Linear-time algorithms for computing 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 segmentA(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⩽jwk, and the density is (∑i⩽k⩽jak)/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(nlogL)-time algorithm by Lin, Jiang and Chao. We then extend this result, providing an O(n)-time algorithm for the case when both L and U are specified.