A Dynamic Programming Approach To Length-Limited Huffman Coding

TL;DR

This paper presents a simple dynamic programming method for length-limited Huffman coding that matches the best known time and space bounds, and introduces a technique to reduce space complexity applicable to other Monge property DPs.

Contribution

It introduces a straightforward DP approach with reduced space complexity for length-limited Huffman coding, extending applicability to other Monge property dynamic programming problems.

Findings

Achieves $ ext{O}(ND)$ time and space complexity for the problem.

Develops a simple space reduction technique for Monge property DPs.

Demonstrates applicability to problems in web-proxy design and wireless paging.

Abstract

The ``state-of-the-art'' in Length Limited Huffman Coding algorithms is the $\Theta(ND)$-time, $\Theta(N)$-space one of Hirschberg and Larmore, where $D\le N$ is the length restriction on the code. This is a very clever, very problem specific, technique. In this note we show that there is a simple Dynamic-Programming (DP) method that solves the problem with the same time and space bounds. The fact that there was an $\Theta(ND)$ time DP algorithm was previously known; it is a straightforward DP with the Monge property (which permits an order of magnitude speedup). It was not interesting, though, because it also required $\Theta(ND)$ space. The main result of this paper is the technique developed for reducing the space. It is quite simple and applicable to many other problems modeled by DPs with the Monge property. We illustrate this with examples from web-proxy design and wireless mobile paging.