Searching for the closest-pair in a query translate

Abstract

We consider a range-search variant of the closest-pair problem. Let $\Gamma$ be a fixed shape in the plane. We are interested in storing a given set of $n$ points in the plane in some data structure such that for any specified translate of $\Gamma$, the closest pair of points contained in the translate can be reported efficiently. We present results on this problem for two important settings: when $\Gamma$ is a polygon (possibly with holes) and when $\Gamma$ is a general convex body whose boundary is smooth. When $\Gamma$ is a polygon, we present a data structure using $O(n)$ space and $O(\log n)$ query time, which is asymptotically optimal. When $\Gamma$ is a general convex body with a smooth boundary, we give a near-optimal data structure using $O(n\log n)$ space and $O({\log }^{2}n)$ query time. Our results settle some open questions posed by Xue et al. [SoCG 2018].