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On range searching with semialgebraic sets

  • Pankaj K. Agarwal
  • Jiří Matoušek
Invited Lectures
Part of the Lecture Notes in Computer Science book series (LNCS, volume 629)

Abstract

Let P be a set of n points in ℝd (d a small fixed positive integer), and let Γ be a collection of subsets of ℝd, each of which is defined by a constant number of bounded degree polynomials. The Γ-range searching problem is defined as: Preprocess P into a data structure, so that all points of P lying in a given γ Γ can be counted (or reported) efficiently. Generalizing the simplex range searching techniques, we construct a data structure for Γ-range searching with nearly linear space and preprocessing time, which can answer a query in time O(n 1−1/b+δ ), where d≤b≤ 2d−3 and δ>0 is an arbitrarily small constant. The actual value of b is related to the problem of partitioning arrangements of algebraic surfaces into constant-complexity cells.

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References

  1. [AHL90]
    A. Aggarwal, M. Hansen, and T.Leighton. Solving query-retrieval problems by compacting Voronoi diagrams. Proc. 21st ACM Symposium on Theory of Computing, 1990, 331–340.Google Scholar
  2. [AHWW87]
    N. Alon, D. Haussler, E. Welzl, and G. Wöginger. Partitioning and geometric embedding of range spaces of finite Vapnik-Chervonenkis dimension. In Proc. 3. ACM Symposium on Computational Geometry, pages 331–340, 1987.Google Scholar
  3. [APS91]
    B. Aronov, M. Pellegrini, and M. Sharir, On the zone of a surface in a hyperplane arrangement. Discrete & Computational Geometry, to appear.Google Scholar
  4. [AS91]
    P. K. Agarwal and M. Sharir. Applications of a new space partitioning scheme. In Proc. 2. Workshop on Algorithms and Data Structures, 1991.Google Scholar
  5. [CEGS89]
    B. Chazelle, H. Edelsbrunner, L. Guibas, and M. Sharir. Point-location in real-algebraic varieties and its applications. In Proc. 16th International Colloquium on Automata, Languages and Programming, pages 179–192, 1989.Google Scholar
  6. [CF90]
    B. Chazelle and J. Friedman. A deterministic view of random sampling and its use in geometry. Combinatorica, 10(3):229–249, 1990.MATHMathSciNetCrossRefGoogle Scholar
  7. [Cha89]
    B. Chazelle. Lower bounds on the complexity of polytope range searching. J. Amer. Math. Soc, 2(4):637–666, 1989.MATHMathSciNetCrossRefGoogle Scholar
  8. [Cha91]
    B. Chazelle. Cutting hyperplanes for divide-and-conquer. Tech. report CS-TR-335-91, Princeton University, 1991. Preliminary version: Proc. 32. IEEE Symposium on Foundations of Computer Science, October 1991.Google Scholar
  9. [CS89]
    K. L. Clarkson and P. Shor. New applications of random sampling in computational geometry II. Discrete & Computational Geometry, 4:387–421, 1989.MATHMathSciNetCrossRefGoogle Scholar
  10. [CSW90]
    B. Chazelle, M. Sharir, and E. Welzl. Quasi-optimal upper bounds for simplex range searching and new zone theorems. In Proc. 6. ACM Symposium on Computational Geometry, pages 23–33, 1990.Google Scholar
  11. [CW89]
    B. Chazelle and E. Welzl. Quasi-optimal range searching in spaces of finite VC-dimension. Discrete & Computational Geometry, 4:467–490, 1989.MATHMathSciNetCrossRefGoogle Scholar
  12. [DE87]
    D. Dobkin and H. Edelsbrunner. Space searching for intersecting objects. Journal of Algorithms, 8:348–361, 1987.MATHMathSciNetCrossRefGoogle Scholar
  13. [Ede87]
    H. Edelsbrunner. Algorithms in Combinatorial Geometry. Springer-Verlag, Berlin-Heidelberg-New York, 1987.MATHGoogle Scholar
  14. [HW87]
    D. Haussler and E. Welzl. ε-nets and simplex range queries. Discrete & Computational Geometry, 2:127–151, 1987.MATHMathSciNetCrossRefGoogle Scholar
  15. [KPW92]
    J. Komlós, J. Pach, and G. Wöginger. Almost tight bounds for epsilon-nets. Discrete & Computational Geometry, 1992. To appear.Google Scholar
  16. [Mat91a]
    J. Matoušek. Approximations and optimal geometric divide-and-conquer. In Proc. 23. ACM Symposium on Theory of Computing, pages 506–511, 1991.Google Scholar
  17. [Mat91b]
    J. Matoušek. Cutting hyperplane arrangements. Discrete & Computational Geometry, 6(5):385–406, 1991.MathSciNetGoogle Scholar
  18. [Mat91c]
    J. Matoušek. Efficient partition trees. In Proc. 7. ACM Symposium on Computational Geometry, pages 1–9, 1991. Also to appear in Discrete & Computational Geometry.Google Scholar
  19. [Mat91d]
    J. Matoušek. Reporting points in halfspaces. Proc. 32nd IEEE Symposium on Foundations of Computer Science, 1991, pp. 207–215.Google Scholar
  20. [Mat92]
    J. Matoušek. Range searching with efficient hierarchical cuttings. In Proc. 8. ACM Symposium on Computational Geometry, 1992. To appear.Google Scholar
  21. [PS85]
    F. Preparata and M. I. Shamos. Computational Geometry — An Introduction. Springer-Verlag, 1985.Google Scholar
  22. [Wil82]
    D. E. Willard. Polygon retrieval. SIAM Journal on Computing, 11:149–165, 1982.MATHMathSciNetCrossRefGoogle Scholar
  23. [YY85]
    F. F. Yao and A. C. Yao. A general approach to geometric queries. In Proc. 17. ACM Symposium on Theory of Computing, pages 163–168, 1985.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1992

Authors and Affiliations

  • Pankaj K. Agarwal
    • 1
  • Jiří Matoušek
    • 2
    • 3
  1. 1.Computer Science DepartmentDuke UniversityDurham
  2. 2.Katedra aplikované matamatikyUniversita KarlovaPraha
  3. 3.Institut für InformatikFreie Universität BerlinGermany

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