New bounds for range closest-pair problems

TL;DR

This paper introduces new data structures for the range closest-pair problem in computational geometry, providing improved bounds for various query types through worst-case and average-case analyses.

Contribution

It proposes novel data structures for RCP queries over quadrants, strips, rectangles, and halfplanes, with significant improvements and new bounds over previous results.

Findings

Improved bounds for RCP queries in quadrants and rectangles.

New data structures for RCP in strips and halfplanes.

Both worst-case and average-case analyses demonstrate enhanced performance.

Abstract

Given a dataset $S$ of points in $\mathbb{R}^2$, the range closest-pair (RCP) problem aims to preprocess $S$ into a data structure such that when a query range $X$ is specified, the closest-pair in $S \cap X$ can be reported efficiently. The RCP problem can be viewed as a range-search version of the classical closest-pair problem, and finds applications in many areas. Due to its non-decomposability, the RCP problem is much more challenging than many traditional range-search problems. This paper revisits the RCP problem, and proposes new data structures for various query types including quadrants, strips, rectangles, and halfplanes. Both worst-case and average-case analyses (in the sense that the data points are drawn uniformly and independently from the unit square) are applied to these new data structures, which result in new bounds for the RCP problem. Some of the new bounds significantly improve the previous results, while the others are entirely new.