k-Maximum Subarrays for Small k: Divide-and-Conquer made simpler

TL;DR

This paper introduces a simplified divide-and-conquer algorithm for the k-maximum subarray problem that is efficient for small k, along with a novel sublinear time method for finding the k largest sums of two sorted arrays.

Contribution

It presents a simpler, more implementable divide-and-conquer solution for small k-maximum subarrays and a novel sublinear algorithm for sum selection.

Findings

Efficient divide-and-conquer algorithm with simplified combine phase.

First sublinear time algorithm for finding k largest sums of two sorted arrays.

Excellent performance demonstrated on test data.

Abstract

Given an array A of n real numbers, the maximum subarray problem is to find a contiguous subarray which has the largest sum. The k-maximum subarrays problem is to find k such subarrays with the largest sums. For the 1-maximum subarray the well known divide-and-conquer algorithm, presented in most textbooks, although suboptimal, is easy to implement and can be made optimal with a simple change that speeds up the combine phase. On the other hand, the only known divide-and-conquer algorithm for k > 1, that is efficient for small values of k, is difficult to implement, due to the intricacies of the combine phase. In this paper we give a divide- and-conquer solution for the k-maximum subarray problem that simplifies the combine phase considerably while preserving the overall running time. In the process of designing the combine phase of the algorithm we provide a simple, sublinear, O($k^{1/2} log^3 k$) time algorithm, for finding the k largest sums of X + Y, where X and Y are sorted arrays of size n and $k <= n^2$. The k largest sums are implicitly represented, and can be enumerated with an additional O(k) time. To our knowledge, this is the first sublinear time algorithm for this well studied problem. Unlike previous solutions, that are fairly complicated and sometimes difficult to implement, ours rely on simple operations such as merging sorted arrays, binary search, and selecting the $k^{th}$ smallest number in an array. We have implemented our algorithms and report excellent performance on test data.