Some words like “Help", "Stop", “Fire", “Run" are short, not because theyare frequently used, but perhaps because time is precious in the situations in which these words are required. With the same token, in designing asource code for the random variable X, the average code length can beweighted 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 thecost per symbol of the codeword when X = i. Therefore, the average costC of the code description of X can be written asmC = EP:cilii=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, namelyqi = qx(i), wheremCiPiand Q=>ciPiQi=1Note that q is a valid probability distribution.(b) How would you use the Huffman code procedure to minimize C (overall possible prefix codes)? Specifically, what distribution you want touse to design a Huffman code?

Answered: Some words like “Help", "Stop", “Fire",…

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

Programming Logic & Design Comprehensive

9th Edition

ISBN:9781337669405

Author:FARRELL

Publisher:FARRELL

Chapter3: Understanding Structure

Section: Chapter Questions

Problem 2GZ

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Question

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?

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