Discrete & Computational Geometry

, Volume 5, Issue 5, pp 449–483

Partitioning arrangements of lines I: An efficient deterministic algorithm

  • Pankaj K. Agarwal
Article

Abstract

In this paper we consider the following problem: Given a set ℒ ofn lines in the plane, partition the plane intoO(r2) triangles so that no triangle meets more thanO(n/r) lines of ℒ. We present a deterministic algorithm for this problem withO(nr logn/r) running time, whereω is a constant <3.33.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [A]
    P. K. Agarwal, Partitioning arrangements of lines, II: Applications,Discrete and Computational Geometry 5 (1990), 533–574.MathSciNetCrossRefMATHGoogle Scholar
  2. [AS]
    P. K. Agarwal and M. Sharir, Red-blue intersection detection algorithms with applications to motion planning and collision detection,SIAM Journal on Computing 19 (1990), 297–322.MathSciNetCrossRefMATHGoogle Scholar
  3. [AKS]
    M. Ajtai, J. Komlos, and E. Szemerédi, Sorting inc logn parallel steps,Proceedings of the 15th Annual ACM Symposium on Theory of Computing, 1983, pp. 1–9.Google Scholar
  4. [B]
    K. E. Batcher, Sorting networks and their applications,Proceedings of the AFIPS Spring Joint Summer Computer Conference, vol. 32 (1968), pp. 307–314.Google Scholar
  5. [CF]
    B. Chazelle and J. Friedman, A deterministic view of random sampling and its use in geometry,Proceedings of the 29th Annual IEEE Symposium on Foundations of Computer Science, 1988, pp. 539–549.Google Scholar
  6. [Cl1]
    K. Clarkson, A probabilistic algorithm for the post office problem,Proceedings of the 17th Annual ACM Symposium on Theory of Computing, 1985, pp. 75–84.Google Scholar
  7. [Cl2]
    K. Clarkson, New applications of random sampling in computational geometry,Discrete and Computational Geometry 2 (1987), 195–222.MathSciNetCrossRefMATHGoogle Scholar
  8. [Cl3]
    K. Clarkson, Applications of random sampling in computational geometry, II,Proceedings of the 4th Annual Symposium on Computational Geometry, 1988, pp. 1–11.Google Scholar
  9. [CS]
    K. Clarkson and P. Shor, Algorithms for diametric pairs and convex hulls that are optimal, randomized and incremental,Proceedings of the 4th Annual Symposium on Computational Geometry, 1988, pp. 12–17.Google Scholar
  10. [CTV]
    K. Clarkson, R. E. Tarjan, and C. J. Van Wyk, A fast Las Vegas algorithm for triangulating a simple polygon,Discrete and Computational Geometry 4 (1989), 423–432.MathSciNetCrossRefMATHGoogle Scholar
  11. [Co]
    R. Cole, Slowing down sorting networks to obtain faster sorting algorithms,Journal of the Association for Computing Machinery 31 (1984), 200–208.Google Scholar
  12. [CSSS]
    R. Cole, J. Salowe, W. Steiger, and E. Szemerédi, An optimal-time algorithm for slope selection,SIAM Journal on Computing 16 (1989), 792–810.CrossRefGoogle Scholar
  13. [CSY]
    R. Cole, M. Sharir, and C. K. Yap, Onk-hulls and related problems,SIAM Journal on Computing 16 (1987), 61–77.MathSciNetCrossRefMATHGoogle Scholar
  14. [E]
    H. Edelsbrunner,Algorithms in Combinatorial Geometry, Springer-Verlag, Heidelberg, 1987.CrossRefMATHGoogle Scholar
  15. [EGH*]
    H. Edelsbrunner, L. Guibas, J. Hershberger, R. Seidel, M. Sharir, J. Snoeyink, and E. Welzl, Implicitly representing arrangements of lines or segments,Discrete and Computational Geometry 4 (1989), 433–466.MathSciNetCrossRefMATHGoogle Scholar
  16. [EGS]
    H. Edelsbrunner, L. Guibas, and M. Sharir, The complexity and construction of many faces in arrangements of lines or of segments,Discrete and Computational Geometry 5 (1990), 161–196.MathSciNetCrossRefMATHGoogle Scholar
  17. [EW]
    H. Edelsbrunner and E. Welzl, Constructing belts in two-dimensional arrangements with applications,SIAM Journal on Computing 15 (1986), 271–284.MathSciNetCrossRefMATHGoogle Scholar
  18. [GOS]
    L. Guibas, M. Overmars, and M. Sharir, Ray shooting, implicit point location, and related queries in arrangements of segments, Technical Report 433, Dept. Computer Science, New York University, March 1989.Google Scholar
  19. [HW]
    D. Haussler and E. Welzl,ɛ-nets and simplex range queries,Discrete and Computational Geometry 2 (1987), 127–151.MathSciNetCrossRefMATHGoogle Scholar
  20. [Ma]
    J. Matoušek, Construction ofɛ-nets,Discrete and Computational Geometry, this issue, 427–448.Google Scholar
  21. [Me]
    N. Megiddo, Applying parallel computation algorithms in design of serial algorithms,Journal of the Association of Computing Machinery 30 (1983), 852–865.MathSciNetCrossRefMATHGoogle Scholar
  22. [RS1]
    J. Reif and S. Sen, Optimal randomized parallel algorithms for computational geometry,Proceedings of the 16th International Conference on Parallel Processing, 1987, pp. 270–277.Google Scholar
  23. [RS2]
    J. Reif and S. Sen, Polling: A new randomized sampling technique for computational geometry,Proceedings of the 21st Annual ACM Symposium on Theory of Computing, 1989, pp. 394–404.Google Scholar
  24. [S]
    S. Suri, A linear algorithm for minimum link paths inside a simple polygon,Computer Vision, Graphics and Image Processing 35 (1986), 99–110.CrossRefMATHGoogle Scholar
  25. [We]
    E. Welzl, More onk-sets of finite sets in the plane,Discrete and Computational Geometry 1 (1986), 83–94.MathSciNetCrossRefGoogle Scholar
  26. [Wo]
    G. Woeginger, Epsilon-nets for half planes, Technical Report B-88-02, Dept. of Mathematics, Free University, Berlin, March 1988.Google Scholar

Copyright information

© Springer-Verlag New York Inc. 1990

Authors and Affiliations

  • Pankaj K. Agarwal
    • 1
  1. 1.Courant Institute of Mathematical SciencesNew York UniversityUSA
  2. 2.Department of Computer ScienceDuke UniversityDurhamUSA

Personalised recommendations