Path Maximum Query and Path Maximum Sum Query in a Tree

Abstract

SUMMARYLet 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 log n) preprocessing time. key words: path maximum query, path maximum sum query, tree

Identifying Clusters in Graph Representations of Genomes

“…The related problems of finding a single segment with maximum score or 𝑘 segments in a sequence was studied by multiple authors [4,12,13]. A single path can also be found on a weighted tree [14,15]. There are also algorithms for the related maximum density segment problem [16].…”