A Work Efficient Parallel Algorithm for Constructing Huffman Codes

Given an tilphabet C = (al! . . . , a,) and a corresponding list of weights [WI, . . . , w,], a Huffman code for this alphabet is a prefix code that minimizes the weighted length of a code string, defined to be Cr.., wili, where li is the length of the code assigned to ai. A. Huffman code can be generated in O(n log n) time for an unsorted list of weights alld in Cl(n) time if the weights are already sorted [Huf52, vL76, MK95]. There are some parallel algorithms for const#ructing Hufhnan codes. The best NC algorithm is due to Atallah et all [AKL+89]. It I’UI~S in O(lo$n) time and requires 0(n2/logn) processors. In [LP95] Larmore and Przytycka show a reduction from the CLWS (Concave Least Weight Subsequece) problem to the Huffman code problem and they present an O(n Ii2 logn)-time algorithm for CLWS problem that requires O(n) processors. In [KP90] Kirkpatrick and Przytycka show several approximated parallel algorithms for constructing Huffman codes. In particular, they propose a l/n” approximate algorithm that runs in O(lc logn log* n) and requires n CREW processors, Yor a given value of Ic. A list of n numbers can be sorted in O(log n,) parallel time, with O(n log n) work [Co188]. Hence, throughout this work, we assume that the list of weights is sorted, that is, w1 < . . . < w,,.