Path Maximum Query and Path Maximum Sum Query in a Tree

Let T be a node-weighted tree with n nodes, and let π(u, v) denote the path between two nodes u and v in T . We address two problems: (i) Path Maximum Query: Preprocess T so that, for a query pair of nodes u and v, the maximum weight on π(u, v) can be found quickly. (ii) Path Maximum Sum Query: Preprocess T so that, for a query pair of nodes u and v, the maximum weight sum subpath of π(u, v) can be found quickly. For the problems we present solutions with O(1) query time and O(n logn) preprocessing time. key words: path maximum query, path maximum sum query, tree.