Point Location in Disconnected Planar Subdivisions

TL;DR

This paper introduces a linear-size data structure for efficient point location in disconnected planar subdivisions, achieving near-optimal expected query times based on information-theoretic lower bounds.

Contribution

It extends previous connected subdivision results to disconnected subdivisions, providing a distribution-sensitive, succinct point location structure with optimal expected query time.

Findings

Expected query time is O(H+1), matching lower bounds.

Data structure size is linear in the number of subdivision elements.

Applicable to disconnected planar subdivisions, broadening previous scope.

Abstract

Let $G$ be a (possibly disconnected) planar subdivision and let $D$ be a probability measure over $\R^2$. The current paper shows how to preprocess $(G,D)$ into an O(n) size data structure that can answer planar point location queries over $G$. The expected query time of this data structure, for a query point drawn according to $D$, is $O(H+1)$, where $H$ is a lower bound on the expected query time of any linear decision tree for point location in $G$. This extends the results of Collette et al (2008, 2009) from connected planar subdivisions to disconnected planar subdivisions. A version of this structure, when combined with existing results on succinct point location, provides a succinct distribution-sensitive point location structure.