🤖 AI Summary
This work addresses the problem of range minimum queries on leaf-to-ancestor paths in trees within the deterministic comparison model, where edge or node weights are accessible only through a comparison oracle. The authors propose a static data structure that achieves worst-case O(1) query time without any oracle calls during queries. This is accomplished by transforming edge weights to node weights, applying deterministic tie-breaking, employing ladder decomposition, binary lifting, and a sparse-table RMQ technique indexed via oracle-based selection. The structure requires O(n log h) preprocessing time, space, and oracle invocations, where n is the number of nodes and h the tree height. Notably, this is the first solution in the deterministic comparison model to achieve constant-time leaf-to-ancestor path minimum queries with zero oracle calls at query time, and its preprocessing oracle complexity matches the theoretical lower bound, establishing optimality.
📝 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.