Improved algorithms for the k maximum-sums problems

Given a sequence of n real numbers and an integer k, 1 <= k <= 1/2n(n-1), the k maximum-sum segments problem is to locate the k segments whose sums are the k largest among all possible segment sums. Recently, Bengtsson and Chen gave an O(min {k+n log(2) n, n root k})-time algorithm for this problem. In this paper, we propose an O(n+k log(min{n, k}))-time algorithm for the same problem which is superior to Bengtsson and Chen's when k is o(n log n). We also give the first optimal algorithm for delivering the k maximum-sum segments in non-decreasing order if k <= n. Then we develop an O(n(2d-1)+k log min {n,k})-time algorithm for the d-dimensional version of the problem, where d>1 and each dimension, without loss of generality, is of the same size n. This improves the best previously known O(n(2d-1) C)-time algorithm, also by Bengtsson and Chen, where C=min{k+n log(2) n, n root k}. It should be pointed out that, given a two-dimensional array of size m x n, our algorithm for finding the k maximum-sum subarrays is the first one achieving cubic time provided that k is O(m(2)n/log n).