BSP/CGM Algorithms for Maximum Subsequence and Maximum Subarray

Abstract

The maximum subsequence problem finds the contiguous subsequence of n real numbers with the highest sum. This problem appears in the analysis of DNA or protein sequences. It can be solved sequentially in O(n) time. In the 2-D version, given an n × n array A, the maximum subarray of A is the contiguous subarray that has the maximum sum. The sequential algorithm for the maximum subarray problem takes O(n ³) time. We present efficient BSP/CGM parallel algorithms that require a constant number of communication rounds for both problems. In the first algorithm, the sequence stored on each processor is reduced to only five numbers, so that the resulting values can be concentrated on a single processor which runs an adaptation of the sequential algorithm to obtain the result. The parallel algorithm requires O(n/p) computing time. In the second algorithm, the input array is partitioned equally among the processors and we first reduce each subarray to a sequence, and then apply the first algorithm to solve it. The parallel algorithm takes O(n ³/p) computing time. The good performance of the parallel algorithms is confirmed by experimental results run on a 64-node Beowulf parallel computer.