Algorithms for finding the weight-constrained k longest paths in a tree and the length-constrained k maximum-sum segments of a sequence

In this work, we obtain the following new results: – Given a tree T = (V, E) with a length function ` : E → R and a weight function w : E → R, a positive integer k, and an interval [L, U ], the Weight-Constrained k Longest Paths problem is to find the k longest paths among all paths in T with weights in the interval [L,U ]. We show that the Weight-Constrained k Longest Paths problem has a lower bound Ω(V log V + k) in the algebraic computation tree model and give an O(V log V + k)-time algorithm for it. – Given a sequence A = (a1, a2, . . . , an) of numbers and an interval [L,U ], we define the sum and length of a segment A[i, j] to be ai + ai+1 + · · · + aj and j − i + 1, respectively. The Length-Constrained k Maximum-Sum Segments problem is to find the k maximum-sum segments among all segments of A with lengths in the interval [L,U ]. We show that the Length-Constrained k Maximum-Sum Segments problem can be solved in O(n + k) time. ∗Corresponding author,[email protected]