Disjoint segments with maximum density

Abstract

Given a sequence A of numbers and two positive integers ℓ and k, we study the problem to find k disjoint segments of A, each has length at least ℓ, such that their sum of densities is maximized. We give the first known polynomial-time algorithm for the problem: For general k, our algorithm runs in O(nℓk] time. For the special case with k = 2 (respectively, k = 3), we also show how to solve the problem in O(n) (respectively, O(n + ℓ2)} time. © Springer-Verlag Berlin Heidelberg 2005.