Encodings of Range Maximum-Sum Segment Queries and Applications

TL;DR

This paper improves data structures for range maximum-sum segment queries, reducing space requirements and establishing tighter lower bounds, while also applying these structures to optimize algorithms for finding k-covers.

Contribution

It introduces space-efficient data structures for range maximum-sum segment queries and improves the lower bound on space complexity, with applications to k-cover problems.

Findings

Reduced space to Θ(n) bits for index-only queries

Improved space lower bound to 1.89113n - Θ(log n) bits

Simplified linear-time algorithm for finding k-covers

Abstract

Given an array A containing arbitrary (positive and negative) numbers, we consider the problem of supporting range maximum-sum segment queries on A: i.e., given an arbitrary range [i,j], return the subrange [i' ,j' ] \subseteq [i,j] such that the sum of the numbers in A[i'..j'] is maximized. Chen and Chao [Disc. App. Math. 2007] presented a data structure for this problem that occupies {\Theta}(n) words, can be constructed in {\Theta}(n) time, and supports queries in {\Theta}(1) time. Our first result is that if only the indices [i',j'] are desired (rather than the maximum sum achieved in that subrange), then it is possible to reduce the space to {\Theta}(n) bits, regardless the numbers stored in A, while retaining the same construction and query time. We also improve the best known space lower bound for any data structure that supports range maximum-sum segment queries from n bits to 1.89113n - {\Theta}(lg n) bits, for sufficiently large values of n. Finally, we provide a new application of this data structure which simplifies a previously known linear time algorithm for finding k-covers: i.e., given an array A of n numbers and a number k, find k disjoint subranges [i_1 ,j_1 ],...,[i_k ,j_k ], such that the total sum of all the numbers in the subranges is maximized.