Discrete & Computational Geometry

, Volume 6, Issue 3, pp 307–338 | Cite as

On levels in arrangements and voronoi diagrams

  • Ketan Mulmuley


This paper gives efficient, randomized algorithms for the following problems: (1) construction of levels of order 1 tok in an arrangement of hyperplanes in any dimension and (2) construction of higher-order Voronoi diagrams of order 1 tok in any dimension. A new combinatorial tool in the form of a mathematical series, called a θ series, is associated with an arrangement of hyperplanes inR d . It is used to study the combinatorial as well as algorithmic complexity of the geometric problems under consideration.


Voronoi Diagram Computational Geometry Horizontal Edge Visibility Distance Valid Path 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. 1.
    A. Agarwal, L. Guibas, S. Saxe, P. Shor, A linear time algorithm for computing the Voronoi diagram of a convex polygon,Proceedings of the Annual ACM Symposium on Theory of Computing, 1987, pp. 39–45.Google Scholar
  2. 2.
    N. Alon, E. Gyori, The number of small semispaces of a finite set of points in the plane,J. Combin. Theory Ser. A 41 (1986), 154–157.MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    K. Brown, Voronoi diagrams from convex hulls,Inform. Process. Lett. 9 (1979), 223–228.CrossRefMATHGoogle Scholar
  4. 4.
    B. Chazelle, H. Edelsbrunner, An improved algorithm for constructingk-th order Voronoi diagram,Proceedings of the Annual Symposium on Computational Geometry, 1985, pp. 228–234.Google Scholar
  5. 5.
    B. Chazelle, F. Preparata, Halfspace range search: an algorithmic application ofk-sets,Discrete Comput. Geom. 1 (1986), 83–93.MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    K. Clarkson, New applications of random sampling in computational geometry,Discrete Comput. Geom. 2 (1987), 195–222.MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    K. Clarkson, Applications of random sampling in computational geometry, II,Proceedings of the Annual Symposium on Computational Geometry, 1988, pp. 1–11.Google Scholar
  8. 8.
    K. Clarkson, P. Shor, Algorithms for diametral pairs and convex hulls that are optimal randomized, and incremental,Proceedings of the Annual Symposium on Computational Geometry, 1988, pp. 12–17.Google Scholar
  9. 9.
    R. Cole, M. Sharir, C. Yap, Onk-hulls and related problems,Proceedings of the 16th Annual SIGACT Symposium, 1984, pp. 154–166.Google Scholar
  10. 10.
    H. Edelsbrunner, Edge skeletons in arrangements with applications,Algorithmica 1 (1986), 93–109.MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    H. Edelsbrunner, Private communication.Google Scholar
  12. 12.
    H. Edelsbrunner, J. O'Rourke, R. Seidel, Constructing arrangements of lines and hyperplanes with applications,SIAM J. Comput. 15 (1986), 341–363.MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    H. Edelsbrunner, R. Seidel, Voronoi diagrams and arrangments,Discrete Comput. Geom. 1 (1986), 25–44.MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    P. Erdös, L. Lovász, A. Simmons, E. Strauss, Dissection graphs of planar point sets, inA Survey of Combinatorial Theory, J. N. Srivastava,et al., eds., North-Holland, Amsterdam, 1973, pp. 139–149.Google Scholar
  15. 15.
    J. E. Goodman, R. Pollack, On the number ofk-subsets of a set ofn points in the plane,J. Combin. Theory Ser. A 36 (1984), 101–104.MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    D. Knuth,The Art of Computer Programming, Vol. 2, Addison Wesley, Reading, MA, 1969.MATHGoogle Scholar
  17. 17.
    D. Lee, Onk-nearest neighbour Voronoi diagrams in the plane,IEEE Trans. Comput. 31 (1982), 478–487.MathSciNetMATHGoogle Scholar
  18. 18.
    K. Mulmuley, A fast planar partition algorithm, I,Proceedings of the 29th Annual Symposium on Foundations of Computer Science, 1988, pp. 580–589. Full version to appear inJ. Symbolic Logic, a special issue on Computational Geometry.Google Scholar
  19. 19.
    K. Mulmuley, A fast planar partition algorithm, II,Proceedings of the Fifth ACM Annual Symposium on Computational Geometry, 1989, pp. 33–43. To appear inJ. Assoc. Comput. Mach.Google Scholar
  20. 20.
    K. Mulmuley, An efficient algorithm for hidden surface removal,Computer Graphics 23(3) (1989), 379–388.CrossRefGoogle Scholar
  21. 21.
    K. Mulmuley, An efficient algorithm for hidden surface removal, II, Technical Report, University of Chicago, August 1989, Invited for publication inJ. Algorithms. (Also see On obstructions in relation to a fixed viewpoint,Proceedings of the 30th Annual Symposium on Foundations of Computer Science, 1989, pp. 592–597.)Google Scholar
  22. 22.
    F. Preparata, M. Shamos,Computational Geometry, Springer-Verlag, New York, 1985.CrossRefGoogle Scholar
  23. 23.
    E. Welzl, More onk-sets of finite sets in the plane,Discrete Comput. Geom. 1 (1986), 95–100.MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag New York Inc 1991

Authors and Affiliations

  • Ketan Mulmuley
    • 1
  1. 1.Department of Computer ScienceUniversity of ChicagoChicagoUSA

Personalised recommendations