Efficient algorithms for k maximum sums

  • Authors:

  • Fredrik Bengtsson;Jingsen Chen

  • Affiliations:

  • Department of Computer Science and Electrical Engineering, Luleå University of Technology, Luleå, Sweden;Department of Computer Science and Electrical Engineering, Luleå University of Technology, Luleå, Sweden

  • Venue:
  • ISAAC'04 Proceedings of the 15th international conference on Algorithms and Computation
  • Year:
  • 2004

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Abstract

We study the problem of computing the k maximum sum subsequences Given a sequence of real numbers 〈x1,x2,...xn〉 and an integer parameter k, $1\leq k \leq \frac{1}{2}n(n-1)$, the problem involves finding the k largest values of $\sum\limits^{j}_{\ell=i}x_{\ell}$ for 1 ≤ i ≤ j ≤ n The problem for fixed k = 1, also known as the maximum sum subsequence problem, has received much attention in the literature and is linear-time solvable Recently, Bae and Takaoka presented a Θ(nk)-time algorithm for the k maximum sum subsequences problem In this paper, we design efficient algorithms that solve the above problem in $O(min\{k+n{\rm log}^{2}n,n\sqrt{k}\})$ time in the worst case Our algorithm is optimal for k ≥ n log2n and improves over the previously best known result for any value of the user-defined parameter k Moreover, our results are also extended to the multi-dimensional versions of the k maximum sum subsequences problem; resulting in fast algorithms as well.