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Latin American Symposium on Theoretical Informatics

LATIN 2008: LATIN 2008: Theoretical Informatics pp 687–698Cite as

I/O-Efficient Point Location in a Set of Rectangles

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Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 4957))

Abstract

In this paper we present an external memory data structure for point location queries in a set of d-dimensional rectangles. Our data structure uses O(N/B) blocks of space and supports point location queries in \(O(\log^{d-1}_B N )\) I/Os, where N is the number of rectangles and B is the block size.

We also present a O((N/B)log B N) space data structure that supports point location queries in a two-dimensional rectangular subdivision of a U×U grid in \(O(\log_2\log_B U+ (\log_2\log_B N)^2)\) I/Os and a \(O((N/B)\log^2_B N)\) space data structure that supports point location queries in a three- dimensional rectangular subdivision in O(log B N) I/Os. As an application of our result, we describe a data structure for three-dimensional orthogonal range reporting queries on a grid of size U with \(O(\log_2\log_B U + (\log_2\log_B N)^2 +T/B)\) I/O operations per query, where T is the number of points in the answer.

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References

  1. Agarwal, P.K., Arge, L., Brodal, G.S., Vitter, J.S.: I/O-Efficient Dynamic Point Location in Monotone Planar Subdivisions. In: Proc. SODA, pp. 11–20 (1999)

    Google Scholar 

  2. Agarwal, P.K., Arge, L., Yi, K.: An Optimal Dynamic Interval Stabbing-Max Data Structure? In: Proc. SODA, pp. 803–812 (2005)

    Google Scholar 

  3. Aggarwal, A., Vitter, J.S.: The Input/Output Complexity of Sorting and Related Problems. Communications of the ACM 31(9), 1116–1127 (1988)

    Article  MathSciNet  Google Scholar 

  4. Arge, L.: External Memory Data Structures. In: Meyer auf der Heide, F. (ed.) ESA 2001. LNCS, vol. 2161, pp. 1–29. Springer, Heidelberg (2001)

    Chapter  Google Scholar 

  5. Arge, L., Procopiuc, O., Ramaswamy, S., Suel, T., Vitter, J.S.: Theory and Practice of I/O-Efficient Algorithms for Multidimensional Batched Searching Problems (Extended Abstract). In: Proc. SODA, pp. 685–694 (1998)

    Google Scholar 

  6. Arge, L., Vengroff, D.E., Vitter, J.S.: External-Memory Algorithms for Processing Line Segments in Geographic Information Systems. Algorithmica 47(1), 1–25 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  7. Arge, L., Vahrenhold, J.: I/O-Efficient Dynamic Planar Point Location. Computational Geometry 29(2), 147–162 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  8. Bender, M.A., Cole, R., Raman, R.: Exponential Structures for Efficient Cache-Oblivious Algorithms. In: Widmayer, P., Triguero, F., Morales, R., Hennessy, M., Eidenbenz, S., Conejo, R. (eds.) ICALP 2002. LNCS, vol. 2380, pp. 195–207. Springer, Heidelberg (2002)

    Chapter  Google Scholar 

  9. de Berg, M., van Kreveld, M.J., Snoeyink, J.: Two- and Three-Dimensional Point Location in Rectangular Subdivisions. J. Algorithms 18(2), 256–277 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  10. Chazelle, B., Guibas, L.J.: Fractional Cascading: I. A Data Structuring Technique. Algorithmica 1, 133–162 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  11. Crauser, A., Ferragina, P., Mehlhorn, K., Meyer, U., Ramos, E.A.: Randomized External-Memory Algorithms for Line Segment Intersection and Other Geometric Problems. Int. J. Comput. Geometry Appl. 11(3), 305–337 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  12. Edelsbrunner, H., Haring, G., Hilbert, D.: Rectangular point location in d dimensions with applications. Computer J 29, 76–82 (1986)

    Article  MathSciNet  Google Scholar 

  13. van Emde Boas, P., Kaas, R., Zijlstra, E.: Design and Implementation of an Efficient Priority Queue. Mathematical Systems Theory 10, 99–127 (1977)

    Article  MATH  Google Scholar 

  14. Frigo, M., Leiserson, C.E., Prokop, H., Ramachandran, S.: Cache-Oblivious Algorithms. In: Proc. FOCS 1999, pp. 285–298 (1999)

    Google Scholar 

  15. Gabow, H., Bentley, J.L., Tarjan, R.E.: Scaling and related techniques for geometry problems. In: Proc. STOC 1984, pp. 135–143 (1984)

    Google Scholar 

  16. Goodrich, M.T., Tsay, J.-J., Vengroff, D.E., Vitter, J.S.: External-Memory Computational Geometry (Preliminary Version). In: Proc. FOCS 1993, pp. 714–723 (1993)

    Google Scholar 

  17. Nekrich, Y.: A Data Structure for Multi-Dimensional Range Reporting. In: Proc. SoCG 2007, pp. 344–353 (2007)

    Google Scholar 

  18. Nekrich, Y.: External Memory Range Reporting on a Grid. In: Tokuyama, T. (ed.) ISAAC 2007. LNCS, vol. 4835, Springer, Heidelberg (2007)

    Chapter  Google Scholar 

  19. Pǎtraşcu, M., Thorup, M.: Time-space Trade-offs for Predecessor Search. In: Proc. STOC 2006, pp. 232–240 (2006)

    Google Scholar 

  20. Smid, M.: Dynamic rectangular point location, with an application to the closest pair problem. Information and Computation 116, 1–9 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  21. Snoeyink, J.: Point Location. In: Handbook of Discrete and Computational Geometry, CRC Press, Inc., Boca Raton (1997)

    Google Scholar 

  22. Subramanian, S., Ramaswamy, S.: The P-range Tree: A New Data Structure for Range Searching in Secondary Memory. In: Proc. SODA 1995, pp. 378–387 (1995)

    Google Scholar 

  23. Vengroff, D.E., Vitter, J.S.: Efficient 3-D Range Searching in External Memory. In: Proc. STOC 1996, pp. 192–201 (1996)

    Google Scholar 

  24. Vitter, J.S.: External Memory Algorithms and Data Structures: Dealing with Massive Data. ACM Computing Surveys 33(2), 209–271 (2001)

    Article  Google Scholar 

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Authors and Affiliations

  1. Dept. of Computer Science, University of Bonn,  

    Yakov Nekrich

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  1. Yakov Nekrich
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Editor information

Eduardo Sany Laber Claudson Bornstein Loana Tito Nogueira Luerbio Faria

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Nekrich, Y. (2008). I/O-Efficient Point Location in a Set of Rectangles. In: Laber, E.S., Bornstein, C., Nogueira, L.T., Faria, L. (eds) LATIN 2008: Theoretical Informatics. LATIN 2008. Lecture Notes in Computer Science, vol 4957. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-78773-0_59

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  • DOI: https://doi.org/10.1007/978-3-540-78773-0_59

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-78772-3

  • Online ISBN: 978-3-540-78773-0

  • eBook Packages: Computer ScienceComputer Science (R0)

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