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 ∩ Q can be reported quickly. RCP search has received attention over years, but the primary focus was only on 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 R d for any constant d. Furthermore, we prove a conditional lower bound for orthogonal RCP search for d ≥ 3.
IntroductionThe closest-pair problem is one of the most fundamental problems in computational geometry and finds numerous applications in various areas, such as collision detection, traffic control, etc. In many scenarios, instead of finding the global closest-pair, people want to know the closest pair contained in some specified ranges. This results in the notion of range closest-pair (RCP) search. 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 ∩ Q can be reported quickly. RCP search has received considerable attention over the years [1,4,9,10,16,17,20,19,21,22].Unlike most traditional range-search problems, RCP search is non-decomposable. That is, if we partition the dataset S into S 1 and S 2 , given a query range Q, the closest pair in S ∩ Q cannot be obtained efficiently from the closest pairs in S 1 ∩ Q and S 2 ∩ Q. Due to the non-decomposability, many traditional range-search techniques are inapplicable to RCP search, which makes the problem quite challenging. As such, despite of much effort made on this topic, most known results are restricted to the plane case, i.e., RCP search in R 2 . Beyond R 2 , only very specific query types have been studied, such as 2-sided box queries.In this paper, we investigate RCP search in higher dimensions. We consider four widely-studied query types: orthogonal queries, simplex queries, halfspace queries, and ball queries. We are interested in designing efficient RCP data structures (in terms of space cost, query time, and preprocessing time) for these kinds of query ranges, and proving conditional lower bounds for these problems.Related work. The closest-pair problem and range search are both well-studied problems in computational geometry; see [2,18] for surveys of these two topics.RCP search was for the first time introduced by Shan et al. [16] and subsequently studied in [1,4,9,10,17,20,19,21,22]. In R 2 , the query types studied include quadrants, strips, rectangles, and halfplanes. RCP search with these query ranges can be solved using near-linear space with poly-logarithmic query time. The best known data structures were given by Xue et al. [21], and we summarize the bounds in Table 1. For fat rectangles queries (i.e., rectangles of constant aspect ratio), Bae and S...
