Optimal Merging Algorithms for Lossless Codes with Generalized Criteria

TL;DR

This paper introduces a new algorithm for designing lossless prefix codes that optimally balance maximum and average codeword lengths using a convex combination, applicable to various coding scenarios.

Contribution

It proposes a novel coding algorithm based on a merging rule that transforms source probabilities, enabling optimization of a combined length criterion.

Findings

The algorithm effectively balances maximum and average codeword lengths.

It generalizes to criteria involving exponential functions and length constraints.

Applicable to source uncertainty and limited length coding problems.

Abstract

This paper presents lossless prefix codes optimized with respect to a pay-off criterion consisting of a convex combination of maximum codeword length and average codeword length. The optimal codeword lengths obtained are based on a new coding algorithm which transforms the initial source probability vector into a new probability vector according to a merging rule. The coding algorithm is equivalent to a partition of the source alphabet into disjoint sets on which a new transformed probability vector is defined as a function of the initial source probability vector and a scalar parameter. The pay-off criterion considered encompasses a trade-off between maximum and average codeword length; it is related to a pay-off criterion consisting of a convex combination of average codeword length and average of an exponential function of the codeword length, and to an average codeword length pay-off criterion subject to a limited length constraint. A special case of the first related pay-off is connected to coding problems involving source probability uncertainty and codeword overflow probability, while the second related pay-off compliments limited length Huffman coding algorithms.