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

Let T be a tree in which every edge is associated with a real number. The sum of a path in T is the sum of the numbers associated with the edges of the path and its length is the number of the edges in it. For two positive integers L1 ≤ L2 and two real numbers S1 ≤ S2, a path is feasible if its length is between L1 and L2 and its sum is between S1 and S2. We address the problem: Given a tree T , and four numbers, L1, L2, S1 and S2, find the longest feasible path of T . We provide an optimal O(n logn) time algorithm for the problem, where n = |T |. key words: length constraint, longest path, sum constraint, tree.