Abstract voronoi diagrams and their applications

Extended abstract
  • Rolf Klein
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 333)


Given a set S of n points in the plane, and for every two of them a separating Jordan curve, the abstract Voronoi diagram V(S) can be defined, provided that the regions obtained as the intersections of all the “halfplanes” containing a fixed point of S are path-connected sets and together form an exhaustive partition of the plane. This definition does not involve any notion of distance. The underlying planar graph, \(\hat V\)(S), turns out to have O(n) edges and vertices. If S=LR is such that the set of edges separating L-faces from R-faces in \(\hat V\)(S) does not contain loops then \(\hat V\)(L) and \(\hat V\)(R) can be merged within O(n) steps giving \(\hat V\)(S). This result implies that for a large class of metrics d in the plane the d-Voronoi diagram of n points can be computed within optimal O(n log n) time. Among these metrics are, for example, the symmetric convex distance functions as well as the metric defined by the city layout of Moscow or Karlsruhe.


Voronoi diagram metric computational geometry 


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. [Br]
    K. Q. Brown, “Voronoi diagrams from convex hulls”, Inf. Proc. Lett. 9, pages 223–228, 1979.Google Scholar
  3. [ChDr]
    L. P. Drysdale, III, “Voronoi diagrams based on convex distance functions”, Proc. 1st ACM Symposium on Computational Geometry, Baltimore, 1985, pages 235–244.Google Scholar
  4. [DuYa]
    C. ó'Dúnlaing and C. K. Yap, ”A retraction method for planning the motion of a disc”, in J. Schwartz, M. Sharir, and J. Hopcroft (eds.), Planning, Geometry, and Complexity of Robot Motion, Ablex Publishing Corp., Norwood, NJ, 1986.Google Scholar
  5. [F]
    S. Fortune, “A sweepline algorithm for Voronoi diagrams”, Algorithmica 2 (2), 1987, pages 153–174.CrossRefGoogle Scholar
  6. [H]
    F. K. Hwang, “An O(n log n) algorithm for rectilinear minimal spanning trees”, JACM 26, 1979, pages 177–182.Google Scholar
  7. [Kl]
    R. Klein, “Voronoi diagrams in the Moscow metric”, Technical Report No 7, Institut für Informatik, Universität Freiburg, to be presented at WG '88, Amsterdam.Google Scholar
  8. [KlWo]
    R. Klein and D. Wood, “Voronoi diagrams based on general metrics in the plane”, in R. Cori and M. Wirsing (eds.), Proc. 5th Annual Symposium on Theoretical Aspects of Computer Science (STACS), Bordeaux, France, 1988, LNCS, pages 281–191.Google Scholar
  9. [L]
    D. T. Lee, “Two-dimensional Voronoi diagrams in the L p metric”, JACM 27, 1980, pages 604–618.CrossRefGoogle 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. [PrSh]
    F. Preparata and I. Shamos, “Computational Geometry” An Introduction, Springer, 1985.Google Scholar
  12. [ShHo]
    M. I. Shamos and D. Hoey, “Closest-point problems”, Proc. 6th IEEE Symposium on Foundations of Computer Science, 1975, pages 151–162.Google Scholar
  13. [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
  1. 1.Institut für InformatikUniversität FreiburgFreiburgFed. Rep. of Germany

Personalised recommendations