# Range closest-pair search in higher dimensions

## TL;DR

This paper extends range closest-pair search data structures to higher dimensions, providing new solutions for various query types and establishing lower bounds for orthogonal queries in dimensions three and above.

## Contribution

It introduces the first nontrivial data structures for RCP in higher dimensions across multiple query types and proves a conditional lower bound for orthogonal RCP search.

## Key findings

- Developed data structures for RCP in higher dimensions for orthogonal, simplex, halfspace, and ball queries.
- Established a conditional lower bound for orthogonal RCP search in dimensions d ≥ 3.
- Extended the scope of RCP search beyond the plane to higher-dimensional spaces.

## 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 $d \geq 3$.