ANN queries: covering Voronoi diagram with hyperboxes

TL;DR

This paper introduces an algorithm for efficient approximate nearest neighbor queries in high-dimensional Euclidean spaces using Voronoi diagrams and hyperbox coverings, with provable preprocessing and query time bounds.

Contribution

It presents a novel method combining Voronoi diagrams and hyperbox trees for approximate nearest neighbor search with explicit complexity analysis.

Findings

Preprocessing constructs box trees of size depending on dimension and volume.

Query answering operates in logarithmic plus polynomial time in epsilon.

Average size of the approximate neighbor set depends on data distribution and epsilon.

Abstract

Given a set $S$ of $n$ points in $d$-dimensional Euclidean metric space $X$ and a small positive real number $\epsilon$, we present an algorithm to preprocess $S$ and answer queries that require finding a set $S' \subseteq S$ of $\epsilon$-approximate nearest neighbors (ANNs) to a given query point $q \in X$. The following are the characteristics of points belonging to set $S'$: - $\forall s \in S'$, $\exists$ a point $p \in X$ such that $|pq| \le \epsilon$ and the nearest neighbor of $p$ is $s$, and - $\exists$ a $s' \in S'$ such that $s'$ is a nearest neighbor of $q$. During the preprocessing phase, from the Voronoi diagram of $S$ we construct a set of box trees of size $O(4^d\frac{V}{\delta}(\frac{\pi}{\epsilon})^{d-1})$ which facilitate in querying ANNs of any input query point in $O(\frac{1}{d}lg \frac{V}{\delta} + (\frac{\pi}{\epsilon})^{d-1})$ time. Here $\delta$ equals to $(\frac{\epsilon}{2\sqrt{d}})^d$, and $V$ is the volume of a large bounding box that contains all the points of set $S$. The average case cardinality of $S'$ is shown to rely on $S$ and $\epsilon$.