Data structures for range-aggregate extent queries

Abstract

A fundamental and well-studied problem in computational geometry is range searching, where the goal is to preprocess a set, S, of geometric objects (e.g., points in the plane) so that the subset S ′ ⊆ S that is contained in a query range (e.g., an axes-parallel rectangle) can be reported efficiently. However, in many situations, what is of interest is to generate a more informative "summary" of the output, obtained by applying a suitable aggregation function on S ′ . Examples of such aggregation functions include count, sum, min, max, mean, median, mode, and top-k that are usually computed on a set of weights defined suitably on the objects. Such range-aggregate query problems have been the subject of much recent research in both the database and the computational geometry communities.In this paper, we further generalize this line of work by considering aggregation functions on point-sets that measure the extent or "spread" of the objects in the retrieved set S ′ . The functions considered here include closest pair, diameter, and width. The challenge here is that these aggregation functions (unlike, say, count) are not efficiently decomposable in the sense that the answer to S ′ cannot be inferred easily from answers to subsets that induce a partition of S ′ . Nevertheless, we have been able to obtain space-and query-time-efficient solutions to several such problems including: closest pair queries with axes-parallel rectangles on point sets in the plane and on random point-sets in R d (d ≥ 2), closest pair queries with disks on random point-sets in the plane, diameter queries on point-sets in the plane, and guaranteedquality approximations for diameter and width queries in the plane. Our results are based on a combination of geometric techniques, including multilevel range trees, Voronoi Diagrams, Euclidean Minimum Spanning Trees, sparse representations of candidate outputs, and proofs of (expected) upper bounds on the sizes of such representations.

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.…”

“…Beyond R 2 , the problem is quite open. To our best knowledge, the only known results are the orthogonal RCP data structure given by Gupta et al [9] which only has guaranteed average-case performance and the approximate colored RCP data structures given by Xue [19] which can only handle restricted query types (dominance query in R 3 and 2-sided box query in R d ).…”

“…Then the data structure computes the closest pair φ in K using some pre-stored information and computes the closest pair φ in L using the standard closest-pair algorithm (which can be done efficiently as L is small). If the two points of the closest pair φ * in S ∩ Q are both in K or both in 1 Gupta et al [9] obtainedÕ( √ n) query time for two-dimensional disks, but only for uniformly distributed point sets; the general problem was left open in their paper. L, we are done.…”

Finding Pairwise Intersections Inside a Query Range

“…Their data structure needs O(n log 3 n) space and is able to answer the desired queries in O(log 2 n + #answers) time. Abam et al [1], Gupta [16], and Gupta et al [17] have presented data structures that return the closest pair inside a query range.…”