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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

Show that if an undirected graph with n vertices has k connected components, then it has at least n - k edges.

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

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.

The following statements may or may not be correct, In each case, either prove it (if it is correct) or give a counter-example (if it isn鈥檛 correct). Always assume that the graph G=(V,E)is undirected. Do not assume that edge weights are distinct unless this is specifically stated.

  1. If a graph G has more than |V|-1edges, and there is a unique heaviest edge, then this edge cannot be part of a minimum spanning tree.
  2. If G has a cycle with a unique heaviest edge e, then e cannot be part of any MST.
  3. Let e be any edge of minimum weight in G. Then e must be part of some MST.
  4. If the lightest edge in a graph is unique, then it must be part of every MST.
  5. If e is part of some MST of G, then it must be a lightest edge across some cut of .
  6. If G has a cycle with a unique lightest edge e must be part of every MST.
  7. The shortest-path tree computed by Dijkstra鈥檚 algorithm is necessarily an MST.
  8. The shortest path between two nodes is necessarily part of some MST.
  9. Prim鈥檚 algorithm works correctly when there are negative edges.
  10. (For any r>0, define an r-path to be a path whose edges all have weight <r). If G contains an r-path from node s to t , then every MST of G must also contain an r-path from node s to node t.

Give the state of the disjoint-sets data structure after the following sequence of operations, starting from singleton sets 1,鈥�,8. Usepath compression. In the case of ties, always make the lower numbered root point to the higher numbered ones.

union1,2,union3,4,union5,6,union7,8

,union1,4,union6,7,union4,5,find1
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