Finding maximum sum segments in sequences with uncertainty

Abstract

Given a sequence of n numbers and two positive integers L and U with L≤U, the maximum sum segment problem is to find a segment of the sequence with length between L and U such that the sum of numbers in this segment is maximized. In this paper, we introduce the concept of uncertainty into the maximum sum segment problem, where each of the n numbers of the sequence is not given as an exact value but as an interval characterizing the possible range of its value. In such a sequence with uncertainty, we are interested in segments that have potentiality to be maximum sum segments. A segment is a potential maximum sum segment if there exists a possible assignment scenario of the uncertain numbers such that the segment has maximum sum under this assignment. We define the maximum sum segment with uncertainty (MSSU) problem, which consists of two sub-problems: (1) reporting all potential maximum sum segments; and (2) counting the total number of those segments. For the case that L=1 and U=n, we propose an O(n+K)-time algorithm for the reporting problem and an O(n)-time algorithm for the counting problem, where K is the number of potential maximum sum segments. For general L and U, we give an O(n(U−L))-time algorithm for either the reporting or the counting problem. Note that we assume the word-RAM model of computation.