# CS代考计算机代写 algorithm DNA flex BU CS 332 – Theory of Computation

BU CS 332 – Theory of Computation

Lecture 18:

• Time Complexity

Reading:

Sipser Ch 7.1‐7.2

• Complexity Class P

Mark Bun April 6, 2020

Where we are in CS 332

Automata & Formal Languages Computability Complexity

Previous unit: Computability theory

What kinds of problems can / can’t computers solve?

Final unit: Complexity theory

What kinds of problems can / can’t computers solve under constraints on their computational resources?

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First topic: Time complexity

Today: Answering the basic questions

1. How do we measure complexity? (as in CS 330)

2. Asymptotic notation (as in CS 330)

3. How robust is the TM model when we care about measuring complexity?

4. How do we mathematically capture our intuitive notion of “efficient algorithms”?

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Running time analysis

Time complexity of a TM (algorithm) = maximum number of steps it takes on a worst‐case input

Formally: Let ∗ . A TM runs in time if on every input , halts on within at most steps

‐ Focus on worst‐case running time: Upper bound of must hold for all inputs of length

‐ Exact running time does not translate well between computational models / real computers. Instead focus on asymptotic complexity.

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Example

How much time does it take for a basic single‐tape TM to decide ?

Let’s analyze one particular TM :

= “On input :

1. Scan input and reject if not of the form 2. While input contains both ’s and ’s:

∗∗

Cross off one and one

3. Accept if no 0’s and no 1’s left. Otherwise, reject.”

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Review of asymptotic notation

‐notation (upper bounds)

means:

There exist constants

such that

Example:

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( , )

6

for every

Caution: does not mean “equals” Not reflexive:

Example:

does not mean

Alternative (better) notation:

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Examples

•

632

• • •

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Review of asymptotic notation

‐notation (lower bounds)

Example:

( , )

9

means:

There exist constants

such that

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for every

When should we use vs. ?

Upper bounds: Use

“The merge‐sort algorithm uses at most

comparisons in the worst case”

Lower bounds: Use

“Every comparison‐based sorting algorithm requires at

least comparisons in the worst case”

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Review of asymptotic notation

‐notation (tight bounds)

Example:

Generally, polynomials are easy:

means:

AND

𝑑𝑑 1 0𝑑 4/6/2020 CS332 ‐ Theory of Computation 11

Little‐oh and little‐omega

‐notation and ‐notation are like and ; ‐notation and ‐notation are like and

means:

For every constant there exists

such that

for every

Example: ( ) 4/6/2020 CS332 ‐ Theory of Computation

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A handy‐dandy chart

Notation f(n)=O(g(n))

… means … ∃c>0,n0>0,∀n>n0 :

Think… Upper bound

Example 100n2

lim ←

f(n)=(g(n)) f(n)=(g(n))

cg(n) < f(n)
both of the above:
Lower bound
= (n100) log(n!)
If it exists, it is > 0

f(n)=o(g(n)) f(n)=(g(n))

f(n) < cg(n)
Strict upper bound
Limit exists, =0
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f(n) < cg(n) ∃c>0,n0>0,∀n>n0 :

= O(n3) 2n

If it exists, it is < ∞
f=(g)andf=O(g) ∀c>0,∃n0>0,∀n>n0 :

Tight bound

=(nlogn) n2 = o(2n)

If it exists, it is>0and< ∞
∀c>0,∃n0>0,∀n>n0 : cg(n) < f(n)
Strict lower bound
n2
= (log n)
Limit exists, =∞
Asymptotic notation within expressions
Asymptotic notation within an expression is shorthand for an unspecified function satisfying the statement
Examples:
•
• •
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FAABs: Frequently asked asymptotic bounds
•Polynomials. 0 1 • Logarithms. 𝑎
𝑑 𝑑 is 𝑑 if 𝑑 𝑏 for all constants
For every • Exponentials. For all
,
and all
,
𝑑𝑛
• Factorial.
By Stirling’s formula,
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Time Complexity
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Time complexity classes
Let
is a class (i.e., set) of languages:
A language if there exists a basic single‐ tape (deterministic) TM that
1) Decides , and 2) Runs in time
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Example
𝐴01 𝑚0 𝑀 = “On input 𝑤:
1. Scan input and reject if not of the form 0∗1∗ 2. While input contains both 0’s and 1’s:
Cross off one 0 and one 1
3. Accept if no 0’s and no 1’s left. Otherwise, reject.”
• runs in time
• Is there a faster algorithm?
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Example
𝐴01 𝑚0 𝑀′ = “On input 𝑤:
1. Scan input and reject if not of the form 0∗1∗
2. While input contains both 0’s and 1’s:
• Reject if the total number of 0’s and 1’s remaining is odd
• Cross off every other 0 and every other 1
3. Accept if no 0’s and no 1’s left. Otherwise, reject.” • Running time of :
• Is there a faster algorithm?
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Example
Running time of :
Theorem (Sipser, Problem 7.49): If can be decided in
time on a 1‐tape TM, then is regular
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Does it matter that we’re using the 1‐tape model for this result?
It matters: 2‐tape TMs can decide faster
𝑀′′ = “On input 𝑤:
1. Scan input and reject if not of the form 0∗1∗
2. Copy 0’s to tape 2
3. Scan tape 1. For each 1 read, cross of a 0 on tape 2
4. If 0’s on tape 2 finish at same time as 1’s on tape 1, accept. Otherwise, reject.”
Analysis: is decided in time on a 2‐tape TM Moral of the story (part 1): Unlike decidability, time
complexity depends on the TM model
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How much does the model matter?
Theorem: Let be a function. Every multi‐tape TM running in time has an equivalent single‐tape TM running in time
Proof idea:
We already saw how to simulate a multi‐tape TM with a
single‐tape TM
Need a runtime analysis of this construction
Moral of the story (part 2): Time complexity doesn’t depend too much on the TM model (as long as it’s deterministic, sequential)
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Simulating Multiple Tapes
Implementation‐Level Description
On input
1. Format tape into
2. For each move of :
Scan left‐to‐right, finding current symbols Scan left‐to‐right, writing new symbols, Scan left‐to‐right, moving each tape head
If a tape head goes off the right end, insert blank If a tape head goes off left end, move back right
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How much does the model matter?
Theorem: Let be a function. Every multi‐tape TM running in time has an equivalent single‐tape TM running in time
Proof: Time analysis of simulation
• Time to initialize (i.e., format tape):
• Time to simulate one step of multi‐tape TM: • Number of steps to simulate:
=> Total time:

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Extended Church‐Turing Thesis

Every “reasonable” model of computation can be simulated by a basic, single‐tape TM with only a polynomial slowdown.

E.g., doubly infinite TMs, multi‐tape TMs, RAM TMs Does not include nondeterministic TMs (not reasonable)

Possible counterexamples? Randomized computation, parallel computation, DNA computing, quantum computation

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Complexity Class

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Complexity class

Definition: is the class of languages decidable in polynomial time on a basic single‐tape (deterministic) TM

• Class doesn’t change if we substitute in another reasonable deterministic model (Extended Church‐Turing)

• Cobham‐Edmonds Thesis: Captures class of problems that are feasible to solve on computers

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Examples of languages in

• 𝑃𝐴𝑇𝐻

𝐺, 𝑠, 𝑡 𝐺 is a directed graph with a directed path from 𝑠 to 𝑡

• 𝐴 𝐷,𝑤 • 𝑅𝐸𝐿𝑃𝑅𝐼𝑀𝐸

𝐷 is a DFA that accepts input 𝑤

𝑥, 𝑦 𝑥 and 𝑦 are relatively prime

• 𝑃𝑅𝐼𝑀𝐸𝑆

𝑥 𝑥 is prime

2006 Gödel Prize citation

• Every context‐free language (section tomorrow)

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The 2006 Gödel Prize for outstanding articles in theoretical computer science is awarded to Manindra Agrawal, Neeraj Kayal, and Nitin Saxena for their paper “PRIMES is in P.”

In August 2002 one of the most ancient computational problems was finally solved….