Point Location in Dynamic Planar Subdivisions

TL;DR

This paper introduces a new data structure for efficiently handling point location queries in dynamic planar subdivisions with insertions and deletions, achieving sublinear update and polylogarithmic query times.

Contribution

The paper presents a linear-size data structure for dynamic planar subdivisions supporting efficient updates and queries, answering an open question in computational geometry.

Findings

Supports insertions and deletions with sublinear update time

Achieves polylogarithmic query time

Handles disconnected subdivisions efficiently

Abstract

We study the point location problem on dynamic planar subdivisions that allows insertions and deletions of edges. In our problem, the underlying graph of a subdivision is not necessarily connected. We present a data structure of linear size for such a dynamic planar subdivision that supports sublinear-time update and polylogarithmic-time query. Precisely, the amortized update time is $O(\sqrt{n}\log n(\log\log n)^{3/2})$ and the query time is $O(\log n(\log\log n)^2)$, where $n$ is the number of edges in the subdivision. This answers a question posed by Snoeyink in the Handbook of Computational Geometry. When only deletions of edges are allowed, the update time and query time are just $O(\alpha(n))$ and $O(\log n)$, respectively.