Approximate Dynamic Nearest Neighbor Searching in a Polygonal Domain

2026-03-12

AI Summary

This work proposes the first data structure for approximate nearest neighbor (ANN) queries in two-dimensional polygonal domains with obstacles that supports dynamic insertions and deletions. The structure efficiently answers $(1+\varepsilon)$-approximate nearest neighbor queries along with the corresponding shortest-path distances. By integrating geometric decomposition, approximate distance fields, a hierarchical layout, and a logarithmic-scale spatial index, the method achieves a query time of $O\left(\frac{1}{\varepsilon^2} \log n + \frac{1}{\varepsilon} \log n \log m + \frac{1}{\varepsilon} \log^2 m\right)$ and amortized update time within the same bound, while using $O\left(\frac{n+m}{\varepsilon} \log n + \frac{m}{\varepsilon} \log m\right)$ space, where $n$ is the complexity of the polygonal domain and $m$ is the number of sites. This design balances spatial efficiency with strong dynamic performance.

Technology Category

• Search and Optimization: Distributed Search
• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies
• Search and Retrieval-Augmented AI: Vertical and domain-specific search

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.

Problem

• approximate nearest neighbor
• dynamic data structure
• polygonal domain
• shortest path query
• geometric approximation

Innovation

• approximate nearest neighbor
• dynamic data structure
• polygonal domain
• shortest path query
• geometric approximation