In-place calculation of minimum-redundancy codes

Abstract

The optimal prefixfree code problem is to determine, for a given array p=[p _i ¦i∈{1...n}] of n weights, an integer array l= [l _i ¦∈{1...n}] of n codeword lengths such that $\sum\nolimits_{i = 1}^n {2^{ - l_i } \leqslant 1}$and $\sum\nolimits_{i = 1}^n {p_i l_i }$is minmized. Huffman's famous greedy algorithm solves this problem in O(n log n) time, if p is unsorted; and can be implemented to execute in O(n) time, if the input array p is sorted. Here we consider the space requirements of the greedy method. We show that if p is sorted then it is possible to calculate the array l in-place, with l _i overwriting p _i , in O(n) time and using O(1) additional space. The new implementation leads directly to an O(n log n)-time and n + O(1) words of extra space implementation for the case when p is not sorted. The proposed method is simple to implement and executes quickly.