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Chapter 5: Q13E (page 162)

A long string consists of the four characters A,C,G,T ; they appear with frequency 31%,20%,9%and40% respectively. What is the Huffman encoding of these four characters?

Short Answer

Expert verified

Huffman encoding of the charactersA,C,G,T is01,001,000,1 respectively.

Step by step solution

01

Frequencies of the characters are sorted in increasing order

Write the given frequency distribution in the form of a table, in increasing order.

02

Represent the frequencies in a full binary tree

A binary tree in which every node has zero or two children is called a full binary tree.

Characters along with their frequencies in sorted order are placed at the leaf nodes, and Huffman encoding is done by following a path from the root to leaf where every left node is represented as 0 and every right node is represented as 1.Full binary tree created is shown:

03

Huffman encoding of alphabets

White the Huffman encoding of the given frequency distribution.

Thus, Huffman encoding of the given characters are 01,000,001,1.

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Most popular questions from this chapter

Question: Suppose the symbols a,b,c,d,e occur with frequencies 12,14,18,116,116,respectively.

(a) What is the Huffman encoding of the alphabet?

(b) If this encoding is applied to a file consisting of1,000,1000 characters with the given frequencies, what is the length of the encoded file in bits?

Prove the following two properties of the Huffman encoding scheme.

(a) If some character occurs with frequency more than 25, then there is guaranteed to be a codeword of length 1 .

(b) If all characters occur with frequency less than13 , then there is guaranteed to be no codeword of length 1 .

Give a linear-time algorithm that takes as input a tree and determines whether it has a perfect matching: a set of edges that touches each node exactly once.

A feedback edge set of an undirected graph G(V,E) is a subset of edgesE'鈯�Ethat intersects every cycle of the graph. Thus, removing the edges will render the graph acyclic.

Give an efficient algorithm for the following problem:

Input: Undirected graph G(V,E) with positive edge weights we.

Output: A feedback edge set E'鈯�Eminimum total weight 鈭�e鈭�E'we.

Let G=(V,E) be an undirected graph. Prove that if all its edge weights are distinct, then it has a unique minimum spanning tree

Consider the following graph.

(a) What is the cost of its minimum spanning tree?

(b) How many minimum spanning trees does it have?

(c) Suppose Kruskal鈥檚 algorithm is run on this graph. In what order are the edges added to the MST? For each edge in this sequence, give a cut that justifies its addition.
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