Dynamic Geodesic Nearest Neighbor Searching in a Simple Polygon

TL;DR

This paper introduces a dynamic data structure for geodesic nearest neighbor queries within a simple polygon, supporting efficient insertions, deletions, and queries with proven time complexities.

Contribution

It presents the first data structure that efficiently handles dynamic geodesic nearest neighbor queries in simple polygons, supporting both insertions and deletions with known operation order.

Findings

Supports queries in O(√n log n log^2 m) time

Supports updates in O(√n log^3 m) time

Uses O(n log m + m) space

Abstract

We present an efficient dynamic data structure that supports geodesic nearest neighbor queries for a set of point sites $S$ in a static simple polygon $P$. Our data structure allows us to insert a new site in $S$, delete a site from $S$, and ask for the site in $S$ closest to an arbitrary query point $q \in P$. All distances are measured using the geodesic distance, that is, the length of the shortest path that is completely contained in $P$. Our data structure supports queries in $O(\sqrt{n}\log n\log^2 m)$ time, where $n$ is the number of sites currently in $S$, and $m$ is the number of vertices of $P$, and updates in $O(\sqrt{n}\log^3 m)$ time. The space usage is $O(n\log m + m)$. If only insertions are allowed, we can support queries in worst-case $O(\log^2 n\log^2 m)$ time, while allowing for $O(\log n\log^3 m)$ amortized time insertions. We can achieve the same running times in case there are both insertions and deletions, but the order of these operations is known in advance.