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

Abstract

In this work, we obtain the following new results: – Given a tree T = ( V , E ) with a length function ℓ : 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 = ( a 1 , a 2 , … , a n ) of numbers and an interval [ L , U ] , we define the sum and length of a segment A [ i , j ] to be a i + a i + 1 + ⋯ + a j 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.