Introduction to Algorithms
3rd Edition
ISBN: 9780262033848
Author: Thomas H. Cormen, Ronald L. Rivest, Charles E. Leiserson, Clifford Stein
Publisher: MIT Press
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Chapter 16.3, Problem 6E
Program Plan Intro
To represent optimal prefix code on C using
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Suppose we have an optimal prefix code on a set C= {0,1,...,n- 1} of characters and we wish to
transmit this code using as few bits as possible. Show how to represent any optimal prefix code
on C using only 2n - 1 + n [lg n] bits. (Hint: Use 2n - 1 bits to specify the structure of the tree, as
discovered by a walk of the tree.)
What is the encoding of the data sample [1.4,2]^T that is obtained using the spanning vector
c=[1.8,1.7]^T?
Write the answer with 2 decimal places.
Use your tabulated solutions from the uploaded pdf to answer the following
questions. Based on your tabulation of executed iteration for the Dijkstra Algorithm
with Customer 6 (C6) connected to Router 6 (R6), which iteration Node Sets consist of
the shortest path for R10? Your answer must correspond to the exact Nodes Sets
indicated in the table. Examples: [6] or [6,3,2,1,4], etc.
Answer:
Chapter 16 Solutions
Introduction to Algorithms
Ch. 16.1 - Prob. 1ECh. 16.1 - Prob. 2ECh. 16.1 - Prob. 3ECh. 16.1 - Prob. 4ECh. 16.1 - Prob. 5ECh. 16.2 - Prob. 1ECh. 16.2 - Prob. 2ECh. 16.2 - Prob. 3ECh. 16.2 - Prob. 4ECh. 16.2 - Prob. 5E
Ch. 16.2 - Prob. 6ECh. 16.2 - Prob. 7ECh. 16.3 - Prob. 1ECh. 16.3 - Prob. 2ECh. 16.3 - Prob. 3ECh. 16.3 - Prob. 4ECh. 16.3 - Prob. 5ECh. 16.3 - Prob. 6ECh. 16.3 - Prob. 7ECh. 16.3 - Prob. 8ECh. 16.3 - Prob. 9ECh. 16.4 - Prob. 1ECh. 16.4 - Prob. 2ECh. 16.4 - Prob. 3ECh. 16.4 - Prob. 4ECh. 16.4 - Prob. 5ECh. 16.5 - Prob. 1ECh. 16.5 - Prob. 2ECh. 16 - Prob. 1PCh. 16 - Prob. 2PCh. 16 - Prob. 3PCh. 16 - Prob. 4PCh. 16 - Prob. 5P
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- I ANALYSIS. Read the following situations and analyze the fundamentals of multimedia elements. 1. A multimedia company uses a compression technique to encode the message before transmitting over the network. Suppose the message contains the following characters with their frequency: Character A Frequency 5 b 9 C d e f 12 13 16 a. Build the Huffman code tree for given message as given. b. Use Huffman tree to find code for each character. 45 c. If the data consists of only these characters, what is the total number of bits to be transmitted? d. Find the number of bits is sent with 8-bit ASCII values without compression?arrow_forwardLet U = l, 2, 3, 4, 5, 6, 7, 8, 9, 1 O, and the elements of U are arranged in increasing order; that is, aj = i. What bit strings represent the subsets of all odd integers in U, all even integers in U, and integers not exceeding 5 in U?arrow_forwardLet's say that the components of U are arranged in ascending order, with U = l, 2, 3, 4, 5, 6, 7, 8, 9, and 1 O, which means that aj = i. What bit strings encompass the subsets of all odd numbers in U, all even integers in U, and all integers in U that do not exceed 5?arrow_forward
- Create an algorithm for the Dijkstra Shortest Weighted Path based on the provided data.G is a weighted (directed or undirected) network, and s is a node in it.post-cond: specifies the shortest weighted route from s to each node of G, and d the lengths of those paths.arrow_forwardLet S be the set of bit strings of length four or more. For example, 10011l e S, 00111100 € S, or 1011 e S. Let R = {(r, y)|r E S,y E S, and r, y agree on their first four bits}. For example, (1010111, 1010010) e R but (10010111, 110010) g R. Show that R is an equivalence relation on S. What are the equivalence classes of the bit string 11110101arrow_forwardWe have discussed Huffman encoding for data compression in lecture and tutorial, now we can implement the Huffman decoding for data extraction to recover the original data. To decode the encoded data, we require the Huffman tree. We iterate through the binary encoded data. To find character corresponding to current bits, we use following simple steps. We start from root and do following until a leaf is found. If current bit is 0, we move to left node of the tree. If the bit is 1, we move to right node of the tree. If during traversal, we encounter a leaf node, we print character of that particular leaf node and then again continue the iteration of the encoded data starting from step 1. Your task is to implement the Huffman decoding algorithm from the above steps in a C++ program with Huffman decoding function and a main function to decode a compressed string based on Huffman encoding and display the original string.arrow_forward
- For the following list of letter frequencies, create a Huffman tree, and use it to determine the encoding for each of the letters. After you’ve written down the encoding for each letter, determine the average number of bits needed to encode ANY letter using this encoding.A: .33 B: .10 C: .08 D: .12 E: .37arrow_forwardProblem 2: Let Un, for n ≥ 0, be the number of binary strings of length n in which no two 0's are at distance two apart. (More precisely, the difference of their positions in the string cannot be equal 2.) For example, U4 = 9, because there are 9 binary strings of length 4 that satisfy this condition: 0011, 0110, 0111, 1001, 1011, 1100, 1101, 1110, 1111. Derive the recurrence equation for Un. You need to give a clear and complete justification for your equation. (You do not need to solve the recurrence.)arrow_forwardLet s = σ1. . . σk be a binary string of length k > 0. We say that a binary string w = w1 . . . wn contains s as a subsequence if there are k indices 1 ≤ i1 < i2, . . . < ik ≤ n such that wik = sr for every 1 ≤ r ≤ k. For example, if s = 11 then 10001, 1010 and 110 contain s as a subsequence whereas 000 and 1000 do not. Prove that the language of all binary strings containing a fixed binary string s of length k as a subsequence is a regular language.arrow_forward
- Use the provided data to create the Dijkstra Shortest Weighted Path method.Precondition: S is a node in the weighted (directed or undirected) network G.Post-cond: specifies the shortest weighted route between each node of G and s, and d specifies the lengths of those paths.arrow_forwardImagine that Huffman Encoding is to be used in the previous example with the same letters, A. E, K and L and the Encoding is represented using a Binary Tree. The following frequencies are given for these respective letters as typically occurring in English text. Frequency(A)-0.35 Frequency(E)-0.24 Frequency(K)-0.21 FrequelicytL)-0.17 Using the Greedy Template in Huffman Encoding, which letters will most appropriately be the leaves of the Binary Tree? (Note: Consider only these letters for now] O Eand K OAand E O Land A O Kand L No new itata to sve. Last checked at 620om Submit Quarrow_forwardOne way to avoid Runge's problem is to choose non-equally spaced data points. A particularly good set to choose are the so-called Chebyshev nodes. To use these, replace the equidistant points xk produced by xeq with xk = COS 2k+1 2n+2 for k=0,..., n. (3.4) Generate data points using (3.4) and (3.3), then plot the resulting Lagrange interpolating polynomial and, hence, show that using (3.4) solves the Runge problem.arrow_forward
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