Efficient BSP/CGM Algorithms for the Maximum Subarray Sum and Related Problems

Abstract

Given an $n \times n$ array A of integers, with at least one positive value, the maximum subarray sum problem consists in finding the maximum sum among the sums of all rectangular subarrays of A. The maximum subarray problem appears in several scientific applications, particularly in Computer Vision. The algorithms that solve this problem have been used to help the identification of the brightest regions of the images used in astronomy and medical diagnosis. The best known sequential algorithm that solves this problem has ${\text {O}}\bigl (n^{3}\bigr )$ time complexity. In this work we revisit the BSP/CGM parallel algorithm that solves this problem and we present BSP/CGM algorithms for the following related problems: the maximum largest subarray sum, the maximum smallest subarray sum, the number of subarrays of maximum sum, the selection of the subarray with k- maximum sum and the location of the subarray with the maximum relative density sum. To the best of our knowledge there are no parallel BSP/CGM algorithms for these related problems. Our algorithms use p processors and require ${\text {O}}\bigl (n^{3}/p\bigr )$ parallel time with a constant number of communication rounds. In order to show the applicability of our algorithms, we have implemented them on a cluster of computers using MPI and on a machine with GPGPU using CUDA and OpenMP. We have obtained good speedup results in both environments. We also tested the maximum relative density sum algorithm with a image of the cancer imaging archive.