Optimal Prefix Codes for Infinite Alphabets with Nonlinear Costs

TL;DR

This paper develops methods for designing optimal prefix codes for infinite alphabets under nonlinear cost functions, specifically exponential means, with applications in communication reliability and queueing systems.

Contribution

It introduces novel algorithms for finding optimal codes for exponential mean costs, extending to geometric, Poisson, and alphabetic distributions, and considers buffer overflow minimization.

Findings

Algorithms for geometric and Poisson distributions are developed.

Extended methods to alphabetic codes and maximum pointwise redundancy.

Application to buffer overflow probability minimization.

Abstract

Let $P = \{p(i)\}$ be a measure of strictly positive probabilities on the set of nonnegative integers. Although the countable number of inputs prevents usage of the Huffman algorithm, there are nontrivial $P$ for which known methods find a source code that is optimal in the sense of minimizing expected codeword length. For some applications, however, a source code should instead minimize one of a family of nonlinear objective functions, $\beta$-exponential means, those of the form $\log_a \sum_i p(i) a^{n(i)}$, where $n(i)$ is the length of the $i$th codeword and $a$ is a positive constant. Applications of such minimizations include a novel problem of maximizing the chance of message receipt in single-shot communications ($a<1$) and a previously known problem of minimizing the chance of buffer overflow in a queueing system ($a>1$). This paper introduces methods for finding codes optimal for such exponential means. One method applies to geometric distributions, while another applies to distributions with lighter tails. The latter algorithm is applied to Poisson distributions and both are extended to alphabetic codes, as well as to minimizing maximum pointwise redundancy. The aforementioned application of minimizing the chance of buffer overflow is also considered.