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arXiv · 1903.03755

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

Shivkumar K. Manickam
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Abstract

Let $\mathcal{P}_n$ be the set of all probability mass functions (PMFs) $(p_1,p_2,\ldots,p_n)$ that satisfy $p_i>0$ for $1\leq i \leq n$. Define the minimum expected length function $\mathcal{L}_D :\mathcal{P}_n \rightarrow \mathbb{R}$ such that $\mathcal{L}_D (P)$ 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 $\mathcal{L}_D$ 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 $\mathcal{L}_D$ 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.

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Shivkumar K. Manickam. 2019-03-09. Probability Mass Functions for which Sources have the Maximum Minimum Expected Length. https://arxiv.org/abs/1903.03755

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