Optimal Algorithms for Finding Density-Constrained Longest and Heaviest Paths in a Tree

Abstract

SUMMARYLet T be a tree with n nodes, in which each edge is associated with a length and a weight. The density-constrained longest (heaviest) path problem is to find a path of T with maximum path length (weight) whose path density is bounded by an upper bound and a lower bound. The path density is the path weight divided by the path length. We show that both problems can be solved in optimal O(n log n) time.

Optimal Algorithms for Finding the Longest Path with Length and Sum Constraints in a Tree

“…The sorted lists Λ and Λ required by the algorithm can be obtained in O(|T |) and O(|T |) time, respectively, by the method of Kim [6].…”