Maximum - Density Sequence Segments with Bioinformatics Applications ⋆

We study an abstract optimization problem arising from biomolecular sequence analysis. For a sequence A of pairs (ai, wi) for i = 1, . . . , n and wi > 0, a segment A(i, j) is a consecutive subsequence of A starting with index i and ending with index j. The width of A(i, j) is w(i, j) = ∑ i≤k≤j wk, and the density is ( ∑ i≤k≤j ak)/w(i, j). The maximum-density segment problem takes A and two values L and U as input and asks for a segment of A with the largest possible density among those of width at least L and at most U . When U is unbounded, we provide a relatively simple, O(n)-time algorithm, improving upon the O(n logL)-time algorithm by Lin, Jiang and Chao. When both L and U are specified, there are no previous nontrivial results. We solve the problem in O(n) time if wi = 1 for all i, and more generally in O(n+ n log(U − L+ 1)) time when wi ≥ 1 for all i.