Tree Path Minimum Query Oracle via Boruvka Trees

TL;DR

This paper introduces a novel Boruvka tree-based oracle for tree path minimum queries that achieves near-optimal query time and construction efficiency, simplifying previous methods and improving performance.

Contribution

It presents the first near-optimal query time algorithm for tree path minimum queries using Boruvka trees, matching optimal trade-offs in construction and comparison complexity.

Findings

Achieves $O(n \log \lambda_k(n))$ construction time and space.

Answers queries in $O(k + \log \lambda_k(n))$ time.

Simplifies previous approaches with a new Boruvka tree-based method.

Abstract

Tree path minimum query problem is a fundamental problem while processing trees, and is used widely in minimum spanning tree verification and randomized minimum spanning tree algorithms. In this paper, we study the possibility of building an oracle in advance, which is able to answer the queries efficiently. We present an algorithm based on Boruvka trees. Our algorithm is the first to achieve a near-optimal bound on query time, while matching the currently optimal trade-off between construction time and the number of comparisons required at query. Particularly, in order to answer each query within $2k$ comparisons, our algorithm requires $O(n \log \lambda_k(n))$ time and space to construct the oracle, and the oracle can answer queries in $O(k + \log \lambda_k(n))$ time. Here $\lambda_k(n)$ is the inverse of the Ackermann function along the $k$-th column. This algorithm not only is simpler than the previous ones, but also gives a completely different method of solving this problem.