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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

In this problem, we will develop a new algorithm for finding minimum spanning trees. It is based upon the following property:

Pick any cycle in the graph, and let e be the heaviest edge in that cycle. Then there is a minimum spanning tree that does not contain e.

(a) Prove this property carefully.

(b) Here is the new MST algorithm. The input is some undirected graph G=(V,E) (in adjacency list format) with edge weights {we}.sort the edges according to their weights for each edge e鈭�E, in decreasing order of we:

if e is part of a cycle of G:

G = G - e (that is, remove e from G )

return G , Prove that this algorithm is correct.

(c) On each iteration, the algorithm must check whether there is a cycle containing a specific edge . Give a linear-time algorithm for this task, and justify its correctness.

(d) What is the overall time taken by this algorithm, in terms of |E|? Explain your answer.

Entropy: Consider a distribution overnpossible outcomes, with probabilities p1,p2,K,pn.

a. Just for this part of the problem, assume that each piis a power of 2 (that is, of the form 1/2k). Suppose a long sequence of msamples is drawn from the distribution and that for all 1鈮�i鈮�n, the ithoutcome occurs exactly times in the sequence. Show that if Huffman encoding is applied to this sequence, the resulting encoding will have length

鈭�i-1nmpilog1pi

b. Now consider arbitrary distributions-that is, the probabilities pi are noy restricted to powers of 2. The most commonly used measure of the amount of randomness in the distribution is the entropy.

鈭�i-1nmpilog1pi

For what distribution (over outcomes) is the entropy the largest possible? The smallest possible?

Suppose you are given a weighted graph G=(V,E) with a distinguished vertex s and where all edge weights are positive and distinct. Is it possible for a tree of shortest paths from s and a minimum spanning tree in G to not share any edges? If so, give an example. If not, give a reason.

Under a Huffman encoding of symbols with frequenciesf1,f2,.....,fn , what is the longest a codeword could possibly be? Give an example set of frequencies that would produce this case.

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.
See all solutions

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