Point Location in Incremental Planar Subdivisions

TL;DR

This paper introduces a novel data structure for point location in incremental planar subdivisions, achieving efficient query and update times, and surpassing previous methods in update speed for dynamic subdivisions.

Contribution

It presents the first data structure with polylogarithmic query and update times for incremental (possibly disconnected) planar subdivisions.

Findings

Supports queries in O(log^2 n) time

Supports updates in O(log n log log n) amortized time

Uses O(n log n) space

Abstract

We study the point location problem in incremental (possibly disconnected) planar subdivisions, that is, dynamic subdivisions allowing insertions of edges and vertices only. Specifically, we present an $O(n\log n)$-space data structure for this problem that supports queries in $O(\log^2 n)$ time and updates in $O(\log n\log\log n)$ amortized time. This is the first result that achieves polylogarithmic query and update times simultaneously in incremental (possibly disconnected) planar subdivisions. Its update time is significantly faster than the update time of the best known data structure for fully-dynamic (possibly disconnected) planar subdivisions.