Algorithmica

, 2:153

A sweepline algorithm for Voronoi diagrams

  • Steven Fortune
Article

Abstract

We introduce a geometric transformation that allows Voronoi diagrams to be computed using a sweepline technique. The transformation is used to obtain simple algorithms for computing the Voronoi diagram of point sites, of line segment sites, and of weighted point sites. All algorithms haveO(n logn) worst-case running time and useO(n) space.

Key words

Voroni diagram Delaunay triangulation Sweepline algorithm 

References

  1. [1]
    F. Aurenhammer and H. Edelsbrunner, An optimal algorithm for constructing the weighted Voronoi diagram in the plane,Pattern Recognition,17 (1984), 251–257.MATHCrossRefMathSciNetGoogle Scholar
  2. [2]
    J. L. Bentley, B. W. Weide, and A. C. Yao, Optimal expected-time algorithms for closest-point problems,ACM Trans. Math. Software,6 (1980), 563–580.MATHCrossRefMathSciNetGoogle Scholar
  3. [3]
    L. P. Chew and R. L. Drysdale, Voronoi diagrams based on convex distance functions,Proceedings of the Symposium on Computational Geometry, 1985, pp. 235–244.Google Scholar
  4. [4]
    J. R. Driscoll, N. Sarnak, D. D. Sleator, and R. E. Tarjan, Making data structures persistent,Proceedings of the 18th Annual ACM Symposium on Theory of Computing, 1986, pp. 109–121.Google Scholar
  5. [5]
    H. Edelsbrunner, private communication, 1985.Google Scholar
  6. [6]
    H. Edelsbrunner, L. J. Guibas, and J. Stolfi, Optimal point location in a monotone subdivision, Technical Report, DEC Systems Research Center, Palo Alto, CA, 1984.Google Scholar
  7. [7]
    S. J. Fortune, Fast algorithms for polygon containment,Automata, Languages, and Programming, 12th Colloquium, Lecture Notes in Computer Science, Vol. 194, Springer-Verlag, New York, pp. 189–198.Google Scholar
  8. [8]
    P. J. Green and R. Sibson, Computing Dirichlet tesselations in the plane,Comput. J.,21 (1977) 168–173.MathSciNetGoogle Scholar
  9. [9]
    D. Kirkpatrick, Efficient computation of continuous skeletons,Proceedings of the 20th Annual Symposium on Foundations of Computer Science, 1979, pp. 18–27.Google Scholar
  10. [10]
    D. T. Lee, Medial axis transformation of a planar shape,IEEE Trans. Pattern Analysis Machine Intel.,4 (1982), 363–369.MATHCrossRefGoogle Scholar
  11. [11]
    D. T. Lee and R. L. Drysdale, Generalizations of Voronoi diagrams in the plane,Siam J. Comput.,10 (1981), 73–87.MATHCrossRefMathSciNetGoogle Scholar
  12. [12]
    D. T. Lee and B. J. Schacter, Two algorithms for constructing a Delauney triangulation,Internat. J. Comput. Inform. Sci.,9 (1980), 219–227.MATHCrossRefMathSciNetGoogle Scholar
  13. [13]
    D. Leven and M. Sharir, Planning a purely translational motion for a convex object in two-dimensional space using generalized Voronoi diagrams, Technical Report 34/85, Tel Aviv University, 1985.Google Scholar
  14. [14]
    D. Leven and M. Sharir, Intersection problems and applications of Voronoi diagrams, inAdvances in Robotics, Vol. I (J. Schwartz and C. K. Yap, eds), Lawrence Erlbaum, 1986.Google Scholar
  15. [15]
    T. Ohya, M. Iri, and K. Murota, Improvements of the incremental method for the Voronoi diagram with computational comparison of various algorithms,J. Oper. Res. Soc. Japan,27 (1984), 306–336.MATHMathSciNetGoogle Scholar
  16. [16]
    F. P. Preparata, The medial axis of a simple polygon,Proceedings of the Sixth Symposium on Mathematical Foundations of Computer Science, Lecture Notes in Computer Science, Vol. 53, Springer-Verlag, New York, 1977, pp. 443–450.Google Scholar
  17. [17]
    F. P. Preparata and M. I. Shamos,Computational Geometry: An Introduction, Springer-Verlag, New York, 1985.Google Scholar
  18. [18]
    M. I. Shamos and D. Hoey, Closest-point problems,Proceedings of the 16th Annual Symposium on Foundations of Computer Science, 1975, pp. 151–162.Google Scholar
  19. [19]
    M. Sharir, Intersection and closest-pair problems for a set of planar discs,SIAM J. Comput.,14 (1985), 448–468.MATHCrossRefMathSciNetGoogle Scholar
  20. [20]
    R. Sedgewick,Algorithms, Addison Wesley, Reading, MA, 1983.MATHGoogle Scholar
  21. [21]
    C. K. Yap, AnO(n logn) algorithm for the Voronoi diagram of a set of simple curve segments, NsYU-Courant Robotics Report No. 43 (submitted toSIAM J. Comput.).Google Scholar

Copyright information

© Springer-Verlag New York Inc. 1987

Authors and Affiliations

  • Steven Fortune
    • 1
  1. 1.AT&T Bell LaboratoriesMurray HillUSA

Personalised recommendations