ANN queries: covering Voronoi diagram with hyperboxes

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 $\epsilon$.