# 程序代写代做代考 scheme algorithm CSE1729 Introduction to Programming April 15, 2017

CSE1729 Introduction to Programming April 15, 2017
Problem Set 10

Constructing Huffman trees

Recall the discussion from class on Huffman trees. In particular, to construct an optimal encoding tree for a
family of symbols �1, . . . ,�k with frequencies f1, . . . , fk, carry out the following algorithm:

1. Place each symbol �i into its own tree; define the weight of this tree to be fi .

2. If all symbols are in a single tree, return this tree (which yields the optimal code).

3. Otherwise, find the two current trees of minimal weight (breaking ties arbitrarily) and combine them into
a new tree by introducing a new root node, and assigning the two light trees to be its subtrees. The weight
of this new tree is defined to be the sum of the weights of the subtrees. Return to step 2.

As an example, consider Huffman encoding a long English text document:

• You would begin by computing the frequencies of each symbol in the document. This would produce a
table, something like the one shown below.

Symbol Frequency
a 2013
b 507
c 711

Here the “frequency” is the number of times the symbol appeared in the document. (If you prefer, you
could divide each of these numbers by the total length of the document; in that case, you could think of the
frequencies as probabilities. This won’t change the results of the Huffman code algorithm.)

• Following this, you can apply the Huffman code algorithm above: this will produce a “Huffman code tree.”
The purpose of this tree is to associate a “codeword” with each symbol. Specifically, the path from the root
to a given symbol can be turned into a codeword by treating every step to a left child as a zero and every
step to a right child as a one. In the figure below, the symbol ↵ would be associated with the codeword 100.

É

É

↵ É

Figure 1: A Huffman tree; the codeword associated with ↵ is 100. The É placeholders represent other symbols.

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• Finally, you can “encode” the document by associating each symbol in the document with the corresponding
codeword. Note that this will turn the document into a long string of 0s and 1s.

• Likewise, you can “decode” the encoded version of a document, by reading the encoded version of the
document from left to right, and following the path it describes in the Huffman tree. Every time a symbol
is reached, the process starts again at the root of the tree.

Strings and characters in SCHEME SCHEME has the facility to work with strings and characters (a string is just
a sequence of characters). In particular, SCHEME treats characters as atomic objects that evaluate to themselves.
They are denoted: #a, #, . . . . Thus, for example,

> #a
#a
> #A
#A
> (eq? #A #a)
#f
> (eq? #a #a)
#t

The “space” character is denoted #space. A “newline” (or carriage return) is denoted #
ewline.
A string in SCHEME is a sequence of characters, but the exact relationship between strings and characters

requires an explanation. A string is denoted, for example, “Hello!”. You can build a string from characters
by using the string command as shown below. An alternate method is to use the list->string command,
which constructs a string from a list of characters, also modeled below. Likewise, you can “explode” a string into
a list of characters by the command string->list:

> (string #S #c #h #e #m #e)
“Scheme”
> (list->string ’(#S #c #h #e #m #e))
“Scheme”
> (string->list “Scheme”)
(#S #c #h #e #m #e)
> “Scheme”
“Scheme”

Note that strings, like characters, numbers, and Boolean values, evaluate to themselves.

1. Write a SCHEME function which, given a list of characters and frequencies, constructs a Huffman encoding
tree. You may assume that the characters and their frequencies are given in a list of pairs: for example, the
list

((#a . 2013) (# . 507) (#c . 711))

represents the 3 characters a, b, and c, with frequencies 2013, 507, and 711, respectively. Given such a list,
you wish to compute the tree that results from the above algorithm. I suggest that you maintain nodes of
the tree as lists: internal nodes can have the form

(internal 0-tree 1-tree)

where internal is a token that indicates that this is an internal node, and 0-tree and 1-tree are the
two subtrees; leaf nodes can have the form

(s () ())

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where s is the character held by the leaf. Note that you will have to use the SCHEME quote command to
construct internal nodes: for example, to construct an internal node with the two subtrees 0-tree and
1-tree, you could use the procedure below

(define (make-internal-node 0-tree 1-tree)
(list ’internal 0-tree 1-tree))

Note that when you traverse a Huffman coding tree, you can determine if a given node is an internal node
by deciding if the car of the list associated with that node is the token internal.

2. Define a SCHEME function that takes, as input, a Huffman coding tree and outputs a list containing the
elements at the leaves of the tree along with their associated encodings as a string over the characters #
and #1. For example, given the tree of Figure 2, your function should return the list

((#a . “0”) (# . “10”) (#c . “11”)) .

a

b c

Figure 2: A Huffman tree yielding an encoding of the three symbols a, b, and c.

3. Define a SCHEME function encode which takes, as input, a string and a list of characters and frequencies
(as in problem 1) and encodes the string into a 0/1-string using Huffman coding. (You may assume that
every character in the string is indeed in the list of character/frequency pairs.)

4. Define a SCHEME function decode which takes, as input, a string over the symbols 0/1 and a Huffman
coding tree and “decodes” the string according to the tree. (It should return a string.)

5. Finally, show how to package all of the functionality you have created in this assignment as a Huffman-code
object. Here’s the idea.

• Define a function H-code-object which takes, as an argument, a list of character/frequency pairs.
It should return a dispatcher that yields two methods: encode, which maps a character string to a
string of zeros and ones, and decode, which maps a string of zeros and ones to a character string.
You might also implement a “reminder” method which simply returns the set of allowable characters
(the ones that appeared in the original character/frequency list).

• Your method should maintain a Huffman encoding

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