Abstract. The maximum subarray problem for a one- or two-dimensional array is to find the array portion that maiximizes the sum of array elements in it. The K-maximum subarray problem is to find the K subarrays with largest sums. We improve the time complexity for the one-dimensional case from O(min{K + n log 2 n, n √ K}) for 0 ≤ K ≤ n(n − 1)/2 to O(n log K + K 2) for K ≤ n. The latter is better when K ≤ √ n log n. If we simply extend this result to the two-dimensional case, we will have the complexity of O(n 3 log K + K 2 n 2).We improve this complexity to O(n 3) for K ≤ √ n. 1

We develop parallel algorithms for both one-dimensional and two-dimensional versions of the maximum sum problem (or max sum for short) on several interconnection networks. These algorithms are all based on a simple scheme that uses prefix sums. To this end, we first show how to compute prefix sums of N elements on a hypercube, a star, and a pancake interconnection network of size p (where p N) in optimal time of O( N p + log p). For the problem of maximum subsequence sum, the 1-D version of the max sum problem, we find an algorithm that computes the maximum sum of N elements on the aforementioned networks of size p, all with a running time of O( N p + log p), which is optimal in view of the trivial\Omega\Gamma N p + log p) lower bound. When p = O( N log N ), our algorithm computes the max sum in O(log N) time, resulting in an optimal cost of O(N ). This result also matches the performance of two previous algorithms that are designed to run on PRAM. Our 1-D max sum algorithm can...

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