Path Queries in Weighted Trees

Abstract

We consider the problem of supporting several different path queries over a tree on n nodes, each having a weight drawn from a set of σ distinct values, where σ≤n. One query we support is the path median query, which asks for the median weight on a path between two given nodes. For this and the more general path selection query, we present a linear space data structure that answers queries in $O(\lg \sigma)$ time under the word RAM model. This greatly improves previous results on the same problem, as previous data structures achieving $O(\lg n)$ query time use $O(n \lg^2 n)$ space, and previous linear space data structures require O(ne) time to answer a query for any positive constant e [16]. Our linear space data structure also supports path counting queries in $O(\lg \sigma)$ time. This matches the result of Chazelle [8] when σ is close to n, but has better performance when σ is significantly smaller than n. Finally, the same data structure can also support path reporting queries in $O(\lg \sigma + occ \lg \sigma)$ time, where occ is the size of output. In addition, we present a data structure that answers path reporting queries in $O(\lg \sigma + occ \lg\lg\sigma)$ time, using $O(n\lg\lg\sigma)$ space. These are the first data structures that answer path reporting queries.