Approximate Dynamic Nearest Neighbor Searching in a Polygonal Domain

TL;DR

This paper introduces efficient data structures for approximate nearest neighbor and shortest path queries in polygonal domains, supporting dynamic updates with provable approximation guarantees and optimized query times.

Contribution

It presents novel space-efficient data structures for dynamic approximate nearest neighbor searches in polygonal domains with provable performance bounds.

Findings

Achieves $O(rac{n}{ ext{epsilon}} ext{log} n)$ space for approximate distance queries.

Supports dynamic updates with $O(rac{1}{ ext{epsilon}}^2 ext{log} n)$ amortized time.

Provides $(1+ ext{epsilon})$-approximate nearest neighbor retrieval with efficient query times.

Abstract

We present efficient data structures for approximate nearest neighbor searching and approximate 2-point shortest path queries in a two-dimensional polygonal domain $P$ with $n$ vertices. Our goal is to store a dynamic set of $m$ point sites $S$ in $P$ so that we can efficiently find a site $s \in S$ closest to an arbitrary query point $q$. We will allow both insertions and deletions in the set of sites $S$. However, as even just computing the distance between an arbitrary pair of points $q,s \in P$ requires a substantial amount of space, we allow for approximating the distances. Given a parameter $\varepsilon > 0$, we build an $O(\frac{n}{\varepsilon}\log n)$ space data structure that can compute a $1+\varepsilon$-approximation of the distance between $q$ and $s$ in $O(\frac{1}{\varepsilon^2}\log n)$ time. Building on this, we then obtain an $O(\frac{n+m}{\varepsilon}\log n + \frac{m}{\varepsilon}\log m)$ space data structure that allows us to report a site $s \in S$ so that the distance between query point $q$ and $s$ is at most $(1+\varepsilon)$-times the distance between $q$ and its true nearest neighbor in $O(\frac{1}{\varepsilon^2}\log n + \frac{1}{\varepsilon}\log n \log m + \frac{1}{\varepsilon}\log^2 m)$ time. Our data structure supports updates in $O(\frac{1}{\varepsilon^2}\log n + \frac{1}{\varepsilon}\log n \log m + \frac{1}{\varepsilon}\log^2 m)$ amortized time.