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

Let 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.