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Some words like “Help", "Stop", “Fire", “Run" are short, not because they are frequently used, but perhaps because time is precious in the situations

Some words like “Help", "Stop", “Fire", “Run" are short, not because they are frequently used, but perhaps because time is precious in the situations

Database System Concepts
Database System Concepts
7th Edition
ISBN:9780078022159
Author:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan
Publisher:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan
Chapter1: Introduction
Section: Chapter Questions
Problem 1PE
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Some words like “Help", "Stop", “Fire", “Run" are short, not because they are frequently used, but perhaps because time is precious in the situations
Transcribed Image Text:Some words like “Help", "Stop", “Fire", “Run" are short, not because they are frequently used, but perhaps because time is precious in the situations
in which these words are required. With the same token, in designing a source code for the random variable X, the average code length can be weighted using costs assigned for different realizations of X. Specifically, let X be a random variables that takes values from the alphabet & {1,2, ... , m} with probabilities p; = Px(i) where i E {1,2, .. , m}. Let l; be the length of the codeword associated with X = i. Also, let c; be the cost per symbol of the codeword when X = i. Therefore, the average cost C of the code description of X can be written as m C = EP:cili i=1 (a) You want to design a prefix code for X with lengths 1, l2,...,lm (ED-li < 1) such that the average description cost C is mini- mized. You need to find the minimizing set of lengths {l†,l;,...,} and the corresponding value of the minimum cost C*. In class, we studied two different methods to solve this problem: (1) the Lagrangian approach and (2) the KL distance approach. Use both methods to verify your answer. Hint. You may wish to define a new distribution over X, namely qi = qx(i), where m CiPi and Q=>ciPi Q i=1 Note that q is a valid probability distribution. (b) How would you use the Huffman code procedure to minimize C (over all possible prefix codes)? Specifically, what distribution you want to use to design a Huffman code?
Transcribed Image Text:in which these words are required. With the same token, in designing a source code for the random variable X, the average code length can be weighted using costs assigned for different realizations of X. Specifically, let X be a random variables that takes values from the alphabet & {1,2, ... , m} with probabilities p; = Px(i) where i E {1,2, .. , m}. Let l; be the length of the codeword associated with X = i. Also, let c; be the cost per symbol of the codeword when X = i. Therefore, the average cost C of the code description of X can be written as m C = EP:cili i=1 (a) You want to design a prefix code for X with lengths 1, l2,...,lm (ED-li < 1) such that the average description cost C is mini- mized. You need to find the minimizing set of lengths {l†,l;,...,} and the corresponding value of the minimum cost C*. In class, we studied two different methods to solve this problem: (1) the Lagrangian approach and (2) the KL distance approach. Use both methods to verify your answer. Hint. You may wish to define a new distribution over X, namely qi = qx(i), where m CiPi and Q=>ciPi Q i=1 Note that q is a valid probability distribution. (b) How would you use the Huffman code procedure to minimize C (over all possible prefix codes)? Specifically, what distribution you want to use to design a Huffman code?
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