Fast Leaf-to-Ancestor Minimum Query in the Oracle Model

TL;DR

This paper introduces a data structure for fast leaf-to-ancestor minimum queries on weighted trees in the oracle model, achieving constant query time after efficient preprocessing.

Contribution

It presents a novel static data structure that combines multiple techniques to answer leaf-to-ancestor queries in constant time with minimal oracle calls.

Findings

Answers queries in O(1) worst-case time after preprocessing.

Preprocessing takes O(n log h) time, space, and oracle calls.

Preprocessing bound is proven to be tight in the deterministic comparison model.

Abstract

We study leaf-to-ancestor path-minimum queries on a rooted, weighted tree in the oracle model, where the only allowed value operation is a comparison oracle on edge (or node) weights. We give a static data structure that, after O(n log h) preprocessing time, space, and oracle calls (where $n$ is the number of nodes and $h$ is the tree height), answers any leaf-to-ancestor query in O(1) worst-case time with zero oracle calls at query time. The method combines (I) an edge-to-node weight conversion with a deterministic tie-break to obtain a total order; (II) ladder (longest-path) decomposition; (III) binary lifting; and (IV) sparse-table RMQ built over ladder arrays, storing indices selected via the oracle during preprocessing. We also show that the preprocessing oracle-comparison bound is tight in the deterministic comparison model.