Closest-pair queries and minimum-weight queries are equivalent for squares

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∩S. In the range minimum weight problem, we want to report the minimum weight of any point in the set R∩S. We show that these two query problems are equivalent for query ranges that are squares, for data structures having Ω(log⁡n) query times. As a result, we obtain new data structures for range closest pair queries with squares.