Algorithms for Locating Constrained Optimal Intervals

Abstract

In this work, we obtain the following new results.-Given a sequence D = ((h 1 , s 1 ), (h 2 , s 2 ) . . . , (h n , s n )) of number pairs, where s i > 0 for all i, and a number L h , we propose an O(n)-time algorithm for finding an index interval [i, j] that maximizes-Given a sequence D = ((h 1 , s 1 ), (h 2 , s 2 ) . . . , (h n , s n )) of number pairs, where s i = 1 for all i, and an integer L s with 1 ≤ L s ≤ n, we propose an O(n T (L 1/2 s ) L 1/2 s )-time algorithm for finding an index interval [i, j] that maximizes P j k=i h k q P j k=i s k subject to j k=i s k ≥ L s , where T (n ′ ) is the time required to solve the all-pairs shortest paths problem on a graph of n ′ nodes. By the latest result of Chan [8], T (n ′ ) = O(n ′3 (log log n ′ ) 3 (log n ′ ) 2 ), so our algorithm runs in subquadratic time O(nL s (log log Ls) 3 (log Ls) 2 ).