An Optimal Algorithm for the Maximum-Density Segment Problem

We address a fundamental problem arising from analysis of biomolecular sequences. The input consists of two numbers wmin and wmax and a sequence S of n number pairs (ai, wi)with wi > 0. Let segment S(i, j) of S be the consecutive subsequence of S between indices i and j. The density of S(i, j) is d(i, j) = (ai+ai+1+· · ·+aj)/(wi+wi+1+· · ·+wj). Themaximum-density segment problem is to find a maximum-density segment over all segments S(i, j) with wmin ≤ wi + wi+1 + · · · + wj ≤ wmax. The best previously known algorithm for the problem, due to Goldwasser, Kao, and Lu, runs inO(n log(wmax−wmin+1)) time. In the present paper, we solve the problem in O(n) time. Our approach bypasses the complicated right-skew decomposition, introduced by Lin, Jiang, and Chao. As a result, our algorithm has the capability to process the input sequence in an online manner, which is an important feature for dealing with genomescale sequences. Moreover, for a type of input sequences S representable in O(m) space, we show how to exploit the sparsity of S and solve the maximum-density segment problem for S in O(m) time.