Improved Redundancy Bounds for Exponential Objectives

TL;DR

This paper derives new tight bounds on the compression rate for binary prefix codes under exponential objectives, improving upon recent bounds and analyzing properties of optimal codes for these objectives.

Contribution

It introduces improved lower and upper bounds for exponential-average length and redundancy, enhancing understanding of optimal coding strategies under these criteria.

Findings

Derived tight bounds for exponential-average length and redundancy.

Improved upon existing bounds with entropy-based expressions.

Analyzed properties of optimal codes for exponential redundancy.

Abstract

We present new lower and upper bounds for the compression rate of binary prefix codes optimized over memoryless sources according to two related exponential codeword length objectives. The objectives explored here are exponential-average length and exponential-average redundancy. The first of these relates to various problems involving queueing, uncertainty, and lossless communications, and it can be reduced to the second, which has properties more amenable to analysis. These bounds, some of which are tight, are in terms of a form of entropy and/or the probability of an input symbol, improving on recently discovered bounds of similar form. We also observe properties of optimal codes over the exponential-average redundancy utility.