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Orthogonal range searching on the RAM, revisited. (English) Zbl 1283.68139
Proceedings of the 27th annual symposium on computational geometry, SoCG 2011, Paris, France, June 13–15, 2011. New York, NY: Association for Computing Machinery (ACM) (ISBN 978-1-4503-0682-9). 1-10 (2011).
Summary: We present a number of new results on one of the most extensively studied topics in computational geometry, orthogonal range searching. All our results are in the standard word RAM model:
We present two data structures for 2-d orthogonal range emptiness. The first achieves $$O(n \lg \lg n)$$ space and $$O(\lg \lg n)$$ query time, assuming that the $$n$$ given points are in rank space. This improves the previous results by S. Alstrup et al. [“New data structures for orthogonal range searching”, in: Proceedings of the 41st annual symposium on foundations of computer science, FOCS ’00. Washington, DC: IEEE Computer Society. 198–207 (2000)], with $$O(n \lg^{\epsilon}n)$$ space and $$O(\lg \lg n)$$ query time, or with $$O(n\lg\lg n)$$ space and $$O(\lg^{2}\lg n)$$ query time.
Our second data structure uses $$O(n)$$ space and answers queries in $$O(\lg^{\epsilon} n)$$ time. The best previous $$O(n)$$-space data structure, due to Y. Nekrich [Lect. Notes Comput. Sci. 4619, 15–26 (2007; Zbl 1209.68162)], answers queries in $$O(\lg n/\lg\lg n)$$ time. We give a data structure for 3-d orthogonal range reporting with $$O(n \lg^{1+{\epsilon}} n)$$ space and $$O(\lg\lg n + k)$$ query time for points in rank space, for any constant $${\epsilon}>0$$. This improves the previous results in [P. Afshani, Lect. Notes Comput. Sci. 5193, 41–51 (2008; Zbl 1158.68363); M. Karpinski and Y. Nekrich, Lect. Notes Comput. Sci. 5609, 215–224 (2009; Zbl 1248.68524); T. M. Chan, “Persistent predecessor search and orthogonal point location on the word RAM”, in: Proceedings of the twenty-second annual ACM-SIAM symposium on discrete algorithms, SODA ’11. New York, NY: ACM Press. 1131–1145 (2011)], with $$O(n \lg^{3} n)$$ space and $$O(\lg\lg n + k)$$ query time, or with $$O(n \lg^{1+\epsilon}n)$$ space and $$O(\lg^{2}\lg n + k)$$ query time. Consequently, we obtain improved upper bounds for orthogonal range reporting in all constant dimensions above 3.
Our approach also leads to a new data structure for 2D orthogonal range minimum queries with $$O(n\lg^{\epsilon} n)$$ space and $$O(\lg \lg n)$$ query time for points in rank space.
3. We give a randomized algorithm for 4-d offline dominance range reporting/emptiness with running time $$O(n \log n)$$ plus the output size. This resolves two open problems (both appeared in F. P. Preparata and M. I. Shamos’s seminal book [Computational geometry. An introduction. New York etc.: Springer-Verlag (1985; Zbl 0575.68059; Zbl 0759.68037)]):
(a) Given a set of $$n$$ axis-aligned rectangles in the plane, we can report all $$k$$ enclosure pairs (i.e., pairs $$(r_{1},r_{2})$$ where rectangle $$r_{1}$$ completely encloses rectangle $$r_{2}$$) in $$O(n \lg n + k)$$ expected time;
(b) Given a set of $$n$$ points in 4-d, we can find all maximal points (points not dominated by any other points) in $$O(n \lg n)$$ expected time.
The most recent previous development on (a) was reported back in SoCG’95 by P. Gupta et al. [Int. J. Comput. Geom. Appl. 7, No. 5, 437–455 (1997; Zbl 0888.68066)], whose main result was an $$O([n \lg n + k] \lg \lg n)$$ algorithm. The best previous result on (b) was an $$O(n \lg n \lg \lg n)$$ algorithm due to H. N. Gabow [“Scaling and related techniques for geometry problems”, in: Proceedings of the sixteenth annual ACM symposium on theory of computing, STOC ’84. New York, NY: ACM Press. 135–143 (1984)]. As a consequence, we also obtain the current-record time bound for the maxima problem in all constant dimensions above.
For the entire collection see [Zbl 1271.68010].
Reviewer: Reviewer (Berlin)

##### MSC:
 68P10 Searching and sorting 68P05 Data structures 68U05 Computer graphics; computational geometry (digital and algorithmic aspects) 68Q17 Computational difficulty of problems (lower bounds, completeness, difficulty of approximation, etc.)
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