Range closest-pair search in higher dimensions

Abstract

Range closest-pair (RCP) search is a range-search variant of the classical closest-pair problem, which aims to store a given set $S$ of points into some space-efficient data structure such that when a query range $Q$ is specified, the closest pair in $S\cap Q$ can be reported quickly. RCP search has received attention over years, but the primary focus was only on ${\mathbb{R}}^2$. In this paper, we study RCP search in higher dimensions. We give the first nontrivial RCP data structures for orthogonal, simplex, halfspace, and ball queries in ${\mathbb{R}}^d$ for any constant $d$. Furthermore, we prove a conditional lower bound for orthogonal RCP search for $d\ge 3$.