Improved Bounds for Orthogonal Point Enclosure Query and Point Location in Orthogonal Subdivisions in ℝ³

Abstract

In this paper, new results for two fundamental problems in the field of computational geometry are presented: orthogonal point enclosure query (OP EQ) in R 3 and point location in orthogonal subdivisions in R 3 . All the results are in the pointer machine model of computation.(1) In an orthogonal point enclosure query, a set S of n axes-parallel rectangles in R 3 is to be preprocessed, so that given a query point q ∈ R 3 , one can efficiently report all the rectangles in S containing (or stabbed by) q. When rectangles are 3-sided (of the form (−∞, x] × (−∞, y] × (−∞, z]), there exists an optimal solution of Afshani (ESA'08) which uses O(n) space and answers the query in O(log n + k) time, where k is the number of rectangles reported. Unfortunately, when the rectangles are 4-sided (of the form, the best result one can achieve using existing techniques is O(n) space and O(log 2 n + k) query time.The key result of this work is an almost optimal solution for 4-sided rectangles. The first data structure uses O(n log * n) space and answers the query in O(log n + k) time. Here log * n is the iterated logarithm of n. The second data structure uses O(n) space and answers the query in O(log n • log (i) n + k) time, for any constant integer i ≥ 1. Here log (1) n = log n and log (i) n = log(log (i−1) n) when i > 1.To handle OP EQ for general 6-sided rectangles (of the form), existing structures in the literature occupy Ω(n log n) space. This work presents the first known near-linear space data structure. It occupies O(n log * n) space and answer the query in O(log 2 n • log log n + k) time. This is almost optimal, since Afshani, Arge and Larsen (SoCG'10 and SoCG'12) proved that with O(n) space, OP EQ on 6-sided rectangles takes Ω(log 2 n + k) time.(2) In point location in orthogonal subdivisions, a set S of n non-overlapping axes-parallel rectangles in * This research was partly supported by a Doctoral Dissertation Fellowship (DDF) from the Graduate School of University of Minnesota.

Orthogonal Point Location and Rectangle Stabbing Queries in 3-d

“…In 2-d this question has been successfully resolved: there exists a linear-space structure with O(log n) query time [26,25,16,36,39] (actually this result holds for nonorthogonal point location). In 3-d there has been work on this problem [17,23,2,32], but the question has not yet been resolved. The currently best known result on the pointer machine model is a linear-space structure with O(log 3/2 n) query time by Rahul [32].…”

“…In 3-d there has been work on this problem [17,23,2,32], but the question has not yet been resolved. The currently best known result on the pointer machine model is a linear-space structure with O(log 3/2 n) query time by Rahul [32]. In this paper,…”

Finding Pairwise Intersections Inside a Query Range

“…Type (i) is easy to handle without using seed sets: we simply store O in a data structure for 3-dimensional point-enclosure queries [19], which allows us to report all boxes b i ∈ O containing a query point in O(log 2 n · log log n + #answers) time. If we query this structure with a corner q of Q and report all pairs of boxes containing q then we have found all intersecting pairs of Type (i).…”

“…Remark The query bound in Lemma 2 can be improved to O(log 2 n + k) at the cost of O(n log n) storage, by using the data structure of Afshani et al [3] instead of that of Rahul [19].…”

Approximate Range Counting Revisited

“…For the reporting version of 5-sided rectangle stabbing in R 3 problem, Rahul [37] presented a structure of size O(n log * n) which can answer a query in O(log n • log log n + k) time. Build this structure on all the rectangles in set S. Given a query point q, query the structure till all the rectangles in S ∩ q have been reported or Cε −2 log n • log log n + 1 rectangles in S ∩ q have been reported.…”