Generalizations of Length Limited Huffman Coding for Hierarchical Memory Settings

TL;DR

This paper introduces generalized Huffman coding schemes optimized for hierarchical memory systems, including a new soft-length limited variant with penalties, and provides efficient algorithms for these problems.

Contribution

It extends length-limited Huffman coding to hierarchical memory settings and proposes algorithms for a generalized penalty-based coding problem.

Findings

Developed an $O(nD)$ algorithm for soft-length limited Huffman coding.

Provided dynamic programming algorithms for generalized penalty functions.

Achieved PTAS algorithms for complex penalty and objective functions.

Abstract

In this paper, we study the problem of designing prefix-free encoding schemes having minimum average code length that can be decoded efficiently under a decode cost model that captures memory hierarchy induced cost functions. We also study a special case of this problem that is closely related to the length limited Huffman coding (LLHC) problem; we call this the {\em soft-length limited Huffman coding} problem. In this version, there is a penalty associated with each of the $n$ characters of the alphabet whose encodings exceed a specified bound $D$($\leq n$), where the penalty increases linearly with the length of the encoding beyond $D$. The goal of the problem is to find a prefix-free encoding having minimum average code length and total penalty within a pre-specified bound ${\cal P}$. This generalizes the LLHC problem. We present an algorithm to solve this problem that runs in time $O( nD )$. We study a further generalization in which the penalty function and the objective function can both be arbitrary monotonically non-decreasing functions of the codeword length. We provide dynamic programming based exact and PTAS algorithms for this setting.