A fast and space-economical algorithm for length-limited coding

Abstract

The minimum-redundancy prefix code problem is to determine a list of integer codeword lengths l=[l _i¦iε{1... n}], given a list of n symbol weights p=[p_i¦iε{1...n}], such that $\sum\nolimits_{i = 1}^n {2^{ - l_i } } \leqslant 1$, and $\sum\nolimits_{i = 1}^n {l_i p_i }$ l _ip_i is minimised. An extension is the minimum-redundancy length-limited prefix code problem, in which the further constraint l _i≤L is imposed, for all i ε {1... n} and some integer L≥[log₂ n]. The package-merge algorithm of Larmore and Hirschberg generates length-limited codes in O(nL) time using O(n) words of auxiliary space. Here we show how the size of the work space can be reduced to O(L²). This represents a useful improvement, since for practical purposes L is Θ(log n).