A General Framework for Codes Involving Redundancy Minimization

TL;DR

This paper introduces a unified framework with two parameters for various prefix coding problems, connecting existing problems and providing new Huffman-like algorithms for minimizing maximum pointwise redundancy.

Contribution

The paper presents a general framework linking multiple coding problems and introduces two efficient algorithms for the minimum maximum pointwise redundancy problem.

Findings

Two linear-time Huffman-like algorithms for the minimum maximum pointwise redundancy problem.

The second algorithm finds the minimum variance solution among common solutions.

Simple redundancy bounds are established.

Abstract

A framework with two scalar parameters is introduced for various problems of finding a prefix code minimizing a coding penalty function. The framework encompasses problems previously proposed by Huffman, Campbell, Nath, and Drmota and Szpankowski, shedding light on the relationships among these problems. In particular, Nath's range of problems can be seen as bridging the minimum average redundancy problem of Huffman with the minimum maximum pointwise redundancy problem of Drmota and Szpankowski. Using this framework, two linear-time Huffman-like algorithms are devised for the minimum maximum pointwise redundancy problem, the only one in the framework not previously solved with a Huffman-like algorithm. Both algorithms provide solutions common to this problem and a subrange of Nath's problems, the second algorithm being distinguished by its ability to find the minimum variance solution among all solutions common to the minimum maximum pointwise redundancy and Nath problems. Simple redundancy bounds are also presented.