Approximate range closest-pair search

The range closest-pair (RCP) problem, as a range-search version of the classical closest-pair problem, aims to store a dataset of points in some data structure such that whenever a query range Q is given, the closest-pair inside Q can be reported efficiently. This paper studies an approximate version of the RCP problem in which the answer pair is allowed to be “approximately” contained in the query range. A general reduction from the approximate RCP problem to the range-minimum and range-reporting problems is given, which works for a general class of query spaces. The reduction is applied to obtain efficient approximate RCP data structures for disk queries in ℝ² and ball queries in higher dimensions. Finally, the paper also shows that for orthogonal queries, the approximate RCP problem is (asymptotically) at least as hard as the orthogonal range-minimum problem.