Closest-Pair Queries and Minimum-Weight Queries are Equivalent for Squares

TL;DR

This paper proves the equivalence of closest-pair and minimum-weight range queries for squares in the plane, leading to new efficient data structures for these problems.

Contribution

It establishes the equivalence of two geometric query problems for square ranges and develops new data structures based on this insight.

Findings

Closest-pair and minimum-weight queries are equivalent for square ranges.

New data structures for range closest pair queries with squares are introduced.

The equivalence holds for data structures with query times of at least logarithmic complexity.

Abstract

Let $S$ be a set of $n$ weighted points in the plane and let $R$ be a query range in the plane. In the range closest pair problem, we want to report the closest pair in the set $R \cap S$. In the range minimum weight problem, we want to report the minimum weight of any point in the set $R \cap S$. We show that these two query problems are equivalent for query ranges that are squares, for data structures having $\Omega(\log n)$ query times. As a result, we obtain new data structures for range closest pair queries with squares.