A Static Optimality Transformation with Applications to Planar Point Location

TL;DR

This paper introduces a new point location data structure for planar triangulations that adapts to query sequences without prior knowledge, achieving asymptotic optimality similar to statically optimal search trees.

Contribution

It presents a novel static optimality data structure for planar point location that does not require prior query distribution knowledge.

Findings

Achieves asymptotic optimality without prior distribution information

Adapts to query sequences similarly to splay trees in 2D

Matches the performance of the best static structures

Abstract

Over the last decade, there have been several data structures that, given a planar subdivision and a probability distribution over the plane, provide a way for answering point location queries that is fine-tuned for the distribution. All these methods suffer from the requirement that the query distribution must be known in advance. We present a new data structure for point location queries in planar triangulations. Our structure is asymptotically as fast as the optimal structures, but it requires no prior information about the queries. This is a 2D analogue of the jump from Knuth's optimum binary search trees (discovered in 1971) to the splay trees of Sleator and Tarjan in 1985. While the former need to know the query distribution, the latter are statically optimal. This means that we can adapt to the query sequence and achieve the same asymptotic performance as an optimum static structure, without needing any additional information.