Improved algorithmms for the k maximum-sums problems

  • Authors:

  • Chih-Huai Cheng;Kuan-Yu Chen;Wen-Chin Tien;Kun-Mao Chao

  • Affiliations:

  • Department of Computer Science and Information Engineering, National Taiwan University, Taipei, Taiwan;Department of Computer Science and Information Engineering, National Taiwan University, Taipei, Taiwan;Department of Computer Science and Information Engineering, National Taiwan University, Taipei, Taiwan;Department of Computer Science and Information Engineering, National Taiwan University, Taipei, Taiwan and Graduate Institute of Networking and Multimedia, National Taiwan University, Taipei, Taiw ...

  • Venue:
  • Theoretical Computer Science
  • Year:
  • 2006

Quantified Score

Hi-index 5.23

Visualization

Abstract

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 log2 n, n√k})-time algorithm for this problem. Bae and Takaoka later proposed a more efficient algorithm for small k. In this paper, we propose an O(n + k log(min{n, k}))-time algorithm for the same problem, which is superior to both of them 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(n2d-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(n2d-1C)-time algorithm, also by Bengtsson and Chen, where C = min{k + n log2 n, n√k}. It should be pointed out that, given a two-dimensional array of size m × n, our algorithm for finding the k maximum-sum subarrays is the first one achieving cubic time provided that k is O(m2n/log n).