Probability Mass Functions for which Sources have the Maximum Minimum Expected Length

03/09/2019

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by   Shivkumar K. Manickam, et al.

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Let P_n be the set of all probability mass functions (PMFs) (p_1,p_2,...,p_n) that satisfy p_i>0 for 1≤ i ≤ n. Define the minimum expected length function L_D :P_n →R such that L_D (P) is the minimum expected length of a prefix code, formed out of an alphabet of size D, for the discrete memoryless source having P as its source distribution. It is well-known that the function L_D attains its maximum value at the uniform distribution. Further, when n is of the form D^m, with m being a positive integer, PMFs other than the uniform distribution at which L_D attains its maximum value are known. However, a complete characterization of all such PMFs at which the minimum expected length function attains its maximum value has not been done so far. This is done in this paper.