Algorithms for Locating Constrained Optimal Intervals

In this work, we obtain the following new results. Given a sequence D = ((h1, s1), (h2, s2) . . . , (hn, sn)) of number pairs, where si > 0 for all i, and a number Lh, we propose an O(n)-time algorithm for finding an index interval [i, j] that maximizes Pj k=i hk Pj k=i sk subject to ∑j k=i hk ≥ Lh. Given a sequence D = ((h1, s1), (h2, s2) . . . , (hn, sn)) of number pairs, where si = 1 for all i, and an integer Ls with 1 ≤ Ls ≤ n, we propose an O(n (L 1/2 s ) L 1/2 s )-time algorithm for finding an index interval [i, j] that maximizes Pj k=i hk