Figure 7.10(b). Note that the edges (x2, y2) and (x3, y3) are used backward, and all other edges are used forward. All augmenting paths must alternate between edges used backward and forward, as all edges of the graph G′ go from X to Y. Augmenting paths are therefore also called alternating paths in the context of finding a maximum matching. The effect of this augmentation is to take the edges used backward out of the matching, and replace them with the edges going forward. Because the augmenting path goes from s to t, there is one more forward edge than backward edge; thus the size of the matching increases by one.

7.5 A First Application: The Bipartite Matching Problem

371

x1 x2 x3 x4 x5

y1 x1 y2 x2 y3 x3 y4 x4 y5 x5

y1 x1 y2 x2 y3 x3 y4 x4 y5 x5

y1 y2 y3 y4 y5

(a)

(b)

(c)

Figure 7.10 (a) A bipartite graph, with a matching M. (b) The augmenting path in the corresponding residual graph. (c) The matching obtained by the augmentation.

Extensions: The Structure of Bipartite Graphs with No Perfect Matching

Algorithmically, we’ve seen how to find perfect matchings: We use the algo- rithm above to find a maximum matching and then check to see if this matching is perfect.

But let’s ask a slightly less algorithmic question. Not all bipartite graphs have perfect matchings. What does a bipartite graph without a perfect match- ing look like? Is there an easy way to see that a bipartite graph does not have a perfect matching—or at least an easy way to convince someone the graph has no perfect matching, after we run the algorithm? More concretely, it would be nice if the algorithm, upon concluding that there is no perfect matching, could produce a short “certificate” of this fact. The certificate could allow someone to be quickly convinced that there is no perfect matching, without having to look over a trace of the entire execution of the algorithm.

One way to understand the idea of such a certificate is as follows. We can decide if the graph G has a perfect matching by checking if the maximum flow in a related graph G′ has value at least n. By the Max-Flow Min-Cut Theorem, there will be an s-t cut of capacity less than n if the maximum-flow value in G′ has value less than n. So, in a way, a cut with capacity less than n provides such a certificate. However, we want a certificate that has a natural meaning in terms of the original graph G.

What might such a certificate look like? For example, if there are nodes x1, x2 ∈ X that have only one incident edge each, and the other end of each edge is the same node y, then clearly the graph has no perfect matching: both x1 and x2 would need to get matched to the same node y. More generally, consider a subset of nodes A ⊆ X, and let (A) ⊆ Y denote the set of all nodes

372

Chapter 7 Network Flow

that are adjacent to nodes in A. If the graph has a perfect matching, then each node in A has to be matched to a different node in (A), so (A) has to be at least as large as A. This gives us the following fact.

(7.39) If a bipartite graph G = (V , E) with two sides X and Y has a perfect matching, then for all A ⊆ X we must have |(A)| ≥ |A|.

This statement suggests a type of certificate demonstrating that a graph does not have a perfect matching: a set A ⊆ X such that |(A)| < |A|. But is the converse of (7.39) also true? Is it the case that whenever there is no perfect matching, there is a set A like this that proves it? The answer turns out to be yes, provided we add the obvious condition that |X| = |Y| (without which there could certainly not be a perfect matching). This statement is known in the literature as Hall’s Theorem, though versions of it were discovered independently by a number of different people—perhaps first by Ko ̈nig—in the early 1900s. The proof of the statement also provides a way to find such a subset A in polynomial time.
(7.40) Assume that the bipartite graph G = (V , E) has two sides X and Y such that |X| = |Y|. Then the graph G either has a perfect matching or there is a subset A ⊆ X such that |(A)| < |A|. A perfect matching or an appropriate subset A can be found in O(mn) time.
Proof. We will use the same graph G′ as in (7.37). Assume that |X| = |Y| = n. By (7.37) the graph G has a maximum matching if and only if the value of the maximum flow in G′ is n.
We need to show that if the value of the maximum flow is less than n, then there is a subset A such that |(A)| < |A|, as claimed in the statement. By the Max-Flow Min-Cut Theorem (7.12), if the maximum-flow value is less than n, then there is a cut (A′, B′) with capacity less than n in G′. Now the set A′ contains s, and may contain nodes from both X and Y as shown in Figure 7.11. We claim that the set A = X ∩ A′ has the claimed property. This will prove both parts of the statement, as we’ve seen in (7.11) that a minimum cut (A′, B′) can also be found by running the Ford-Fulkerson Algorithm.
First we claim that one can modify the minimum cut (A′,B′) so as to ensure that (A) ⊆ A′, where A = X ∩ A′ as before. To do this, consider a node y ∈ (A) that belongs to B′ as shown in Figure 7.11(a). We claim that by moving y from B′ to A′, we do not increase the capacity of the cut. For what happens when we move y from B′ to A′? The edge (y, t) now crosses the cut, increasing the capacity by one. But previously there was at least one edge (x, y) with x ∈ A, since y ∈ (A); all edges from A and y used to cross the cut, and don’t anymore. Thus, overall, the capacity of the cut cannot increase. (Note that we
7.6 Disjoint Paths in Directed and Undirected Graphs
373
Node y can be moved to the s-side of the cut.
A A
A
sxytsxyt
(a) (b)
Figure 7.11 (a) A minimum cut in proof of (7.40). (b) The same cut after moving node y to the A′ side. The edges crossing the cut are dark.
don’t have to be concerned about nodes x ∈ X that are not in A. The two ends of the edge (x, y) will be on different sides of the cut, but this edge does not add to the capacity of the cut, as it goes from B′ to A′.)
Next consider the capacity of this minimum cut (A′, B′) that has (A) ⊆ A′ as shown in Figure 7.11(b). Since all neighbors of A belong to A′, we see that the only edges out of A′ are either edges that leave the source s or that enter the sink t. Thus the capacity of the cut is exactly
c(A′,B′)=|X ∩B′|+|Y ∩A′|.
Notice that |X ∩ B′| = n − |A|, and |Y ∩ A′| ≥ |(A)|. Now the assumption that
c(A′, B′) < n implies that
n − |A| + |(A)| ≤ |X ∩ B′| + |Y ∩ A′| = c(A′, B′) < n.
Comparing the first and the last terms, we get the claimed inequality |A| > |(A)|.

7.6 Disjoint Paths in Directed and Undirected Graphs

In Section 7.1, we described a flow f as a kind of “traffic” in the network. But our actual definition of a flow has a much more static feel to it: For each edge e, we simply specify a number f (e) saying the amount of flow crossing e. Let’s see if we can revive the more dynamic, traffic-oriented picture a bit, and try formalizing the sense in which units of flow “travel” from the source to

374

Chapter 7 Network Flow

the sink. From this more dynamic view of flows, we will arrive at something called the s-t Disjoint Paths Problem.

The Problem

In defining this problem precisely, we will deal with two issues. First, we will make precise this intuitive correspondence between units of flow traveling along paths, and the notion of flow we’ve studied so far. Second, we will extend the Disjoint Paths Problem to undirected graphs. We’ll see that, despite the fact that the Maximum-Flow Problem was defined for a directed graph, it can naturally be used also to handle related problems on undirected graphs.

We say that a set of paths is edge-disjoint if their edge sets are disjoint, that is, no two paths share an edge, though multiple paths may go through some of the same nodes. Given a directed graph G = (V , E) with two distinguished nodes s, t ∈ V, the Directed Edge-Disjoint Paths Problem is to find the maximum number of edge-disjoint s-t paths in G. The Undirected Edge-Disjoint Paths Problem is to find the maximum number of edge-disjoint s-t paths in an undirected graph G. The related question of finding paths that are not only edge-disjoint, but also node-disjoint (of course, other than at nodes s and t) will be considered in the exercises to this chapter.

Designing the Algorithm

Both the directed and the undirected versions of the problem can be solved very naturally using flows. Let’s start with the directed problem. Given the graph G = (V , E), with its two distinguished nodes s and t, we define a flow network in which s and t are the source and sink, respectively, and with a capacity of 1 on each edge. Now suppose there are k edge-disjoint s-t paths. We can make each of these paths carry one unit of flow: We set the flow to be f(e) = 1 for each edge e on any of the paths, and f(e′) = 0 on all other edges, and this defines a feasible flow of value k.

(7.41) If there are k edge-disjoint paths in a directed graph G from s to t, then the value of the maximum s-t flow in G is at least k.

Suppose we could show the converse to (7.41) as well: If there is a flow of value k, then there exist k edge-disjoint s-t paths. Then we could simply compute a maximum s-t flow in G and declare (correctly) this to be the maximum number of edge-disjoint s-t paths.

We now proceed to prove this converse statement, confirming that this approach using flow indeed gives us the correct answer. Our analysis will also provide a way to extract k edge-disjoint paths from an integer-valued flow sending k units from s to t. Thus computing a maximum flow in G will

7.6 Disjoint Paths in Directed and Undirected Graphs

375

not only give us the maximum number of edge-disjoint paths, but the paths as well.

Analyzing the Algorithm

Proving the converse direction of (7.41) is the heart of the analysis, since it will immediately establish the optimality of the flow-based algorithm to find disjoint paths.

To prove this, we will consider a flow of value at least k, and construct k edge-disjoint paths. By (7.14), we know that there is a maximum flow f with integer flow values. Since all edges have a capacity bound of 1, and the flow is integer-valued, each edge that carries flow under f has exactly one unit of flow on it. Thus we just need to show the following.

(7.42) If f is a 0-1 valued flow of value ν, then the set of edges with flow value f (e) = 1 contains a set of ν edge-disjoint paths.

Proof. We prove this by induction on the number of edges in f that carry flow. If ν = 0, there is nothing to prove. Otherwise, there must be an edge (s, u) that carries one unit of flow. We now “trace out” a path of edges that must also carry flow: Since (s, u) carries a unit of flow, it follows by conservation that there is some edge (u, v) that carries one unit of flow, and then there must be an edge (v, w) that carries one unit of flow, and so forth. If we continue in this way, one of two things will eventually happen: Either we will reach t, or we will reach a node v for the second time.

If the first case happens—we find a path P from s to t—then we’ll use this path as one of our ν paths. Let f ′ be the flow obtained by decreasing the flow values on the edges along P to 0. This new flow f′ has value ν − 1, and it has fewer edges that carry flow. Applying the induction hypothesis for f′, we get ν − 1 edge-disjoint paths, which, along with path P, form the ν paths claimed.

If P reaches a node v for the second time, then we have a situation like the one pictured in Figure 7.12. (The edges in the figure all carry one unit of flow, and the dashed edges indicate the path traversed so far, which has just reached a node v for the second time.) In this case, we can make progress in a different way.

Consider the cycle C of edges visited between the first and second appear- ances of v. We obtain a new flow f′ from f by decreasing the flow values on the edges along C to 0. This new flow f ′ has value ν, but it has fewer edges that carry flow. Applying the induction hypothesis for f ′, we get the ν edge-disjoint paths as claimed.

376

Chapter 7

Network Flow

Flow around a cycle can be zeroed out.

v

P s

t

Figure 7.12 The edges in the figure all carry one unit of flow. The path P of dashed edges is one possible path in the proof of (7.42).

We can summarize (7.41) and (7.42) in the following result.

(7.43) There are k edge-disjoint paths in a directed graph G from s to t if and only if the value of the maximum value of an s-t flow in G is at least k.

Notice also how the proof of (7.42) provides an actual procedure for constructing the k paths, given an integer-valued maximum flow in G. This procedure is sometimes referred to as a path decomposition of the flow, since it “decomposes” the flow into a constituent set of paths. Hence we have shown that our flow-based algorithm finds the maximum number of edge-disjoint s-t paths and also gives us a way to construct the actual paths.

Bounding the Running Time For this flow problem, C = c ≤ eoutofs e

|V| = n, as there are at most |V| edges out of s, each of which has capac- ity 1. Thus, by using the O(mC) bound in (7.5), we get an integer maximum flow in O(mn) time.

The path decomposition procedure in the proof of (7.42), which produces the paths themselves, can also be made to run in O(mn) time. To see this, note that this procedure, with a little care, can produce a single path from s to t using at most constant work per edge in the graph, and hence in O(m) time. Since there can be at most n − 1 edge-disjoint paths from s to t (each must use a different edge out of s), it therefore takes time O(mn) to produce all the paths.

In summary, we have shown

(7.44) The Ford-Fulkerson Algorithm can be used to find a maximum set of edge-disjoint s-t paths in a directed graph G in O(mn) time.

A Version of the Max-Flow Min-Cut Theorem for Disjoint Paths The Max- Flow Min-Cut Theorem (7.13) can be used to give the following characteri-

7.6 Disjoint Paths in Directed and Undirected Graphs

377

zation of the maximum number of edge-disjoint s-t paths. We say that a set F ⊆ E of edges separates s from t if, after removing the edges F from the graph G, no s-t paths remain in the graph.

(7.45) In every directed graph with nodes s and t, the maximum number of edge-disjoint s-t paths is equal to the minimum number of edges whose removal separates s from t.

Proof. If the removal of a set F ⊆ E of edges separates s from t, then each s-t path must use at least one edge from F, and hence the number of edge-disjoint s-t paths is at most |F|.

To prove the other direction, we will use the Max-Flow Min-Cut Theorem (7.13). By (7.43) the maximum number of edge-disjoint paths is the value ν of the maximum s-t flow. Now (7.13) states that there is an s-t cut (A, B) with capacity ν. Let F be the set of edges that go from A to B. Each edge has capacity 1, so |F | = ν and, by the definition of an s-t cut, removing these ν edges from G separates s from t.

This result, then, can be viewed as the natural special case of the Max- Flow Min-Cut Theorem in which all edge capacities are equal to 1. In fact, this special case was proved by Menger in 1927, much before the full Max- Flow Min-Cut Theorem was formulated and proved; for this reason, (7.45) is often called Menger’s Theorem. If we think about it, the proof of Hall’s Theorem (7.40) for bipartite matchings involves a reduction to a graph with unit-capacity edges, and so it can be proved using Menger’s Theorem rather than the general Max-Flow Min-Cut Theorem. In other words, Hall’s Theorem is really a special case of Menger’s Theorem, which in turn is a special case of the Max-Flow Min-Cut Theorem. And the history follows this progression, since they were discovered in this order, a few decades apart.2

Extensions: Disjoint Paths in Undirected Graphs

Finally, we consider the disjoint paths problem in an undirected graph G. Despite the fact that our graph G is now undirected, we can use the maximum- flow algorithm to obtain edge-disjoint paths in G. The idea is quite simple: We replace each undirected edge (u, v) in G by two directed edges (u, v) and

2 In fact, in an interesting retrospective written in 1981, Menger relates his version of the story of how he first explained his theorem to K ̈onig, one of the independent discoverers of Hall’s Theorem. You might think that K ̈onig, having thought a lot about these problems, would have immediately grasped why Menger’s generalization of his theorem was true, and perhaps even considered it obvious. But, in fact, the opposite happened; K ̈onig didn’t believe it could be right and stayed up all night searching for a counterexample. The next day, exhausted, he sought out Menger and asked him for the proof.

378

Chapter 7 Network Flow

(v, u), and in this way create a directed version G′ of G. (We may delete the edges into s and out of t, since they are not useful.) Now we want to use the Ford-Fulkerson Algorithm in the resulting directed graph. However, there is an important issue we need to deal with first. Notice that two paths P1 and P2 may be edge-disjoint in the directed graph and yet share an edge in the undirected graph G: This happens if P1 uses directed edge (u, v) while P2 uses edge (v, u). However, it is not hard to see that there always exists a maximum flow in any network that uses at most one out of each pair of oppositely directed edges.

(7.46) In any flow network, there is a maximum flow f where for all opposite directed edges e=(u,v) and e′=(v,u), either f(e)=0 or f(e′)=0. If the capacities of the flow network are integral, then there also is such an integral maximum flow.

Proof. We consider any maximum flow f, and we modify it to satisfy the claimed condition. Assume e = (u, v) and e′ = (v, u) are opposite directed edges, and f (e) ̸= 0, f (e′) ̸= 0. Let δ be the smaller of these values, and modify f by decreasing the flow value on both e and e′ by δ. The resulting flow f′ is feasible, has the same value as f, and its value on one of e and e′ is 0.

Now we can use the Ford-Fulkerson Algorithm and the path decomposition procedure from (7.42) to obtain edge-disjoint paths in the undirected graph G.

(7.47) There are k edge-disjoint paths in an undirected graph G from s to t if and only if the maximum value of an s-t flow in the directed version G′ of G is at least k. Furthermore, the Ford-Fulkerson Algorithm can be used to find a maximum set of disjoint s-t paths in an undirected graph G in O(mn) time.

The undirected analogue of (7.45) is also true, as in any s-t cut, at most one of the two oppositely directed edges can cross from the s-side to the t- side of the cut (for if one crosses, then the other must go from the t-side to the s-side).

(7.48) In every undirected graph with nodes s and t, the maximum number of edge-disjoint s-t paths is equal to the minimum number of edges whose removal separates s from t.

7.7 Extensions to the Maximum-Flow Problem

Much of the power of the Maximum-Flow Problem has essentially nothing to do with the fact that it models traffic in a network. Rather, it lies in the fact that many problems with a nontrivial combinatorial search component can

7.7 Extensions to the Maximum-Flow Problem

379

be solved in polynomial time because they can be reduced to the problem of finding a maximum flow or a minimum cut in a directed graph.

Bipartite Matching is a natural first application in this vein; in the coming sections, we investigate a range of further applications. To begin with, we stay with the picture of flow as an abstract kind of “traffic,” and look for more general conditions we might impose on this traffic. These more general conditions will turn out to be useful for some of our further applications.

In particular, we focus on two generalizations of maximum flow. We will see that both can be reduced to the basic Maximum-Flow Problem.

The Problem: Circulations with Demands

One simplifying aspect of our initial formulation of the Maximum-Flow Prob- lem is that we had only a single source s and a single sink t. Now suppose that there can be a set S of sources generating flow, and a set T of sinks that can absorb flow. As before, there is an integer capacity on each edge.

With multiple sources and sinks, it is a bit unclear how to decide which source or sink to favor in a maximization problem. So instead of maximizing the flow value, we will consider a problem where sources have fixed supply values and sinks have fixed demand values, and our goal is to ship flow from nodes with available supply to those with given demands. Imagine, for example, that the network represents a system of highways or railway lines in which we want to ship products from factories (which have supply) to retail outlets (which have demand). In this type of problem, we will not be seeking to maximize a particular value; rather, we simply want to satisfy all the demand using the available supply.

Thus we are given a flow network G = (V , E) with capacities on the edges. Now, associated with each node v ∈ V is a demand dv. If dv > 0, this indicates that the node v has a demand of dv for flow; the node is a sink, and it wishes to receive dv units more flow than it sends out. If dv < 0, this indicates that v has a supply of −dv; the node is a source, and it wishes to send out −dv units more flow than it receives. If dv = 0, then the node v is neither a source nor a sink. We will assume that all capacities and demands are integers.
We use S to denote the set of all nodes with negative demand and T to denote the set of all nodes with positive demand. Although a node v in S wants to send out more flow than it receives, it will be okay for it to have flow that enters on incoming edges; it should just be more than compensated by the flow that leaves v on outgoing edges. The same applies (in the opposite direction) to the set T.
380
Chapter 7 Network Flow
3
–3 s* –3 33333
22
–3 2 –3 2
22 22
2
3
1
2
1
2
3
2
2
2
2
2
2
2
4 4t* 4
(a) (b)
Figure7.13 (a)AninstanceoftheCirculationProblemtogetherwithasolution:Numbers inside the nodes are demands; numbers labeling the edges are capacities and flow values, with the flow values inside boxes. (b) The result of reducing this instance to an equivalent instance of the Maximum-Flow Problem.
In this setting, we say that a circulation with demands {dv} is a function f that assigns a nonnegative real number to each edge and satisfies the following two conditions.
(i) (Capacityconditions)Foreache∈E,wehave0≤f(e)≤ce.
(ii) (Demandconditions)Foreachv∈V,wehavev,fin(v)−fout(v)=dv.
Now, instead of considering a maximization problem, we are concerned with a feasibility problem: We want to know whether there exists a circulation that meets conditions (i) and (ii).
For example, consider the instance in Figure 7.13(a). Two of the nodes are sources, with demands −3 and −3; and two of the nodes are sinks, with demands 2 and 4. The flow values in the figure constitute a feasible circulation, indicating how all demands can be satisfied while respecting the capacities.
If we consider an arbitrary instance of the Circulation Problem, here is a simple condition that must hold in order for a feasible circulation to exist: The total supply must equal the total demand.
(7.49) If there exists a feasible circulation with demands {d }, then d = 0. vvv
4
Proof. Suppose there exists a feasible circulation f in this setting. Then d = fin(v)−fout(v).Now,inthislatterexpression,thevaluef(e)for
vvv
each edge e = (u, v) is counted exactly twice: once in f out(u) and once in f in(v).
These two terms cancel out; and since this holds for all values f (e), the overall sum is 0.
7.7 Extensions to the Maximum-Flow Problem
381
s* supplies sources with flow.
s*
S
T
uv
t* siphons flow out of sinks.
t*
Figure 7.14 Reducing the Circulation Problem to the Maximum-Flow Problem.
Thanks to (7.49), we know that
v:dv>0 Let D denote this common value.

dv = −dv. v:dv<0
Designing and Analyzing an Algorithm for Circulations
It turns out that we can reduce the problem of finding a feasible circulation with demands {dv} to the problem of finding a maximum s-t flow in a different network, as shown in Figure 7.14.
The reduction looks very much like the one we used for Bipartite Matching: we attach a “super-source” s∗ to each node in S, and a “super-sink” t∗ to each node in T. More specifically, we create a graph G′ from G by adding new nodes s∗ and t∗ to G. For each node v ∈ T—that is, each node v with dv > 0—we add an edge (v, t∗) with capacity dv. For each node u ∈ S—that is, each node with du < 0—we add an edge (s∗, u) with capacity −du. We carry the remaining structure of G over to G′ unchanged.
In this graph G′, we will be seeking a maximum s∗-t∗ flow. Intuitively, we can think of this reduction as introducing a node s∗ that “supplies” all the sources with their extra flow, and a node t∗ that “siphons” the extra flow out of the sinks. For example, part (b) of Figure 7.13 shows the result of applying this reduction to the instance in part (a).
Note that there cannot be an s∗-t∗ flow in G′ of value greater than D, since the cut (A, B) with A = {s∗} only has capacity D. Now, if there is a feasible circulation f with demands {dv} in G, then by sending a flow value of −dv on each edge (s∗, v), and a flow value of dv on each edge (v, t∗), we obtain an s∗- t∗ flow in G′ of value D, and so this is a maximum flow. Conversely, suppose there is a (maximum) s∗-t∗ flow in G′ of value D. It must be that every edge
382
Chapter 7 Network Flow
out of s∗, and every edge into t∗, is completely saturated with flow. Thus, if we delete these edges, we obtain a circulation f in G with f in(v) − f out(v) = dv for each node v. Further, if there is a flow of value D in G′, then there is such a flow that takes integer values.
In summary, we have proved the following.
(7.50) There is a feasible circulation with demands {dv} in G if and only if the maximum s∗-t∗ flow in G′ has value D. If all capacities and demands in G are integers, and there is a feasible circulation, then there is a feasible circulation that is integer-valued.
At the end of Section 7.5, we used the Max-Flow Min-Cut Theorem to derive the characterization (7.40) of bipartite graphs that do not have perfect matchings. We can give an analogous characterization for graphs that do not have a feasible circulation. The characterization uses the notion of a cut, adapted to the present setting. In the context of circulation problems with demands, a cut (A, B) is any partition of the node set V into two sets, with no restriction on which side of the partition the sources and sinks fall. We include the characterization here without a proof.
(7.51) The graph G has a feasible circulation with demands {dv} if and only
if for all cuts (A, B),
dv ≤ c(A, B). v∈B
It is important to note that our network has only a single “kind” of flow. Although the flow is supplied from multiple sources, and absorbed at multiple sinks, we cannot place restrictions on which source will supply the flow to which sink; we have to let our algorithm decide this. A harder problem is the Multicommodity Flow Problem; here sink ti must be supplied with flow that originated at source si, for each i. We will discuss this issue further in Chapter 11.
The Problem: Circulations with Demands and Lower Bounds
Finally, let us generalize the previous problem a little. In many applications, we not only want to satisfy demands at various nodes; we also want to force the flow to make use of certain edges. This can be enforced by placing lower bounds on edges, as well as the usual upper bounds imposed by edge capacities.
Consider a flow network G = (V , E) with a capacity ce and a lower bound le on each edge e. We will assume 0 ≤ le ≤ ce for each e. As before, each node v will also have a demand dv, which can be either positive or negative. We will assume that all demands, capacities, and lower bounds are integers.
7.7 Extensions to the Maximum-Flow Problem
383
The given quantities have the same meaning as before, and now a lower bound le means that the flow value on e must be at least le. Thus a circulation in our flow network must satisfy the following two conditions.
(i) (Capacityconditions)Foreache∈E,wehavele≤f(e)≤ce.
(ii) (Demandconditions)Foreveryv∈V,wehavefin(v)−fout(v)=dv.
As before, we wish to decide whether there exists a feasible circulation—one that satisfies these conditions.
Designing and Analyzing an Algorithm with Lower Bounds
Our strategy will be to reduce this to the problem of finding a circulation with demands but no lower bounds. (We’ve seen that this latter problem, in turn, can be reduced to the standard Maximum-Flow Problem.) The idea is as follows. We know that on each edge e, we need to send at least le units of flow. So suppose that we define an initial circulation f0 simply by f0(e) = le. f0 satisfies all the capacity conditions (both lower and upper bounds); but it presumably does not satisfy all the demand conditions. In particular,
f0in(v) − f0out(v) = le − le. e into v e out of v
Let us denote this quantity by Lv. If Lv = dv, then we have satisfied the demand condition at v; but if not, then we need to superimpose a circulation f1 on top of f0 that will clear the remaining “imbalance” at v. So we need f1in(v) − f1out(v) = dv − Lv. And how much capacity do we have with which to do this? Having already sent le units of flow on each edge e, we have ce − le more units to work with.
These considerations directly motivate the following construction. Let the graph G′ have the same nodes and edges, with capacities and demands, but no lower bounds. The capacity of edge e will be ce − le. The demand of node v will be dv −Lv.
For example, consider the instance in Figure 7.15(a). This is the same as the instance we saw in Figure 7.13, except that we have now given one of the edges a lower bound of 2. In part (b) of the figure, we eliminate this lower bound by sending two units of flow across the edge. This reduces the upper bound on the edge and changes the demands at the two ends of the edge. In the process, it becomes clear that there is no feasible circulation, since after applying the construction there is a node with a demand of −5, and a total of only four units of capacity on its outgoing edges.
We now claim that our general construction produces an equivalent in- stance with demands but no lower bounds; we can therefore use our algorithm for this latter problem.
384
Chapter 7 Network Flow
Eliminating a lower bound from an edge
Lower bound of 2
–3
–1
3313 22
–3 2–5 2
2222 44
(a) (b)
Figure 7.15 (a) An instance of the Circulation Problem with lower bounds: Numbers inside the nodes are demands, and numbers labeling the edges are capacities. We also assign a lower bound of 2 to one of the edges. (b) The result of reducing this instance to an equivalent instance of the Circulation Problem without lower bounds.
(7.52) There is a feasible circulation in G if and only if there is a feasible circulation in G′. If all demands, capacities, and lower bounds in G are integers, and there is a feasible circulation, then there is a feasible circulation that is integer-valued.
Proof. First suppose there is a circulation f′ in G′. Define a circulation f in G by f (e) = f ′(e) + le. Then f satisfies the capacity conditions in G, and
fin(v)−fout(v)= (le +f′(e))− (le +f′(e))=Lv +(dv −Lv)=dv, e into v e out of v
so it satisfies the demand conditions in G as well.
Conversely, suppose there is a circulation f in G, and define a circulation
f ′ in G′ by f ′(e) = f (e) − le. Then f ′ satisfies the capacity conditions in G′, and (f′)in(v) − (f′)out(v) = (f(e) − le) − (f(e) − le) = dv − Lv,
e into v e out of v so it satisfies the demand conditions in G′ as well.
7.8 Survey Design
Many problems that arise in applications can, in fact, be solved efficiently by a reduction to Maximum Flow, but it is often difficult to discover when such a reduction is possible. In the next few sections, we give several paradigmatic examples of such problems. The goal is to indicate what such reductions tend
to look like and to illustrate some of the most common uses of flows and cuts in the design of efficient combinatorial algorithms. One point that will emerge is the following: Sometimes the solution one wants involves the computation of a maximum flow, and sometimes it involves the computation of a minimum cut; both flows and cuts are very useful algorithmic tools.
We begin with a basic application that we call survey design, a simple version of a task faced by many companies wanting to measure customer satisfaction. More generally, the problem illustrates how the construction used to solve the Bipartite Matching Problem arises naturally in any setting where we want to carefully balance decisions across a set of options—in this case, designing questionnaires by balancing relevant questions across a population of consumers.
The Problem
A major issue in the burgeoning field of data mining is the study of consumer preference patterns. Consider a company that sells k products and has a database containing the purchase histories of a large number of customers. (Those of you with “Shopper’s Club” cards may be able to guess how this data gets collected.) The company wishes to conduct a survey, sending customized questionnaires to a particular group of n of its customers, to try determining which products people like overall.
Here are the guidelines for designing the survey.
. Each customer will receive questions about a certain subset of the products.
. A customer can only be asked about products that he or she has pur- chased.
. To make each questionnaire informative, but not too long so as to dis- courage participation, each customer i should be asked about a number of products between ci and ci′.
. Finally, to collect sufficient data about each product, there must be between pj and pj′ distinct customers asked about each product j.
More formally, the input to the Survey Design Problem consists of a bipartite graph G whose nodes are the customers and the products, and there is an edge between customer i and product j if he or she has ever purchased product j. Further, for each customer i = 1, . . . , n, we have limits ci ≤ ci′ on the number of products he or she can be asked about; for each product j=1,...,k, we have limits pj ≤ pj′ on the number of distinct customers that have to be asked about it. The problem is to decide if there is a way to design a questionnaire for each customer so as to satisfy all these conditions.
7.8 Survey Design
385
386
Chapter 7
Network Flow
Customers
ij
p j , p j st
0,1
Products
c i , c i
Figure 7.16 The Survey Design Problem can be reduced to the problem of finding a feasible circulation: Flow passes from customers (with capacity bounds indicating how many questions they can be asked) to products (with capacity bounds indicating how many questions should be asked about each product).
Designing the Algorithm
We will solve this problem by reducing it to a circulation problem on a flow network G′ with demands and lower bounds as shown in Figure 7.16. To obtain the graph G′ from G, we orient the edges of G from customers to products, add nodes s and t with edges (s,i) for each customer i=1,...,n, edges (j,t) for each product j = 1, . . . , k, and an edge (t, s). The circulation in this network will correspond to the way in which questions are asked. The flow on the edge (s, i) is the number of products included on the questionnaire for customer i, so this edge will have a capacity of ci′ and a lower bound of ci. The flow on the edge (j, t) will correspond to the number of customers who were asked about product j, so this edge will have a capacity of pj′ and a lower bound of pj. Each edge (i, j) going from a customer to a product he or she bought has capacity 1, and 0 as the lower bound. The flow carried by the edge (t , s) corresponds to the overall number of questions asked. We can give this edge a capacity of c′ andalowerboundof c.Allnodeshavedemand0.
ii ii
Our algorithm is simply to construct this network G′ and check whether
it has a feasible circulation. We now formulate a claim that establishes the correctness of this algorithm.
Analyzing the Algorithm
(7.53) The graph G′ just constructed has a feasible circulation if and only if there is a feasible way to design the survey.
Proof. The construction above immediately suggests a way to turn a survey design into the corresponding flow. The edge (i, j) will carry one unit of flow if customer i is asked about product j in the survey, and will carry no flow otherwise. The flow on the edges (s, i) is the number of questions asked from customer i, the flow on the edge (j, t) is the number of customers who were asked about product j, and finally, the flow on edge (t, s) is the overall number of questions asked. This flow satisfies the 0 demand, that is, there is flow conservation at every node. If the survey satisfies these rules, then the corresponding flow satisfies the capacities and lower bounds.
Conversely, if the Circulation Problem is feasible, then by (7.52) there is a feasible circulation that is integer-valued, and such an integer-valued circulation naturally corresponds to a feasible survey design. Customer i will be surveyed about product j if and only if the edge (i, j) carries a unit of flow.
7.9 Airline Scheduling
The computational problems faced by the nation’s large airline carriers are almost too complex to even imagine. They have to produce schedules for thou- sands of routes each day that are efficient in terms of equipment usage, crew allocation, customer satisfaction, and a host of other factors—all in the face of unpredictable issues like weather and breakdowns. It’s not surprising that they’re among the largest consumers of high-powered algorithmic techniques.
Covering these computational problems in any realistic level of detail would take us much too far afield. Instead, we’ll discuss a “toy” problem that captures, in a very clean way, some of the resource allocation issues that arise in a context such as this. And, as is common in this book, the toy problem will be much more useful for our purposes than the “real” problem, for the solution to the toy problem involves a very general technique that can be applied in a wide range of situations.
The Problem
Suppose you’re in charge of managing a fleet of airplanes and you’d like to create a flight schedule for them. Here’s a very simple model for this. Your market research has identified a set of m particular flight segments that would be very lucrative if you could serve them; flight segment j is specified by four parameters: its origin airport, its destination airport, its departure time, and its arrival time. Figure 7.17(a) shows a simple example, consisting of six flight segments you’d like to serve with your planes over the course of a single day:
(1) Boston (depart 6 A.M.) – Washington DC (arrive 7 A.M.) (2) Philadelphia (depart 7 A.M.) – Pittsburgh (arrive 8 A.M.)
7.9 Airline Scheduling
387
388
Chapter 7 Network Flow
BOS 6
DCA 7
PHL 7
DCA 7
PHL 7
DCA 8
PIT 8
DCA 8
PIT 8
LAX 11
PHL 11
(a)
LAX 11
PHL 11
(b)
LAS 5
SFO SEA 2:15 3:15
LAS 5
SFO SEA 2:15 3:15
SEA 6
SFO 2
BOS 6
SEA 6
SFO 2
Figure 7.17 (a) A small instance of our simple Airline Scheduling Problem. (b) An expanded graph showing which flights are reachable from which others.
(3) Washington DC (depart 8 A.M.) – Los Angeles (arrive 11 A.M.) (4) Philadelphia (depart 11 A.M.) – San Francisco (arrive 2 P.M.) (5) San Francisco (depart 2:15 P.M.) – Seattle (arrive 3:15 P.M.) (6) Las Vegas (depart 5 P.M.) – Seattle (arrive 6 P.M.)
Note that each segment includes the times you want the flight to serve as well as the airports.
It is possible to use a single plane for a flight segment i, and then later for a flight segment j, provided that
(a) thedestinationofiisthesameastheoriginofj,andthere’senoughtime to perform maintenance on the plane in between; or
(b) you can add a flight segment in between that gets the plane from the destination of i to the origin of j with adequate time in between.
For example, assuming an hour for intermediate maintenance time, you could use a single plane for flights (1), (3), and (6) by having the plane sit in Washington, DC, between flights (1) and (3), and then inserting the flight
Los Angeles (depart 12 noon) – Las Vegas (1 P.M.)
in between flights (3) and (6).
Formulating the Problem We can model this situation in a very general way as follows, abstracting away from specific rules about maintenance times and intermediate flight segments: We will simply say that flight j is reachable from flight i if it is possible to use the same plane for flight i, and then later for flight j as well. So under our specific rules (a) and (b) above, we can easily determine for each pair i, j whether flight j is reachable from flight i. (Of course, one can easily imagine more complex rules for reachability. For example, the length of maintenance time needed in (a) might depend on the airport; or in (b) we might require that the flight segment you insert be sufficiently profitable on its own.) But the point is that we can handle any set of rules with our definition: The input to the problem will include not just the flight segments, but also a specification of the pairs (i, j) for which a later flight j is reachable from an earlier flight i. These pairs can form an arbitrary directed acyclic graph.
The goal in this problem is to determine whether it’s possible to serve all m flights on your original list, using at most k planes total. In order to do this, you need to find a way of efficiently reusing planes for multiple flights.
For example, let’s go back to the instance in Figure 7.17 and assume we have k = 2 planes. If we use one of the planes for flights (1), (3), and (6) as proposed above, we wouldn’t be able to serve all of flights (2), (4), and (5) with the other (since there wouldn’t be enough maintenance time in San Francisco between flights (4) and (5)). However, there is a way to serve all six flights using two planes, via a different solution: One plane serves flights (1), (3), and (5) (splicing in an LAX–SFO flight), while the other serves (2), (4), and (6) (splicing in PIT–PHL and SFO–LAS).
Designing the Algorithm
We now discuss an efficient algorithm that can solve arbitrary instances of the Airline Scheduling Problem, based on network flow. We will see that flow techniques adapt very naturally to this problem.
The solution is based on the following idea. Units of flow will correspond to airplanes. We will have an edge for each flight, and upper and lower capacity bounds of 1 on these edges to require that exactly one unit of flow crosses this edge. In other words, each flight must be served by one of the planes. If (ui , vi) is the edge representing flight i, and (uj,vj) is the edge representing flight j, and flight j is reachable from flight i, then we will have an edge from vi to uj
7.9 Airline Scheduling
389
390
Chapter 7 Network Flow
with capacity 1; in this way, a unit of flow can traverse (ui, vi) and then move directly to (uj , vj ). Such a construction of edges is shown in Figure 7.17(b).
We extend this to a flow network by including a source and sink; we now give the full construction in detail. The node set of the underlying graph G is defined as follows.
. For each flight i, the graph G will have the two nodes ui and vi.
. G will also have a distinct source node s and sink node t. The edge set of G is defined as follows.
. For each i, there is an edge (ui , vi) with a lower bound of 1 and a capacity of 1. (Each flight on the list must be served.)
. For each i and j so that flight j is reachable from flight i, there is an edge (vi, uj) with a lower bound of 0 and a capacity of 1. (The same plane can perform flights i and j.)
. For each i, there is an edge (s, ui) with a lower bound of 0 and a capacity of 1. (Any plane can begin the day with flight i.)
. For each j, there is an edge (vj , t) with a lower bound of 0 and a capacity of 1. (Any plane can end the day with flight j.)
. There is an edge (s, t) with lower bound 0 and capacity k. (If we have extra planes, we don’t need to use them for any of the flights.)
Finally, the node s will have a demand of −k, and the node t will have a demand of k. All other nodes will have a demand of 0.
Our algorithm is to construct the network G and search for a feasible circulation in it. We now prove the correctness of this algorithm.
Analyzing the Algorithm
(7.54) There is a way to perform all flights using at most k planes if and only if there is a feasible circulation in the network G.
Proof. First, suppose there is a way to perform all flights using k′ ≤ k planes. The set of flights performed by each individual plane defines a path P in the network G, and we send one unit of flow on each such path P. To satisfy the full demands at s and t, we send k − k′ units of flow on the edge (s, t). The resulting circulation satisfies all demand, capacity, and lower bound conditions.
Conversely, consider a feasible circulation in the network G. By (7.52), we know that there is a feasible circulation with integer flow values. Suppose that k′ units of flow are sent on edges other than (s, t). Since all other edges have a capacity bound of 1, and the circulation is integer-valued, each such edge that carries flow has exactly one unit of flow on it.
We now convert this to a schedule using the same kind of construction we saw in the proof of (7.42), where we converted a flow to a collection of paths. In fact, the situation is easier here since the graph has no cycles. Consider an edge (s, ui) that carries one unit of flow. It follows by conservation that (ui, vi) carries one unit of flow, and that there is a unique edge out of vi that carries one unit of flow. If we continue in this way, we construct a path P from s to t, so that each edge on this path carries one unit of flow. We can apply this construction to each edge of the form (s, uj) carrying one unit of flow; in this way, we produce k′ paths from s to t, each consisting of edges that carry one unit of flow. Now, for each path P we create in this way, we can assign a single plane to perform all the flights contained in this path.
Extensions: Modeling Other Aspects of the Problem
Airline scheduling consumes countless hours of CPU time in real life. We mentioned at the beginning, however, that our formulation here is really a toy problem; it ignores several obvious factors that would have to be taken into account in these applications. First of all, it ignores the fact that a given plane can only fly a certain number of hours before it needs to be temporarily taken out of service for more significant maintenance. Second, we are making up an optimal schedule for a single day (or at least for a single span of time) as though there were no yesterday or tomorrow; in fact we also need the planes to be optimally positioned for the start of day N + 1 at the end of day N . Third, all these planes need to be staffed by flight crews, and while crews are also reused across multiple flights, a whole different set of constraints operates here, since human beings and airplanes experience fatigue at different rates. And these issues don’t even begin to cover the fact that serving any particular flight segment is not a hard constraint; rather, the real goal is to optimize revenue, and so we can pick and choose among many possible flights to include in our schedule (not to mention designing a good fare structure for passengers) in order to achieve this goal.
Ultimately, the message is probably this: Flow techniques are useful for solving problems of this type, and they are genuinely used in practice. Indeed, our solution above is a general approach to the efficient reuse of a limited set of resources in many settings. At the same time, running an airline efficiently in real life is a very difficult problem.
7.10 Image Segmentation
A central problem in image processing is the segmentation of an image into various coherent regions. For example, you may have an image representing a picture of three people standing in front of a complex background scene. A
7.10 Image Segmentation
391
392
Chapter 7 Network Flow
natural but difficult goal is to identify each of the three people as coherent objects in the scene.
The Problem
One of the most basic problems to be considered along these lines is that of foreground/background segmentation: We wish to label each pixel in an image as belonging to either the foreground of the scene or the background. It turns out that a very natural model here leads to a problem that can be solved efficiently by a minimum cut computation.
Let V be the set of pixels in the underlying image that we’re analyzing. We will declare certain pairs of pixels to be neighbors, and use E to denote the set of all pairs of neighboring pixels. In this way, we obtain an undirected graph G = (V , E). We will be deliberately vague on what exactly we mean by a “pixel,” or what we mean by the “neighbor” relation. In fact, any graph G will yield an efficiently solvable problem, so we are free to define these notions in any way that we want. Of course, it is natural to picture the pixels as constituting a grid of dots, and the neighbors of a pixel to be those that are directly adjacent to it in this grid, as shown in Figure 7.18(a).
s
(a) t (b)
Figure 7.18 (a) A pixel graph. (b) A sketch of the corresponding flow graph. Not all edges from the source or to the sink are drawn.
For each pixel i, we have a likelihood ai that it belongs to the foreground, and a likelihood bi that it belongs to the background. For our purposes, we will assume that these likelihood values are arbitrary nonnegative numbers provided as part of the problem, and that they specify how desirable it is to have pixel i in the background or foreground. Beyond this, it is not crucial precisely what physical properties of the image they are measuring, or how they were determined.
In isolation, we would want to label pixel i as belonging to the foreground if ai > bi, and to the background otherwise. However, decisions that we make about the neighbors of i should affect our decision about i. If many of i’s neighbors are labeled “background,” for example, we should be more inclined to label i as “background” too; this makes the labeling “smoother” by minimizing the amount of foreground/background boundary. Thus, for each pair (i, j) of neighboring pixels, there is a separation penalty pij ≥ 0 for placing one of i or j in the foreground and the other in the background.

We can now specify our Segmentation Problem precisely, in terms of the likelihood and separation parameters: It is to find a partition of the set of pixels into sets A and B (foreground and background, respectively) so as to maximize

q(A,B)= ai + bj − pij. i∈A j∈B (i,j)∈E

|A∩{i,j}|=1

Thus we are rewarded for having high likelihood values and penalized for having neighboring pairs (i, j) with one pixel in A and the other in B. The problem, then, is to compute an optimal labeling—a partition (A, B) that maximizes q(A, B).

Designing and Analyzing the Algorithm

We notice right away that there is clearly a resemblance between the minimum- cut problem and the problem of finding an optimal labeling. However, there are a few significant differences. First, we are seeking to maximize an objective function rather than minimizing one. Second, there is no source and sink in the labeling problem; and, moreover, we need to deal with values ai and bi on the nodes. Third, we have an undirected graph G, whereas for the minimum-cut problem we want to work with a directed graph. Let’s address these problems in order.

We deal with the fact that our Segmentation Problem is a maximization problem through the following observation. Let Q = (a + b ). The sum

iii

a + b isthesameasthesumQ− b − a,sowecan

i∈A i j∈B j i∈A i j∈B j write

7.10 Image Segmentation

393

394

Chapter 7

Network Flow

q(A,B)=Q−

bi − aj − pij. i∈A j∈B (i,j)∈E

|A∩{i,j}|=1

Thus we see that the maximization of q(A, B) is the same problem as the minimization of the quantity

′

q (A, B) = bi + aj + pij.

i∈A j∈B (i,j)∈E |A∩{i,j}|=1

As for the missing source and the sink, we work by analogy with our con- structions in previous sections: We create a new “super-source” s to represent the foreground, and a new “super-sink” t to represent the background. This also gives us a way to deal with the values ai and bi that reside at the nodes (whereas minimum cuts can only handle numbers associated with edges). Specifically, we will attach each of s and t to every pixel, and use ai and bi to define appropriate capacities on the edges between pixel i and the source and sink respectively.

Finally, to take care of the undirected edges, we model each neighboring pair (i, j) with two directed edges, (i, j) and (j, i), as we did in the undirected Disjoint Paths Problem. We will see that this works very well here too, since in any s-t cut, at most one of these two oppositely directed edges can cross from the s-side to the t-side of the cut (for if one does, then the other must go from the t-side to the s-side).

Specifically, we define the following flow network G′ = (V′, E′) shown in Figure 7.18(b). The node set V′ consists of the set V of pixels, together with two additional nodes s and t. For each neighboring pair of pixels i and j, we add directed edges (i, j) and (j, i), each with capacity pij. For each pixel i, we add an edge (s, i) with capacity ai and an edge (i, t) with capacity bi.

Now, an s-t cut (A, B) corresponds to a partition of the pixels into sets A and B. Let’s consider how the capacity of the cut c(A, B) relates to the quantity q′(A, B) that we are trying to minimize. We can group the edges that cross the cut (A, B) into three natural categories.

. Edges (s, j), where j ∈ B; this edge contributes aj to the capacity of the cut.

. Edges (i, t), where i ∈ A; this edge contributes bi to the capacity of the cut.

. Edges (i, j) where i ∈ A and j ∈ B; this edge contributes pij to the capacity of the cut.

Figure 7.19 illustrates what each of these three kinds of edges looks like relative to a cut, on an example with four pixels.

aw

ax

uw puw

bu

vx pvx

bv

7.10 Image Segmentation

395

s

Figure 7.19 An s-t cut on a graph constructed from four pixels. Note how the three types of terms in the expression for q′(A, B) are captured by the cut.

If we add up the contributions of these three kinds of edges, we get

c(A,B)= bi + aj + pij i∈A j∈B (i,j)∈E

|A∩{i,j}|=1

= q′(A, B).

So everything fits together perfectly. The flow network is set up so that the capacity of the cut (A, B) exactly measures the quantity q′(A, B): The three kinds of edges crossing the cut (A, B), as we have just defined them (edges from the source, edges to the sink, and edges involving neither the source nor the sink), correspond to the three kinds of terms in the expression for q′(A, B).

Thus, if we want to minimize q′(A, B) (since we have argued earlier that this is equivalent to maximizing q(A, B)), we just have to find a cut of minimum capacity. And this latter problem, of course, is something that we know how to solve efficiently.

Thus, through solving this minimum-cut problem, we have an optimal algorithm in our model of foreground/background segmentation.

(7.55) The solution to the Segmentation Problem can be obtained by a minimum-cut algorithm in the graph G′ constructed above. For a minimum cut (A′, B′), the partition (A, B) obtained by deleting s∗ and t∗ maximizes the segmentation value q(A, B).

t

396

Chapter 7 Network Flow

7.11 Project Selection

Large (and small) companies are constantly faced with a balancing act between projects that can yield revenue, and the expenses needed for activities that can support these projects. Suppose, for example, that the telecommunications giant CluNet is assessing the pros and cons of a project to offer some new type of high-speed access service to residential customers. Marketing research shows that the service will yield a good amount of revenue, but it must be weighed against some costly preliminary projects that would be needed in order to make this service possible: increasing the fiber-optic capacity in the core of their network, and buying a newer generation of high-speed routers.

What makes these types of decisions particularly tricky is that they interact in complex ways: in isolation, the revenue from the high-speed access service might not be enough to justify modernizing the routers; however, once the company has modernized the routers, they’ll also be in a position to pursue a lucrative additional project with their corporate customers; and maybe this additional project will tip the balance. And these interactions chain together: the corporate project actually would require another expense, but this in turn would enable two other lucrative projects—and so forth. In the end, the question is: Which projects should be pursued, and which should be passed up? It’s a basic issue of balancing costs incurred with profitable opportunities that are made possible.

The Problem

Here’s a very general framework for modeling a set of decisions such as this. There is an underlying set P of projects, and each project i ∈ P has an associated revenue pi, which can either be positive or negative. (In other words, each of the lucrative opportunities and costly infrastructure-building steps in our example above will be referred to as a separate project.) Certain projects are prerequisites for other projects, and we model this by an underlying directed acyclic graph G = (P, E). The nodes of G are the projects, and there is an edge (i, j) to indicate that project i can only be selected if project j is selected as well. Note that a project i can have many prerequisites, and there can be many projects that have project j as one of their prerequisites. A set of projects A ⊆ P is feasible if the prerequisite of every project in A also belongs to A: for each i ∈ A, and each edge (i, j) ∈ E, we also have j ∈ A. We will refer to requirements of this form as precedence constraints. The profit of a set of projects is defined to be

profit(A) =

pi.

i∈A

The Project Selection Problem is to select a feasible set of projects with maxi- mum profit.

This problem also became a hot topic of study in the mining literature, starting in the early 1960s; here it was called the Open-Pit Mining Problem.3 Open-pit mining is a surface mining operation in which blocks of earth are extracted from the surface to retrieve the ore contained in them. Before the mining operation begins, the entire area is divided into a set P of blocks, and the net value pi of each block is estimated: This is the value of the ore minus the processing costs, for this block considered in isolation. Some of these net values will be positive, others negative. The full set of blocks has precedence constraints that essentially prevent blocks from being extracted before others on top of them are extracted. The Open-Pit Mining Problem is to determine the most profitable set of blocks to extract, subject to the precedence constraints. This problem falls into the framework of project selection—each block corresponds to a separate project.

Designing the Algorithm

Here we will show that the Project Selection Problem can be solved by reducing it to a minimum-cut computation on an extended graph G′, defined analogously to the graph we used in Section 7.10 for image segmentation. The idea is to construct G′ from G in such a way that the source side of a minimum cut in G′ will correspond to an optimal set of projects to select.

To form the graph G′, we add a new source s and a new sink t to the graph

G as shown in Figure 7.20. For each node i∈P with pi >0, we add an edge

(s,i) with capacity pi. For each node i∈P with pi <0, we add an edge (i,t)
with capacity −pi. We will set the capacities on the edges in G later. However,
we can already see that the capacity of the cut ({s}, P ∪ {t}) is C = p , i∈P:pi>0 i

so the maximum-flow value in this network is at most C.

We want to ensure that if (A′, B′) is a minimum cut in this graph, then A = A′ − {s} obeys the precedence constraints; that is, if the node i ∈ A has an edge (i, j) ∈ E, then we must have j ∈ A. The conceptually cleanest way to ensure this is to give each of the edges in G capacity of ∞. We haven’t previously formalized what an infinite capacity would mean, but there is no problem in doing this: it is simply an edge for which the capacity condition imposes no upper bound at all. The algorithms of the previous sections, as well as the Max-Flow Min-Cut Theorem, carry over to handle infinite capacities. However, we can also avoid bringing in the notion of infinite capacities by

3 In contrast to the field of data mining, which has motivated several of the problems we considered earlier, we’re talking here about actual mining, where you dig things out of the ground.

7.11 Project Selection

397

398

Chapter 7 Network Flow

tt

Projects with negative value

Projects

Projects

s

Projects with positive value

An

optimal

subset of s projects

Figure 7.20 The flow graph used to solve the Project Selection Problem. A possible minimum-capacity cut is shown on the right.

simply assigning each of these edges a capacity that is “effectively infinite.” In our context, giving each of these edges a capacity of C + 1 would accomplish this: The maximum possible flow value in G′ is at most C, and so no minimum cut can contain an edge with capacity above C. In the description below, it will not matter which of these options we choose.

We can now state the algorithm: We compute a minimum cut (A′, B′) in G′, and we declare A′−{s} to be the optimal set of projects. We now turn to proving that this algorithm indeed gives the optimal solution.

Analyzing the Algorithm

First consider a set of projects A that satisfies the precedence constraints. Let A′ =A∪{s} and B′ =(P−A)∪{t}, and consider the s-t cut (A′,B′). If the set A satisfies the precedence constraints, then no edge (i, j) ∈ E crosses this cut, as shown in Figure 7.20. The capacity of the cut can be expressed as follows.

(7.56) The capacity of the cut (A′,B′), as defined from a project set A satisfying the precedence constraints, is c(A′, B′) = C − p .

Proof. Edges of G′ can be divided into three categories: those corresponding to the edge set E of G, those leaving the source s, and those entering the sink t. Because A satisfies the precedence constraints, the edges in E do not cross the cut (A′, B′), and hence do not contribute to its capacity. The edges entering the sink t contribute

−pi i∈A and pi<0
to the capacity of the cut, and the edges leaving the source s contribute
pi. i̸∈A and pi>0

Using the definition of C, we can rewrite this latter quantity as C−

i∈A and pi>0 which is

p . The capacity of the cut (A′, B′) is the sum of these two terms, i

⎛⎞ ⎝⎠

as claimed.

(−pi) + C − i∈A and pi<0
i∈A and pi>0

pi = C − pi, i∈A

7.11 Project Selection

399

i∈A i

Next, recall that edges of G have capacity more than C = p , and i∈P:pi>0 i

so these edges cannot cross a cut of capacity at most C. This implies that such cuts define feasible sets of projects.

(7.57) If (A′, B′) is a cut with capacity at most C, then the set A = A′−{s} satisfies the precedence constraints.

Now we can prove the main goal of our construction, that the minimum cut in G′ determines the optimum set of projects. Putting the previous two claims together, we see that the cuts (A′, B′) of capacity at most C are in one- to-one correspondence with feasible sets of project A = A′ − {s}. The capacity of such a cut (A′, B′) is

c(A′, B′) = C − profit(A).

The capacity value C is a constant, independent of the cut (A′, B′), so the cut with minimum capacity corresponds to the set of projects A with maximum profit. We have therefore proved the following.

(7.58) If (A′, B′) is a minimum cut in G′ then the set A = A′−{s} is an optimum solution to the Project Selection Problem.

400

Chapter 7 Network Flow

7.12 Baseball Elimination

Over on the radio side the producer’s saying, “See that thing in the paper last week about Einstein? . . . Some reporter asked him to figure out the mathematics of the pennant race. You know, one team wins so many of their remaining games, the other teams win this number or that number. What are the myriad possibilities? Who’s got the edge?”

“The hell does he know?”

“Apparently not much. He picked the Dodgers to eliminate the Giants last Friday.”

—Don DeLillo, Underworld The Problem

Suppose you’re a reporter for the Algorithmic Sporting News, and the following situation arises late one September. There are four baseball teams trying to finish in first place in the American League Eastern Division; let’s call them New York, Baltimore, Toronto, and Boston. Currently, each team has the following number of wins:

New York: 92, Baltimore: 91, Toronto: 91, Boston: 90.

There are five games left in the season: These consist of all possible pairings of the four teams above, except for New York and Boston.

The question is: Can Boston finish with at least as many wins as every other team in the division (that is, finish in first place, possibly in a tie)?

If you think about it, you realize that the answer is no. One argument is the following. Clearly, Boston must win both its remaining games and New York must lose both its remaining games. But this means that Baltimore and Toronto will both beat New York; so then the winner of the Baltimore-Toronto game will end up with the most wins.

Here’s an argument that avoids this kind of cases analysis. Boston can finish with at most 92 wins. Cumulatively, the other three teams have 274 wins currently, and their three games against each other will produce exactly three more wins, for a final total of 277. But 277 wins over three teams means that one of them must have ended up with more than 92 wins.

So now you might start wondering: (i) Is there an efficient algorithm to determine whether a team has been eliminated from first place? And (ii) whenever a team has been eliminated from first place, is there an “averaging” argument like this that proves it?

In more concrete notation, suppose we have a set S of teams, and for each x ∈ S, its current number of wins is wx. Also, for two teams x, y ∈ S, they still

have to play gxy games against one another. Finally, we are given a specific team z.

We will use maximum-flow techniques to achieve the following two things. First, we give an efficient algorithm to decide whether z has been eliminated from first place—or, to put it in positive terms, whether it is possible to choose outcomes for all the remaining games in such a way that the team z ends with at least as many wins as every other team in S. Second, we prove the following clean characterization theorem for baseball elimination—essentially, that there is always a short “proof” when a team has been eliminated.

(7.59) Suppose that team z has indeed been eliminated. Then there exists a “proof” of this fact of the following form:

. z can finish with at most m wins.

. There is a set of teams T ⊆ S so that

As a second, more complex illustration of how the averaging argument in (7.59) works, consider the following example. Suppose we have the same four teams as before, but now the current number of wins is

New York: 90, Baltimore: 88, Toronto: 87, Boston: 79.

The remaining games are as follows. Boston still has four games against each of the other three teams. Baltimore has one more game against each of New York and Toronto. And finally, New York and Toronto still have six games left to play against each other. Clearly, things don’t look good for Boston, but is it actually eliminated?

The answer is yes; Boston has been eliminated. To see this, first note

that Boston can end with at most 91 wins; and now consider the set of teams

T = {New York, Toronto}. Together New York and Toronto already have 177

wins; their six remaining games will result in a total of 183; and 183 > 91. 2

This means that one of them must end up with more than 91 wins, and so

Boston can’t finish in first. Interestingly, in this instance the set of all three

teams ahead of Boston cannot constitute a similar proof: All three teams taken

togeher have a total of 265 wins with 8 games left among them; this is a total

of 273, and 273 = 91 — not enough by itself to prove that Boston couldn’t end 3

up in a multi-way tie for first. So it’s crucial for the averaging argument that we choose the set T consisting just of New York and Toronto, and omit Baltimore.

7.12 Baseball Elimination

401

wx + gxy > m|T|. x,y∈T

x∈T

(And hence one of the teams in T must end with strictly more than m

wins.)

402

Chapter 7 Network Flow

Designing and Analyzing the Algorithm

We begin by constructing a flow network that provides an efficient algorithm for determining whether z has been eliminated. Then, by examining the minimum cut in this network, we will prove (7.59).

Clearly, if there’s any way for z to end up in first place, we should have z win all its remaining games. Let’s suppose that this leaves it with m wins. We now want to carefully allocate the wins from all remaining games so that no other team ends with more than m wins. Allocating wins in this way can be solved by a maximum-flow computation, via the following basic idea. We have a source s from which all wins emanate. The ith win can pass through one of the two teams involved in the ith game. We then impose a capacity constraint saying that at most m − wx wins can pass through team x.

More concretely, we construct the following flow network G, as shown in ′∗

Figure 7.21. First, let S = S − {z}, and let g = x,y∈S′ gxy—the total number of games left between all pairs of teams in S′. We include nodes s and t, a node vx for each team x∈S′, and a node uxy for each pair of teams x,y∈S′ with a nonzero number of games left to play against each other. We have the following edges.

. Edges (s, uxy) (wins emanate from s);

. Edges (uxy , vx) and (uxy , vy) (only x or y can win a game that they play

against each other); and

. Edges (vx, t) (wins are absorbed at t).

Let’s consider what capacities we want to place on these edges. We want gxy wins to flow from s to uxy at saturation, so we give (s, uxy) a capacity of gxy. We want to ensure that team x cannot win more than m − wx games, so we

The set T={NY, Toronto} proves Boston is eliminated.

6 NY–Tor NY 1 14

st

NY–Balt Tor 13

Balt–Tor Balt

Figure 7.21 The flow network for the second example. As the minimum cut indicates, there is no flow of value g∗, and so Boston has been eliminated.

give the edge (vx, t) a capacity of m − wx. Finally, an edge of the form (uxy, vy) should have at least gxy units of capacity, so that it has the ability to transport all the wins from uxy on to vx; in fact, our analysis will be the cleanest if we give it infinite capacity. (We note that the construction still works even if this edge is given only gxy units of capacity, but the proof of (7.59) will become a little more complicated.)

Now, if there is a flow of value g∗, then it is possible for the outcomes of all remaining games to yield a situation where no team has more than m wins; and hence, if team z wins all its remaining games, it can still achieve at least a tie for first place. Conversely, if there are outcomes for the remaining games in which z achieves at least a tie, we can use these outcomes to define a flow of value g∗. For example, in Figure 7.21, which is based on our second example, the indicated cut shows that the maximum flow has value at most 7, whereas g∗ =6+1+1=8.

In summary, we have shown

(7.60) Team z has been eliminated if and only if the maximum flow in G has value strictly less than g∗. Thus we can test in polynomial time whether z has been eliminated.

Characterizing When a Team Has Been Eliminated

Our network flow construction can also be used to prove (7.59). The idea is that the Max-Flow Min-Cut Theorem gives a nice “if and only if” characterization for the existence of flow, and if we interpret this characterization in terms of our application, we get the comparably nice characterization here. This illustrates a general way in which one can generate characterization theorems for problems that are reducible to network flow.

Proof of (7.59). Suppose that z has been eliminated from first place. Then the maximum s-t flow in G has value g′

Thus we have established the following conclusion, based on the fact that (A, B) is a minimum cut: uxy ∈ A if and only if both x, y ∈ T.

Now we just need to work out the minimum-cut capacity c(A, B) in terms of its constituent edge capacities. By the conclusion in the previous paragraph, we know that edges crossing from A to B have one of the following two forms:

. edges of the form (vx, t), where x ∈ T, and

. edges of the form (s, uxy), where at least one of x or y does not belong

to T (in other words, {x, y} ̸⊂ T).

Thus we have

(m − wx) + gxy

{x,y}̸⊂T ∗

c(A, B) = =m|T|−

x∈T

Since we know that c(A, B) = g′ < g∗, this last inequality implies
x∈T
wx+(g − gxy). x,y∈T
wx − gxy < 0, x,y∈T
gxy > m|T|.

x,y∈T

For example, applying the argument in the proof of (7.59) to the instance in Figure 7.21, we see that the nodes for New York and Toronto are on the source side of the minimum cut, and, as we saw earlier, these two teams indeed constitute a proof that Boston has been eliminated.

* 7.13 A Further Direction: Adding Costs to the Matching Problem

Let’s go back to the first problem we discussed in this chapter, Bipartite Matching. Perfect matchings in a bipartite graph formed a way to model the problem of pairing one kind of object with another—jobs with machines, for example. But in many settings, there are a large number of possible perfect matchings on the same set of objects, and we’d like a way to express the idea that some perfect matchings may be “better” than others.

and hence

x∈T

m|T| −

wx +

x∈T

7.13 A Further Direction: Adding Costs to the Matching Problem

405

The Problem

A natural way to formulate a problem based on this notion is to introduce costs. It may be that we incur a certain cost to perform a given job on a given machine, and we’d like to match jobs with machines in a way that minimizes the total cost. Or there may be n fire trucks that must be sent to n distinct houses; each house is at a given distance from each fire station, and we’d like a matching that minimizes the average distance each truck drives to its associated house. In short, it is very useful to have an algorithm that finds a perfect matching of minimum total cost.

Formally, we consider a bipartite graph G = (V , E) whose node set, as

usual, is partitioned as V = X ∪ Y so that every edge e ∈ E has one end in X

and the other end in Y. Furthermore, each edge e has a nonnegative cost ce.

For a matching M, we say that the cost of the matching is the total cost of all

edges in M, that is, cost(M) = c . The Minimum-Cost Perfect Matching e∈M e

Problem assumes that |X| = |Y| = n, and the goal is to find a perfect matching of minimum cost.

Designing and Analyzing the Algorithm

We now describe an efficient algorithm to solve this problem, based on the idea of augmenting paths but adapted to take the costs into account. Thus, the algorithm will iteratively construct matchings using i edges, for each value of i from 1 to n. We will show that when the algorithm concludes with a matching of size n, it is a minimum-cost perfect matching. The high-level structure of the algorithm is quite simple. If we have a minimum-cost matching of size i, then we seek an augmenting path to produce a matching of size i + 1; and rather than looking for any augmenting path (as was sufficient in the case without costs), we use the cheapest augmenting path so that the larger matching will also have minimum cost.

Recall the construction of the residual graph used for finding augmenting paths. Let M be a matching. We add two new nodes s and t to the graph. We add edges (s, x) for all nodes x ∈ X that are unmatched and edges (y, t) for all nodes y∈Y that are unmatched. An edge e=(x,y)∈E is oriented from x to yifeisnotinthematchingMandfromytoxife∈M.WewilluseGM to denote this residual graph. Note that all edges going from Y to X are in the matching M, while the edges going from X to Y are not. Any directed s-t path P in the graph GM corresponds to a matching one larger than M by swapping edges along P, that is, the edges in P from X to Y are added to M and all edges in P that go from Y to X are deleted from M. As before, we will call a path P in GM an augmenting path, and we say that we augment the matching M using the path P.

406

Chapter 7 Network Flow

Now we would like the resulting matching to have as small a cost as possible. To achieve this, we will search for a cheap augmenting path with respect to the following natural costs. The edges leaving s and entering t will have cost 0; an edge e oriented from X to Y will have cost ce (as including this edge in the path means that we add the edge to M); and an edge e oriented from Y to X will have cost −ce (as including this edge in the path means that we delete the edge from M). We will use cost(P) to denote the cost of a path P in GM. The following statement summarizes this construction.

(7.61) LetMbeamatchingandPbeapathinGM fromstot.LetM′bethe matching obtained from M by augmenting along P. Then |M′| = |M| + 1 and cost(M′) = cost(M) + cost(P).

Given this statement, it is natural to suggest an algorithm to find a minimum-cost perfect matching: We iteratively find minimum-cost paths in GM, and use the paths to augment the matchings. But how can we be sure that the perfect matching we find is of minimum cost? Or even worse, is this algorithm even meaningful? We can only find minimum-cost paths if we know that the graph GM has no negative cycles.

Analyzing Negative Cycles In fact, understanding the role of negative cycles in GM is the key to analyzing the algorithm. First consider the case in which M is a perfect matching. Note that in this case the node s has no leaving edges, and t has no entering edges in GM (as our matching is perfect), and hence no cycle in GM contains s or t.

(7.62) Let M be a perfect matching. If there is a negative-cost directed cycle C in GM, then M is not minimum cost.

Proof. To see this, we use the cycle C for augmentation, just the same way we used directed paths to obtain larger matchings. Augmenting M along C involves swapping edges along C in and out of M. The resulting new perfect matching M′ has cost cost(M′) = cost(M) + cost(C); but cost(C) < 0, and hence M is not of minimum cost.
More importantly, the converse of this statement is true as well; so in fact a perfect matching M has minimum cost precisely when there is no negative cycle in GM.
(7.63) Let M be a perfect matching. If there are no negative-cost directed cycles C in GM, then M is a minimum-cost perfect matching.
Proof. Suppose the statement is not true, and let M′ be a perfect matching of smaller cost. Consider the set of edges in one of M and M′ but not in both.
7.13 A Further Direction: Adding Costs to the Matching Problem
407
Observe that this set of edges corresponds to a set of node-disjoint directed cycles in GM. The cost of the set of directed cycles is exactly cost(M′) − cost(M). Assuming M′ has smaller cost than M, it must be that at least one of these cycles has negative cost.
Our plan is thus to iterate through matchings of larger and larger size, maintaining the property that the graph GM has no negative cycles in any iteration. In this way, our computation of a minimum-cost path will always be well defined; and when we terminate with a perfect matching, we can use (7.63) to conclude that it has minimum cost.
Maintaining Prices on the Nodes It will help to think about a numerical price p(v) associated with each node v. These prices will help both in understanding how the algorithm runs, and they will also help speed up the implementation. One issue we have to deal with is to maintain the property that the graph GM has no negative cycles in any iteration. How do we know that after an augmentation, the new residual graph still has no negative cycles? The prices will turn out to serve as a compact proof to show this.
To understand prices, it helps to keep in mind an economic interpretation of them. For this purpose, consider the following scenario. Assume that the set X represents people who need to be assigned to do a set of jobs Y. For an edge e = (x, y), the cost ce is a cost associated with having person x doing job y. Now we will think of the price p(x) as an extra bonus we pay for person x to participate in this system, like a “signing bonus.” With this in mind, the cost for assigning person x to do job y will become p(x) + ce. On the other hand, we will think of the price p(y) for nodes y ∈ Y as a reward, or value gained by taking care of job y (no matter which person in X takes care of it). This way the “net cost” of assigning person x to do job y becomes p(x) + ce − p(y): this is the cost of hiring x for a bonus of p(x), having him do job y for a cost of ce, and then cashing in on the reward p(y). We will call this the reduced cost of an edge e = (x, y) and denote it by cep = p(x) + ce − p(y). However, it is important to keep in mind that only the costs ce are part of the problem description; the prices (bonuses and rewards) will be a way to think about our solution.
Specifically, we say that a set of numbers {p(v) : v ∈ V} forms a set of compatible prices with respect to a matching M if
(i) forallunmatchednodesx∈Xwehavep(x)=0(thatis,peoplenotasked to do any job do not need to be paid);
(ii) for all edges e = (x, y) we have p(x) + ce ≥ p(y) (that is, every edge has a nonnegative reduced cost); and
(iii) foralledgese=(x,y)∈M wehavep(x)+ce =p(y)(everyedgeusedin the assignment has a reduced cost of 0).
408
Chapter 7 Network Flow
Why are such prices useful? Intuitively, compatible prices suggest that the matching is cheap: Along the matched edges reward equals cost, while on all other edges the reward is no bigger than the cost. For a partial matching, this may not imply that the matching has the smallest possible cost for its size (it may be taking care of expensive jobs). However, we claim that if M is any matching for which there exists a set of compatible prices, then GM has no negative cycles. For a perfect matching M, this will imply that M is of minimum cost by (7.63).
To see why GM can have no negative cycles, we extend the definition of reduced cost to edges in the residual graph by using the same expression cep = p(v) + ce − p(w) for any edge e = (v, w). Observe that the definition of compatible prices implies that all edges in the residual graph GM have nonnegative reduced costs. Now, note that for any cycle C, we have
p cost(C) = ce = ce ,
e∈C e∈C
since all the terms on the right-hand side corresponding to prices cancel out. We know that each term on the right-hand side is nonnegative, and so clearly cost(C) is nonnegative.
There is a second, algorithmic reason why it is useful to have prices on the nodes. When you have a graph with negative-cost edges but no negative cycles, you can compute shortest paths using the Bellman-Ford Algorithm in O(mn) time. But if the graph in fact has no negative-cost edges, then you can use Dijkstra’s Algorithm instead, which only requires time O(m log n)—almost a full factor of n faster.
In our case, having the prices around allows us to compute shortest paths with respect to the nonnegative reduced costs cep, arriving at an equivalent answer. Indeed, suppose we use Dijkstra’s Algorithm to find the minimum cost dp,M(v) of a directed path from s to every node v ∈ X ∪ Y subject to the costs cep. Given the minimum costs dp,M(y) for an unmatched node y ∈ Y, the (nonreduced) cost of the path from s to t through y is dp,M(y) + p(y), and so we find the minimum cost in O(n) additional time. In summary, we have the following fact.
(7.64) Let M be a matching, and p be compatible prices. We can use one run of Dijkstra’s Algorithm and O(n) extra time to find the minimum-cost path from s to t.
Updating the Node Prices We took advantage of the prices to improve one iteration of the algorithm. In order to be ready for the next iteration, we need not only the minimum-cost path (to get the next matching), but also a way to produce a set of compatible prices with respect to the new matching.
7.13 A Further Direction: Adding Costs to the Matching Problem
409
x
e
s xey t y
Figure 7.22 A matching M (the dark edges), and a residual graph used to increase the size of the matching.
To get some intuition on how to do this, consider an unmatched node x with respect to a matching M, and an edge e = (x, y), as shown in Figure 7.22. If the new matching M′ includes edge e (that is, if e is on the augmenting path we use to update the matching), then we will want to have the reduced cost of this edge to be zero. However, the prices p we used with matching M may result in a reduced cost cep > 0 — that is, the assignment of person x to job y, in our economic interpretation, may not be viewed as cheap enough. We can arrange the zero reduced cost by either increasing the price p(y) (y’s reward) by cep, or by decreasing the price p(x) by the same amount. To keep prices nonnegative, we will increase the price p(y). However, node y may be matched in the matching M to some other node x′ via an edge e′ = (x′, y), as shown in Figure 7.22. Increasing the reward p(y) decreases the reduced cost of edge e′ to negative, and hence the prices are no longer compatible. To keep things compatible, we can increase p(x′) by the same amount. However, this change might cause problems on other edges. Can we update all prices and keep the matching and the prices compatible on all edges? Surprisingly, this can be done quite simply by using the distances from s to all other nodes computed by Dijkstra’s Algorithm.

(7.65) Let M be a matching, let p be compatible prices, and let M′ be a matching obtained by augmenting along the minimum-cost path from s to t. Then p′(v) = dp,M(v) + p(v) is a compatible set of prices for M′.

Proof. To prove compatibility, consider first an edge e = (x′, y) ∈ M. The only edge entering x′ is the directed edge (y, x′), and hence dp,M(x′) = dp,M(y) − cep, where cep = p(y) + ce − p(x′), and thus we get the desired equation on such edges. Next consider edges (x, y) in M′−M. These edges are along the minimum-cost path from s to t, and hence they satisfy dp,M(y) = dp,M(x) + cep as desired. Finally, we get the required inequality for all other edges since all edges e = (x, y) ̸∈ M must satisfy dp,M(y) ≤ dp,M(x) + cep.

410

Chapter 7 Network Flow

Finally, we have to consider how to initialize the algorithm, so as to get it underway. We initialize M to be the empty set, define p(x) = 0 for all x ∈ X, and define p(y), for y ∈ Y, to be the minimum cost of an edge entering y. Note that these prices are compatible with respect to M = φ.

We summarize the algorithm below.

Start with M equal to the empty set

Define p(x)=0 for x∈X, and p(y)= min ce for y∈Y

e into y While M is not a perfect matching

Find a minimum-cost s-t path P in GM using (7.64) with prices p Augment along P to produce a new matching M′

Find a set of compatible prices with respect to M′ via (7.65)

Endwhile

The final set of compatible prices yields a proof that GM has no negative cycles; and by (7.63), this implies that M has minimum cost.

(7.66) The minimum-cost perfect matching can be found in the time required for n shortest-path computations with nonegative edge lengths.

Extensions: An Economic Interpretation of the Prices

To conclude our discussion of the Minimum-Cost Perfect Matching Problem, we develop the economic interpretation of the prices a bit further. We consider the following scenario. Assume X is a set of n people each looking to buy a house, and Y is a set of n houses that they are all considering. Let v(x, y) denote the value of house y to buyer x. Since each buyer wants one of the houses, one could argue that the best arrangement would be to find a perfect matching

M that maximizes if e = (x, y).

v(x, y). We can find such a perfect matching by using our minimum-cost perfect matching algorithm with costs ce = −v(x, y)

(x,y)∈M

The question we will ask now is this: Can we convince these buyers to buy the house they are allocated? On her own, each buyer x would want to buy the house y that has maximum value v(x, y) to her. How can we convince her to buy instead the house that our matching M allocated? We will use prices to change the incentives of the buyers. Suppose we set a price P(y) for each house y, that is, the person buying the house y must pay P(y). With these prices in mind, a buyer will be interested in buying the house with maximum net value, that is, the house y that maximizes v(x, y) − P(y). We say that a

perfect matching M and house prices P are in equilibrium if, for all edges (x, y) ∈ M and all other houses y′, we have

v(x, y) − P(y) ≥ v(x, y′) − P(y′).

But can we find a perfect matching and a set of prices so as to achieve this state of affairs, with every buyer ending up happy? In fact, the minimum-cost perfect matching and an associated set of compatible prices provide exactly what we’re looking for.

(7.67) Let M be a perfect matching of minimum cost, where ce = −v(x, y) for each edge e = (x, y), and let p be a compatible set of prices. Then the matching M and the set of prices {P(y) = −p(y) : y ∈ Y} are in equilibrium.

Proof. Consideranedgee=(x,y)∈M,andlete′=(x,y′).SinceMandpare compatible, we have p(x) + ce = p(y) and p(x) + ce′ ≥ p(y′). Subtracting these two inequalities to cancel p(x), and substituting the values of p and c, we get the desired inequality in the definition of equilibrium.

Solved Exercises

Solved Exercise 1

Suppose you are given a directed graph G = (V , E), with a positive integer capacity ce on each edge e, a designated source s ∈ V, and a designated sink t ∈ V. You are also given an integer maximum s-t flow in G, defined by a flow value fe on each edge e.

Now suppose we pick a specific edge e ∈ E and increase its capacity by one unit. Show how to find a maximum flow in the resulting capacitated graph in time O(m + n), where m is the number of edges in G and n is the number of nodes.

Solution The point here is that O(m + n) is not enough time to compute a new maximum flow from scratch, so we need to figure out how to use the flow f that we are given. Intuitively, even after we add 1 to the capacity of edge e, the flow f can’t be that far from maximum; after all, we haven’t changed the network very much.

In fact, it’s not hard to show that the maximum flow value can go up by at most 1.

(7.68) Consider the flow network G′ obtained by adding 1 to the capacity of e. The value of the maximum flow in G′ is either ν(f) or ν(f) + 1.

Solved Exercises

411

412

Chapter 7 Network Flow

Proof. The value of the maximum flow in G′ is at least ν(f), since f is still a feasible flow in this network. It is also integer-valued. So it is enough to show thatthemaximum-flowvalueinG′ isatmostν(f)+1.

By the Max-Flow Min-Cut Theorem, there is some s-t cut (A, B) in the original flow network G of capacity ν(f). Now we ask: What is the capacity of (A, B) in the new flow network G′? All the edges crossing (A, B) have the same capacity in G′ that they did in G, with the possible exception of e (in case e crosses (A, B)). But ce only increased by 1, and so the capacity of (A, B) in the newflownetworkG′ isatmostν(f)+1.

Statement (7.68) suggests a natural algorithm. Starting with the feasible flow f in G′, we try to find a single augmenting path from s to t in the residual graph Gf′ . This takes time O(m + n). Now one of two things will happen. Either we will fail to find an augmenting path, and in this case we know that f is a maximum flow. Otherwise the augmentation succeeds, producing a flow f ′ of value at least ν(f)+1. In this case, we know by (7.68) that f′ must be a maximum flow. So either way, we produce a maximum flow after a single augmenting path computation.

Solved Exercise 2

You are helping the medical consulting firm Doctors Without Weekends set up the work schedules of doctors in a large hospital. They’ve got the regular daily schedules mainly worked out. Now, however, they need to deal with all the special cases and, in particular, make sure that they have at least one doctor covering each vacation day.

Here’s how this works. There are k vacation periods (e.g., the week of Christmas, the July 4th weekend, the Thanksgiving weekend, . . . ), each spanning several contiguous days. Let Dj be the set of days included in the jth vacation period; we will refer to the union of all these days, ∪jDj, as the set of all vacation days.

There are n doctors at the hospital, and doctor i has a set of vacation days Si when he or she is available to work. (This may include certain days from a given vacation period but not others; so, for example, a doctor may be able to work the Friday, Saturday, or Sunday of Thanksgiving weekend, but not the Thursday.)

Give a polynomial-time algorithm that takes this information and deter- mines whether it is possible to select a single doctor to work on each vacation day, subject to the following constraints.

. For a given parameter c, each doctor should be assigned to work at most c vacation days total, and only days when he or she is available.

. For each vacation period j, each doctor should be assigned to work at most one of the days in the set Dj. (In other words, although a particular doctor may work on several vacation days over the course of a year, he or she should not be assigned to work two or more days of the Thanksgiving weekend, or two or more days of the July 4th weekend, etc.)

The algorithm should either return an assignment of doctors satisfying these constraints or report (correctly) that no such assignment exists.

Solution This is a very natural setting in which to apply network flow, since at a high level we’re trying to match one set (the doctors) with another set (the vacation days). The complication comes from the requirement that each doctor can work at most one day in each vacation period.

So to begin, let’s see how we’d solve the problem without that require- ment, in the simpler case where each doctor i has a set Si of days when he or she can work, and each doctor should be scheduled for at most c days total. The construction is pictured in Figure 7.23(a). We have a node ui representing each doctor attached to a node vl representing each day when he or she can

Holidays

Holidays

Solved Exercises

413

Gadgets

Doctors

Doctors

Source

Sink

Source Sink

(a)

(b)

Figure 7.23 (a) Doctors are assigned to holiday days without restricting how many days in one holiday a doctor can work. (b) The flow network is expanded with “gadgets” that prevent a doctor from working more than one day from each vacation period. The shaded sets correspond to the different vacation periods.

414

Chapter 7 Network Flow

work; this edge has a capacity of 1. We attach a super-source s to each doctor node ui by an edge of capacity c, and we attach each day node vl to a super- sink t by an edge with upper and lower bounds of 1. This way, assigned days can “flow” through doctors to days when they can work, and the lower bounds on the edges from the days to the sink guarantee that each day is covered. Fi- nally, suppose there are d vacation days total; we put a demand of +d on the sink and −d on the source, and we look for a feasible circulation. (Recall that once we’ve introduced lower bounds on some edges, the algorithms in the text are phrased in terms of circulations with demands, not maximum flow.)

But now we have to handle the extra requirement, that each doctor can work at most one day from each vacation period. To do this, we take each pair (i, j) consisting of a doctor i and a vacation period j, and we add a “vacation gadget” as follows. We include a new node wij with an incoming edge of capacity 1 from the doctor node ui, and with outgoing edges of capacity 1 to each day in vacation period j when doctor i is available to work. This gadget serves to “choke off” the flow from ui into the days associated with vacation period j, so that at most one unit of flow can go to them collectively. The construction is pictured in Figure 7.23(b). As before, we put a demand of +d on the sink and −d on the source, and we look for a feasible circulation. The total running time is the time to construct the graph, which is O(nd), plus the time to check for a single feasible circulation in this graph.

The correctness of the algorithm is a consequence of the following claim.

(7.69) There is a way to assign doctors to vacation days in a way that respects all constraints if and only if there is a feasible circulation in the flow network we have constructed.

Proof. First, if there is a way to assign doctors to vacation days in a way that respects all constraints, then we can construct the following circulation. If doctor i works on day l of vacation period j, then we send one unit of flow along the path s, ui, wij, vl, t; we do this for all such (i, l) pairs. Since the assignment of doctors satisfied all the constraints, the resulting circulation respects all capacities; and it sends d units of flow out of s and into t, so it meets the demands.

Conversely, suppose there is a feasible circulation. For this direction of the proof, we will show how to use the circulation to construct a schedule for all the doctors. First, by (7.52), there is a feasible circulation in which all flow values are integers. We now construct the following schedule: If the edge (wij , vl) carries a unit of flow, then we have doctor i work on day l. Because of the capacities, the resulting schedule has each doctor work at most c days, at most one in each vacation period, and each day is covered by one doctor.

Exercises

u

Exercises

415

11 ure 7.24. The capacity of each edge appears as a label next to the s1t

1. (a)

List all the minimum s-t cuts in the flow network pictured in Fig- edge.

(b) What is the minimum capacity of an s-t cut in the flow network in Figure 7.25? Again, the capacity of each edge appears as a label next to the edge.

2. Figure7.26showsaflownetworkonwhichans-tflowhasbeencomputed. The capacity of each edge appears as a label next to the edge, and the numbers in boxes give the amount of flow sent on each edge. (Edges without boxed numbers—specifically, the four edges of capacity 3—have no flow being sent on them.)

11 v

Figure 7.24 What are the minimum s-t cuts in this flow network?

u

24 s6t

(a) What is the value of this flow? Is this a maximum (s,t) flow in this

graph? 42

(b) Find a minimum s-t cut in the flow network pictured in Figure 7.26, and also say what its capacity is.

3. Figure7.27showsaflownetworkonwhichans-tflowhasbeencomputed. The capacity of each edge appears as a label next to the edge, and the numbers in boxes give the amount of flow sent on each edge. (Edges without boxed numbers have no flow being sent on them.)

(a) What is the value of this flow? Is this a maximum (s,t) flow in this graph?

v

Figure 7.25 What is the min- imum capacity of an s-t cut in this flow network?

5

5 33

10

8 10 8

st

5

33

10

d

Figure 7.26 What is the value of the depicted flow? Is it a maximum flow? What is the minimum cut?

8

5

8

8

5

5

416

Chapter 7

Network Flow

a

5 10 2

1 s3b6c5t

33

1 10

d

Figure 7.27 What is the value of the depicted flow? Is it a maximum flow? What is the minimum cut?

(b) Find a minimum s-t cut in the flow network pictured in Figure 7.27, and also say what its capacity is.

4. Decidewhetheryouthinkthefollowingstatementistrueorfalse.Ifitis true, give a short explanation. If it is false, give a counterexample.

Let G be an arbitrary flow network, with a source s, a sink t, and a positive integer capacity ce on every edge e. If f is a maximum s-t flow in G, then f saturates every edge out of s with flow (i.e., for all edges e out of s, we have f(e)=ce).

5. Decidewhetheryouthinkthefollowingstatementistrueorfalse.Ifitis true, give a short explanation. If it is false, give a counterexample.

Let G be an arbitrary flow network, with a source s, a sink t, and a positive integer capacity ce on every edge e; and let (A, B) be a mimimum s-t cut with respect to these capacities {ce : e ∈ E}. Now suppose we add 1 to every capacity; then (A, B) is still a minimum s-t cut with respect to these new capacities {1+ce :e∈E}.

6. Supposeyou’reaconsultantfortheErgonomicArchitectureCommission, and they come to you with the following problem.

They’re really concerned about designing houses that are “user- friendly,” and they’ve been having a lot of trouble with the setup of light fixtures and switches in newly designed houses. Consider, for example, a one-floor house with n light fixtures and n locations for light switches mounted in the wall. You’d like to be able to wire up one switch to control each light fixture, in such a way that a person at the switch can see the light fixture being controlled.

6

5

1

3

1

5

5

1

Exercises

417

a

2

1

b

a

1

b

3

c

23 c

(a) Ergonomic (b) Not ergonomic

Figure 7.28 The floor plan in (a) is ergonomic, because we can wire switches to fixtures in such a way that each fixture is visible from the switch that controls it. (This can be done by wiring switch 1 to a, switch 2 to b, and switch 3 to c.) The floor plan in (b) is not ergonomic, because no such wiring is possible.

Sometimes this is possible and sometimes it isn’t. Consider the two simple floor plans for houses in Figure 7.28. There are three light fixtures (labeled a, b, c) and three switches (labeled 1, 2, 3). It is possible to wire switches to fixtures in Figure 7.28(a) so that every switch has a line of sight to the fixture, but this is not possible in Figure 7.28(b).

Let’s call a floor plan, together with n light fixture locations and n switch locations, ergonomic if it’s possible to wire one switch to each fixture so that every fixture is visible from the switch that controls it. A floor plan will be represented by a set of m horizontal or vertical line segments in the plane (the walls), where the ith wall has endpoints (xi , yi), (xi′ , yi′). Each of the n switches and each of the n fixtures is given by its coordinates in the plane. A fixture is visible from a switch if the line segment joining them does not cross any of the walls.

Give an algorithm to decide if a given floor plan is ergonomic. The running time should be polynomial in m and n. You may assume that you have a subroutine with O(1) running time that takes two line segments as input and decides whether or not they cross in the plane.

7. Consider a set of mobile computing clients in a certain town who each need to be connected to one of several possible base stations. We’ll suppose there are n clients, with the position of each client specified by its (x, y) coordinates in the plane. There are also k base stations; the position of each of these is specified by (x, y) coordinates as well.

For each client, we wish to connect it to exactly one of the base stations. Our choice of connections is constrained in the following ways.

418

Chapter 7 Network Flow

There is a range parameter r—a client can only be connected to a base station that is within distance r. There is also a load parameter L—no more than L clients can be connected to any single base station.

Your goal is to design a polynomial-time algorithm for the following problem. Given the positions of a set of clients and a set of base stations, as well as the range and load parameters, decide whether every client can be connected simultaneously to a base station, subject to the range and load conditions in the previous paragraph.

8. Statistically, the arrival of spring typically results in increased accidents and increased need for emergency medical treatment, which often re- quires blood transfusions. Consider the problem faced by a hospital that is trying to evaluate whether its blood supply is sufficient.

The basic rule for blood donation is the following. A person’s own blood supply has certain antigens present (we can think of antigens as a kind of molecular signature); and a person cannot receive blood with a particular antigen if their own blood does not have this antigen present. Concretely, this principle underpins the division of blood into four types: A, B, AB, and O. Blood of type A has the A antigen, blood of type B has the B antigen, blood of type AB has both, and blood of type O has neither. Thus, patients with type A can receive only blood types A or O in a transfusion, patients with type B can receive only B or O, patients with type O can receive only O, and patients with type AB can receive any of the four types.4

(a) Let sO, sA, sB, and sAB denote the supply in whole units of the different blood types on hand. Assume that the hospital knows the projected demand for each blood type dO, dA, dB, and dAB for the coming week. Give a polynomial-time algorithm to evaluate if the blood on hand would suffice for the projected need.

(b) Consider the following example. Over the next week, they expect to need at most 100 units of blood. The typical distribution of blood types in U.S. patients is roughly 45 percent type O, 42 percent type A, 10 percent type B, and 3 percent type AB. The hospital wants to know if the blood supply it has on hand would be enough if 100 patients arrive with the expected type distribution. There is a total of 105 units of blood on hand. The table below gives these demands, and the supply on hand.

4 The Austrian scientist Karl Landsteiner received the Nobel Prize in 1930 for his discovery of the blood types A, B, O, and AB.

blood type

O A B AB

supply demand

50 45 36 42 11 8 8 3

Exercises

419

Is the 105 units of blood on hand enough to satisfy the 100 units of demand? Find an allocation that satisfies the maximum possible number of patients. Use an argument based on a minimum-capacity cut to show why not all patients can receive blood. Also, provide an explanation for this fact that would be understandable to the clinic administrators, who have not taken a course on algorithms. (So, for example, this explanation should not involve the words flow, cut, or graph in the sense we use them in this book.)

9. Networkflowissuescomeupindealingwithnaturaldisastersandother crises, since major unexpected events often require the movement and evacuation of large numbers of people in a short amount of time.

Consider the following scenario. Due to large-scale flooding in a re- gion, paramedics have identified a set of n injured people distributed across the region who need to be rushed to hospitals. There are k hos- pitals in the region, and each of the n people needs to be brought to a hospital that is within a half-hour’s driving time of their current location (so different people will have different options for hospitals, depending on where they are right now).

At the same time, one doesn’t want to overload any one of the hospitals by sending too many patients its way. The paramedics are in touch by cell phone, and they want to collectively work out whether they can choose a hospital for each of the injured people in such a way that the load on the hospitals is balanced: Each hospital receives at most ⌈n/k⌉ people.

Give a polynomial-time algorithm that takes the given information about the people’s locations and determines whether this is possible.

10. SupposeyouaregivenadirectedgraphG=(V,E),withapositiveinteger capacityce oneachedgee,asources∈V,andasinkt∈V.Youarealso given a maximum s-t flow in G, defined by a flow value fe on each edge e. The flow f is acyclic: There is no cycle in G on which all edges carry positive flow. The flow f is also integer-valued.

420

Chapter 7 Network Flow

11.

Now suppose we pick a specific edge e∗ ∈ E and reduce its capacity by 1 unit. Show how to find a maximum flow in the resulting capacitated graph in time O(m + n), where m is the number of edges in G and n is the number of nodes.

Your friends have written a very fast piece of maximum-flow code based on repeatedly finding augmenting paths as in Section 7.1. However, after you’ve looked at a bit of output from it, you realize that it’s not always finding a flow of maximum value. The bug turns out to be pretty easy to find; your friends hadn’t really gotten into the whole backward-edge thing when writing the code, and so their implementation builds a variant of the residual graph that only includes the forward edges. In other words, it searches for s-t paths in a graph G ̃f consisting only of edges e for which f (e) < ce, and it terminates when there is no augmenting path consisting entirely of such edges. We’ll call this the Forward-Edge-Only Algorithm. (Note that we do not try to prescribe how this algorithm chooses its forward-edge paths; it may choose them in any fashion it wants, provided that it terminates only when there are no forward-edge paths.)
It’s hard to convince your friends they need to reimplement the code. In addition to its blazing speed, they claim, in fact, that it never returns a flow whose value is less than a fixed fraction of optimal. Do you believe this? The crux of their claim can be made precise in the following statement.
There is an absolute constant b > 1 (independent of the particular input flow network), so that on every instance of the Maximum-Flow Problem, the Forward-Edge-Only Algorithm is guaranteed to find a flow of value at least 1/b times the maximum-flow value (regardless of how it chooses its forward-edge paths).

Decide whether you think this statement is true or false, and give a proof of either the statement or its negation.

Considerthefollowingproblem.Youaregivenaflownetworkwithunit- capacity edges: It consists of a directed graph G = (V, E), a source s ∈ V, andasinkt∈V;andce =1foreverye∈E.Youarealsogivenaparameterk.

The goal is to delete k edges so as to reduce the maximum s-t flow in G by as much as possible. In other words, you should find a set of edges F ⊆ E so that |F| = k and the maximum s-t flow in G′ = (V, E − F) is as small as possible subject to this.

Give a polynomial-time algorithm to solve this problem.

In a standard s-t Maximum-Flow Problem, we assume edges have capaci- ties, and there is no limit on how much flow is allowed to pass through a

12.

13.

node. In this problem, we consider the variant of the Maximum-Flow and Minimum-Cut problems with node capacities.

Let G = (V, E) be a directed graph, with source s ∈ V, sink t ∈ V, and nonnegative node capacities {cv ≥ 0} for each v ∈ V. Given a flow f in this graph, the flow though a node v is defined as fin(v). We say that a flow is feasible if it satisfies the usual flow-conservation constraints and the node-capacity constraints: f in(v) ≤ cv for all nodes.

Give a polynomial-time algorithm to find an s-t maximum flow in such a node-capacitated network. Define an s-t cut for node-capacitated networks, and show that the analogue of the Max-Flow Min-Cut Theorem holds true.

14. We define the Escape Problem as follows. We are given a directed graph G = (V , E) (picture a network of roads). A certain collection of nodes X ⊂ V are designated as populated nodes, and a certain other collection S ⊂ V are designated as safe nodes. (Assume that X and S are disjoint.) In case of an emergency, we want evacuation routes from the populated nodes to the safe nodes. A set of evacuation routes is defined as a set of paths in G so that (i) each node in X is the tail of one path, (ii) the last node on each path lies in S, and (iii) the paths do not share any edges. Such a set of paths gives a way for the occupants of the populated nodes to “escape” to S, without overly congesting any edge in G.

(a) Given G, X, and S, show how to decide in polynomial time whether such a set of evacuation routes exists.

(b) Suppose we have exactly the same problem as in (a), but we want to enforce an even stronger version of the “no congestion” condition (iii). Thus we change (iii) to say “the paths do not share any nodes.”

With this new condition, show how to decide in polynomial time whether such a set of evacuation routes exists.

Also, provide an example with the same G, X, and S, in which the answer is yes to the question in (a) but no to the question in (b).

15. SupposeyouandyourfriendAlanislive,togetherwithn−2otherpeople, at a popular off-campus cooperative apartment, the Upson Collective. Over the next n nights, each of you is supposed to cook dinner for the co-op exactly once, so that someone cooks on each of the nights.

Of course, everyone has scheduling conflicts with some of the nights (e.g., exams, concerts, etc.), so deciding who should cook on which night becomes a tricky task. For concreteness, let’s label the people

{p1, . . . , pn},

Exercises

421

422

Chapter 7 Network Flow

the nights

{d1, . . . , dn};

and for person pi, there’s a set of nights Si ⊂ {d1, . . . , dn} when they are

not able to cook.

A feasible dinner schedule is an assignment of each person in the co- op to a different night, so that each person cooks on exactly one night, there is someone cooking on each night, and if pi cooks on night dj, then dj ̸∈ Si.

(a) Describe a bipartite graph G so that G has a perfect matching if