Finding Maximum Sum Segments in Sequences with Uncertainty

In this paper, we propose to study the famous maximum sum segment problem on a sequence consisting of n uncertain numbers, where each number is given as an interval characterizing its possible value. Given two integers L and U , a segment of length between L and U is called a potential maximum sum segment if there exists a possible assignment of the uncertain numbers such that, under the assignment, the segment has maximum sum over all segments of length between L and U . We define the maximum sum segment with uncertainty problem, which consists of two sub-problems: (1) reporting all potential maximum sum segments; (2) counting the total number of those segments. For L =1 and U =n , we propose an O (n +K )-time algorithm and an O (n )-time algorithm, respectively, 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 problem.