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

In this paper, new results for two fundamental problems in the field of computational geometry are presented: orthogonal point enclosure query (OPEQ) in R and point location in orthogonal subdivisions in R. 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 is to be preprocessed, so that given a query point q ∈ R, 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 [x1, x2] × (−∞, y] × (−∞, z]), the best result one can achieve using existing techniques is O(n) space and O(log 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 n + k) time, for any constant integer i ≥ 1. Here log n = log n and log n = log(log(i−1) n) when i > 1. To handle OPEQ for general 6-sided rectangles (of the form [x1, x2]× [y1, y2]× [z1, z2]), 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 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, OPEQ on 6-sided rectangles takes Ω(log 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. †Department of Computer Science and Engineering, University of Minnesota, Twin Cities [email protected] R(d ≥ 3) is to be preprocessed, so that given a query point q ∈ R, one can efficiently report the rectangle in S containing q. In a pointer machine model, the first known solution by Edelsbrunner, Haring and Hilbert in 1986 uses O(n) space and answers the query in O(logd−1 n) query time. After a long gap, Afshani, Arge and Larsen (SoCG’10) improved the query time to O(log n(log n/ log log n)d−2). This work presents a data structure which occupies O(n) space and answers the query in O(logd−3/2 n) time, improving the previously best known query time by roughly a √ log n factor.