Efficient Algorithms for k Maximum Sums

Abstract

AbstractWe study the problem of computing the k maximum sum subsequences. Given a sequence of real numbers 〈x 1,x 2,...x n 〉 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 log2 n 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.KeywordsAssociation RuleLarge ElementGood ElementLeft BlockCurrent ColumnThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.