Searching for the closest-pair in a query translate

TL;DR

This paper develops efficient data structures for quickly finding the closest pair of points within any translated shape in the plane, addressing both polygonal and smooth convex bodies, and resolving open questions in the field.

Contribution

It introduces optimal and near-optimal data structures for range-search closest pair problems for polygons and smooth convex bodies, improving query efficiency.

Findings

Polygon case: $O(n)$ space, $O( ext{log } n)$ query time

Convex body case: $O(n ext{ log } n)$ space, $O( ext{log}^2 n)$ query time

Results settle open questions from prior research

Abstract

We consider a range-search variant of the closest-pair problem. Let $\varGamma$ 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 $\varGamma$, the closest pair of points contained in the translate can be reported efficiently. We present results on this problem for two important settings: when $\varGamma$ is a polygon (possibly with holes) and when $\varGamma$ is a general convex body whose boundary is smooth. When $\varGamma$ is a polygon, we present a data structure using $O(n)$ space and $O(\log n)$ query time, which is asymptotically optimal. When $\varGamma$ 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].