Voronoi diagrams based on general metrics in the plane

Extended abstract
  • Rolf Klein
  • Derick Wood
Contributed Papers Geometrical Algorithms
Part of the Lecture Notes in Computer Science book series (LNCS, volume 294)


Voronoi diagrams based on metrics others than the Euclidean metric or convex distance functions have recently received considerable interest in robotics and in computational geometry. Since the number of relevant metrics is large (and likely to increase, as new applications come up) a general theory should be developed, leading to results on the structure and on the computation of Voronoi diagrams that hold for large classes of metrics, rather than investigating each case separately.

In this paper we make the first steps in this direction. First we propose a uniform way of defining the Voronoi diagram of n points; the result is always a partition of the plane into n Voronoi regions, even if the bisectors of two points are two-dimensional regions. Then we present a simple axiom for metrics that guarantees that each possible Voronoi region is connected. In this case the normalized Voronoi diagram is a planar graph with n faces and O(n) edges and vertices, if the bisectors behave well. Next we pose two additional axioms each of which ensures that all Voronoi regions are simply-connected. One of them also implies that the bisector of two sets of points divided by a suitable line, contains no loops. Then the normalized Voronoi diagram can be computed within optimal O(n log n) steps, using the divide-and-conquer algorithm. The class of metrics thereby specified does not only contain all symmetric convex distance functions but also many composite metrics like, for example, the combination of the “grid” metric L1 in midtown Manhattan and the Euclidean metric in Central Park and on the Hudson River.


Computational geometry convex distance function metric norm robotics shortest path straight curve Voronoi diagram 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [Ar]
    B. Aronov, “On the geodesic Voronoi diagram of point sites in a simple polygon”, Proc. 3rd ACM Symposium on Computational Geometry, Waterloo, 1987, pages 39–49.Google Scholar
  2. [B]
    H. Busemann, “The geometry of geodesics”, Academic Press Inc., New York, 1955.Google Scholar
  3. [Br]
    K. Q. Brown, “Voronoi diagrams from convex hulls”, Inf. Proc. Lett. 9, pages 223–228, 1979.Google Scholar
  4. [ChDr]
    L. P. Chew and R. L. Drysdale, III, “Voronoi diagrams based on convex distance functions”, Proc. 1st ACM Symposium on Computational Geometry, Baltimore, 1985, pages 235–244.Google Scholar
  5. [DeKl]
    F. Dehne and R. Klein, “A sweepcircle algorithm for Voronoi diagrams”, presented at the Workshop on Graph-Theoretic Concepts in Computer Science (WG 87), Staffelstein, 1987.Google Scholar
  6. [F]
    S. Fortune, “A sweepline algorithm for Voronoi diagrams”, Algorithmica 2(2), 1987, pages 153–174.Google Scholar
  7. [H]
    F. K. Hwang, “An O(n log n) algorithm for rectilinear minimal spanning trees”, JACM 26, 1979, pages 177–182.Google Scholar
  8. [L]
    D. T. Lee, “Two-dimensional Voronoi diagrams in the L p metric”, JACM 27, 1980, pages 604–618.Google Scholar
  9. [LePr]
    D. T. Lee and F. P. Preparata, “Euclidean shortest paths in the presence of rectilinear barriers”, Networks 14(3), pages 393–410.Google Scholar
  10. [LeWo]
    D. T. Lee and C. K. Wong, “Voronoi diagrams in L 1 (L ) metrics with 2-dimensional storage applications”, SIAM J. COMPUT. 9, 1980, pages 200–211.Google Scholar
  11. [M]
    K. Menger, “Untersuchungen über allgemeine Metrik, I, II, III”, Mathematische Annalen 100, 1928, pages 75–163.Google Scholar
  12. [MiPa]
    J. S. B. Mitchell and Ch. H. Papadimitriou, “The weighted region problem”, Proc. 3rd ACM Symposium on Computational Geometry, Waterloo, 1987, pages 30–38.Google Scholar
  13. [R]
    W. Rinow, “Die innere Geometrie der Metrischen Räume”, Grundlehren der Mathematischen Wissenschaften in Einzeldarstellungen, Band 105, Springer-Verlag, Berlin, 1961.Google Scholar
  14. [ShHo]
    M. I. Shamos and D. Hoey, “Closest-point problems”, Proc 16th IEEE Symposium on Foundations of Computer Science, 1975, pages 151–162.Google Scholar
  15. [S]
    Y. A. Shreider, “What is distance?”, Popular Lectures in Mathematics, The University of Chicago Press, Chicago, 1974.Google Scholar
  16. [WiWuWo]
    P. Widmayer, Y. F. Wu, and C. K. Wong, “Distance problems in computational geometry for fixed orientations”, Proc. 1st ACM Symposium on Computational Geometry, Baltimore, 1985, pages 186–195.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1988

Authors and Affiliations

  • Rolf Klein
    • 1
  • Derick Wood
    • 2
  1. 1.Institut für InformatikUniversität FreiburgFreiburgWest Germany
  2. 2.Data Structuring Group, Department of Computer ScienceUniversity of WaterlooWaterlooCanada

Personalised recommendations