Algorithmica

, Volume 7, Issue 1–6, pp 381–413 | Cite as

Randomized incremental construction of Delaunay and Voronoi diagrams

  • Leonidas J. Guibas
  • Donald E. Knuth
  • Micha Sharir
Article

Abstract

In this paper we give a new randomized incremental algorithm for the construction of planar Voronoi diagrams and Delaunay triangulations. The new algorithm is more “on-line” than earlier similar methods, takes expected timeO(nℝgn) and spaceO(n), and is eminently practical to implement. The analysis of the algorithm is also interesting in its own right and can serve as a model for many similar questions in both two and three dimensions. Finally we demonstrate how this approach for constructing Voronoi diagrams obviates the need for building a separate point-location structure for nearest-neighbor queries.

Key words

Delaunay triangulation Voronoi diagram randomized algorithms 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    A. Aggarwal, H. Imai, N. Katoh, and S. Suri, Findingk points with minimum diameter and related problems,Proc. 5th ACM Symp. on Computational Geometry, 1989, pp. 283–291.Google Scholar
  2. [2]
    G. Andrews,The Theory of Partitions, Addison-Wesley, Reading, MA, 1976. (Encyclopedia of Mathematics and Its Applications, Volume 2.)MATHGoogle Scholar
  3. [3]
    I. Bárány, J. Schmerl, S. Sidney, and J. Urrutia, A combinatorial result about points and balls in Euclidean space,Discrete Comput. Geom.,3 (1988), 259–262.Google Scholar
  4. [4]
    J. Bentley, B. Weide, and A. Yao, Optimal expected time algorithms for closest point problems,ACM Trans. Math. Software,6 (1980), 563–580.MATHCrossRefMathSciNetGoogle Scholar
  5. [5]
    J.-D. Boissonnat, and M. Teillaud, A hierarchical representation of objects: the Delaunay tree,Proc. 2nd ACM Symp. on Computational Geometry, 1986, pp. 260–268.Google Scholar
  6. [6]
    B. Chazelle, H. Edelsbrunner, L. Guibas, R. Seidel, and M. Sharir, Selecting multiply covered points and reducing the size of Delaunay triangulations, Manuscript, 1989.Google Scholar
  7. [7]
    P. Chew, The simplest Voronoi diagram algorithm takes linear expected time, Manuscript, 1988.Google Scholar
  8. [8]
    K. Clarkson, Personal communication, 1989.Google Scholar
  9. [9]
    K. Clarkson and P. Shor, Applications of random sampling in computational geometry, II,Discrete Comput. Geom.,4 (1989), 387–421.MATHCrossRefMathSciNetGoogle Scholar
  10. [10]
    B. Delaunay, Neue Darstellung der geometrischen Krystallographie,Z. Kryst.,84 (1932), 109–149.MATHGoogle Scholar
  11. [11]
    B. Delaunay, Sur la sphère vide,Izv. Akad. Nauk SSSR. Otdel. Mat. Estestv. Nauk,7 (1934), 793–800.Google Scholar
  12. [12]
    R. Dwyer, Higher dimensional Voronoi diagrams in linear expected time,Proc. 5th ACM Symp. on Computational Geometry, 1989, pp. 326–333.Google Scholar
  13. [13]
    H. Edelsbrunner,Algorithms in Combinatorial Geometry, Springer-Verlag, Heidelberg, 1987.MATHGoogle Scholar
  14. [14]
    H. Edelsbrunner, L. Guibas, and J. Stolfi, Optimal point location in a monotone subdivision,SIAM J. Comput.,15 (1986), 317–340.MATHCrossRefMathSciNetGoogle Scholar
  15. [15]
    H. Edelsbrunner, N. Hasan, R. Seidel, and X. Shen, Circles through two points that always enclose many points, Technical Report UIUCDCS-R-88-1400, University of Illinois at Urbana, January 1988.Google Scholar
  16. [16]
    S. Fortune, A sweepline algorithm for Voronoi diagrams,Algorithmica,2 (1987), 153–174.MATHCrossRefMathSciNetGoogle Scholar
  17. [17]
    S. Fortune, A note on Delaunay diagonal flips, Unpublished manuscript.Google Scholar
  18. [18]
    P. Green and R. Sibson, Computing Dirichlet tesselation in the plane,Comput. J.,21 (1977), 168–173.MathSciNetGoogle Scholar
  19. [19]
    L. Guibas and M. Sharir, History helps queries, In preparation.Google Scholar
  20. [20]
    L. Guibas and J. Stolfi, Primitives for the manipulation of general subdivisions and the computation of Voronoi diagrams,ACM Trans. Graphics,4 (1985), 74–123.MATHCrossRefGoogle Scholar
  21. [21]
    J. Jaromczyk and G. Swiatek, Degenerate cases do not require more memory, Manuscript, 1989.Google Scholar
  22. [22]
    D. Kirkpatrick, Optimal search in planar subdivisions,SIAM J. Comput.,12 (1983), 28–35.MATHCrossRefMathSciNetGoogle Scholar
  23. [23]
    D. E. Knuth,The Art of Computer Programming, Vol. 3, Addison-Wesley, Reading, MA, 1973.Google Scholar
  24. [24]
    K. Mehlhorn, S. Meiser, and C. Ó'Dúnlaing, On the construction of abstract Voronoi diagrams, Manuscript, 1989.Google Scholar
  25. [25]
    K. Mulmuley, A fast planar partition algorithm, Technical Report 88-107, Department of Computer Science, University of Chicago, May 1988.Google Scholar
  26. [26]
    F. Preparata and M. Shamos,Computational Geometry-An Introduction, Springer-Verlag, Berlin, 1985.Google Scholar
  27. [27]
    F. Preparata and R. Tamassia, Fully dynamic techniques for point location and transitive closure in planar structures,Proc. 29th IEEE Symp. on Foundations of Computer Science, 1988, pp. 558–567.Google Scholar
  28. [28]
    R. Seidel, Private communication.Google Scholar
  29. [29]
    G. Voronoi, Nouvelles applications des paramètres continus à la théorie des formes quadratiques. Premier Mémoire: Sur quelques proprieteés des formes quadratiques positives parfaites,J. Reine Angew. Math.,133 (1907), 97–178.Google Scholar
  30. [30]
    G. Voronoi, Nouvelles applications des paramètres continus à la théorie des formes quadratiques. Deuxième Mémoire: Recherches sur les parallélloèdres primitifs,J. Reine Angew. Math.,134 (1908), 198–287;136 (1909), 67–181.Google Scholar

Copyright information

© Springer-Verlag New York Inc. 1992

Authors and Affiliations

  • Leonidas J. Guibas
    • 1
    • 2
  • Donald E. Knuth
    • 1
  • Micha Sharir
    • 3
    • 4
  1. 1.Computer Science DepartmentStanford UniversityStanfordUSA
  2. 2.DEC Systems Research CenterPalo AltoUSA
  3. 3.Courant Institute of Mathematical SciencesNew York UniversityNew YorkUSA
  4. 4.School of Mathematical SciencesTel Aviv UniversityTel AvivIsrael

Personalised recommendations