Approximate range closest-pair queries

Searching for the closest-pair in a query translate

“…The closest-pair problem and range search are both classical topics; some surveys can be found in [3,15]. The RCP problem in R 2 has been studied in prior work [1,4,8,9,13,14,16,17,18]. State-of-the-art RCP data structures for quadrant, strip, rectangle, and halfplane queries were given in the recent work [18].…”

“…The RCP problem in R 2 has been well-studied over years [1,4,8,9,13,14,16,17,18]. Despite of much effort, the query ranges considered are still restricted to very simple shapes, typically orthogonal rectangles and halfplanes.…”

Closest-pair queries in fat rectangles

“…Xue et al [13] introduced an approximate version of the range closest pair problem and gave a solution for the d-dimensional case, for any constant dimension d ≥ 1. The results in [13] imply that a data structure of size O(n log d−1 n) can be constructed in O(n log d−1 n) time, such that the following holds: For any axes-parallel query rectangle R in R d and any query value > 0, a pair p, q of distinct points in S can be computed, such that (i) p ∈ R, (ii) q is contained in the expanded rectangle obtained by scaling R by a factor of 1 + with respect to its center, and (iii) the distance between p and q is at most the closest-pair distance in R ∩ S. Such a pair p, q can be computed in O(log d−1 n + (f / ) d log(f / )) time, where f denotes the aspect ratio of the query rectangle R. Observe that, since the point q can be outside of R, the distance between p and q may be much smaller than the closest-pair distance in R ∩ S.…”

Range Closest-Pair Search in Higher Dimensions

“…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.…”

“…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.…”