Algorithms for Problems on Maximum Density Segment

Let A be a sequence of n ordered pairs of real numbers (ai, li) (i = 1, . . . , n) with li > 0, and L and U be two positive real numbers with 0 < L U . A segment, denoted by A[i, j], 1 i j n, of A is a consecutive subsequence of A between the indices i and j (i and j included). The length l[i, j], sum s[i, j] and density d[i, j] of a segment A[i, j] are l[i, j] = ∑j t=i lt, s[i, j] = ∑j t=i at and d[i, j] = s[i,j] l[i,j] respectively. A segment A[i, j] is feasible if L l[i, j] U . The lengthconstrained maximum density segment problem is to find a feasible segment of maximum density. We present a simple geometric algorithm for this problem for the uniform length case (li = 1 for all i), with time and space complexities in O(n) and O(U − L + 1) respectively. The k length-constrained maximum density segments problem is to find the k most dense length-constrained segments. For the uniform length case, we propose an algorithm for this problem with time complexity in O(min{nk, n lg(U − L+ 1) + k lg(U − L+ 2), n(U − L+ 1)}).