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.

Let S be a set of n points in R d , where each point has t ≥ 1 real-valued attributes called features. A range-skyline query on S takes as input a query box q ∈ R d and returns the skyline of the points of q ∩ S, computed w.r.t. their features (not their coordinates in R d ). Efficient algorithms are given for computing range-skylines and a related hardness result is established.

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