Adaptive Planar Point Location

TL;DR

This paper introduces self-adjusting data structures for efficient point location queries in convex and connected subdivisions, achieving near-optimal query times with linear space and adaptive processing based on query sequences.

Contribution

It presents novel self-adjusting data structures that adapt to query sequences, providing near-optimal query times for convex and connected subdivisions with linear space.

Findings

Achieves $O( ext{OPT} + n)$ query time for convex subdivisions.

Achieves $O( ext{OPT} + n + |\sigma| ext{log}( ext{log}^* n))$ for connected subdivisions.

Uses $O(n)$ space and includes $O(n)$ preprocessing time.

Abstract

We present self-adjusting data structures for answering point location queries in convex and connected subdivisions. Let $n$ be the number of vertices in a convex or connected subdivision. Our structures use $O(n)$ space. For any convex subdivision $S$, our method processes any online query sequence $\sigma$ in $O(\mathrm{OPT} + n)$ time, where $\mathrm{OPT}$ is the minimum time required by any linear decision tree for answering point location queries in $S$ to process $\sigma$. For connected subdivisions, the processing time is $O(\mathrm{OPT} + n + |\sigma|\log(\log^* n))$. In both cases, the time bound includes the $O(n)$ preprocessing time.