Efficient Algorithms for k Maximum Sums

TL;DR: The algorithm is optimal for k = \Omega(n \log^2 n) and improves over the previously best known result for any value of the user-defined parameter k < 1, resulting in fast algorithms as well.

Abstract: We study the problem of computing the k maximum sum subsequences. Given a sequence of real numbers $\left\langle x_{1},x_{2},\ldots ,x_{n}\right\rangle $ and an integer parameter k, $1\leq k\leq \frac{1}{2}n(n-1),$ the problem involves finding the k largest values of $\sum_{\ell =i}^{j}x_{\ell }$ for $1\leq i\leq j\leq n.$ 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 $\Theta(nk)$ -time algorithm for the k maximum sum subsequences problem. In this paper we design an efficient algorithm that solves the above problem in $O( \min \{k+n\log^{2}n,n\sqrt{k}\}) $ time in the worst case. Our algorithm is optimal for $k = \Omega(n \log^2 n)$ and improves over the previously best known result for any value of the user-defined parameter k < 1. Moreover, our results are also extended to the multi-dimensional versions of the k maximum sum subsequences problem; resulting in fast algorithms as well.