Algorithms for K disjoint maximum subarrays

Abstract

The maximum subarray problem is to find the array portion that maximizes the sum of array elements in it. For K disjoint maximum subarrays, Ruzzo and Tompa gave an O(n) time solution for one-dimension. This solution is, however, difficult to extend to twodimensions. While a trivial solution of O(Kn³) time is easily obtainable for two-dimensions, little study has been undertaken to better this. We first propose an O(n + K log K) time solution for one-dimension. This is equivalent to Ruzzo and Tompa’s when order is considered. Based on this, we achieve O n³+Kn² log n) time for two-dimensions. This is cubic time when K ≤ n / log n.