Hamiltonian abstract Voronoi diagrams in linear time

  • Rolf Klein
  • Andrzej Lingas
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 834)


Let V(S) be an abstract Voronoi diagram, and let H be an unbounded simple curve that visits each of its regions exactly once. Suppose that each bisector B(p, q), where p and q are in S, intersects H only once. We show that such a “Hamiltonian” diagram V(S) can be constructed in linear time, given the order of Voronoi regions of V(S) along H. This result generalizes the linear time algorithm for the Voronoi diagram of the vertices of a convex polygon. We also provide, for any δ > log60/29 2, an O(nδ)-time parallelization of the construction of the V(S) optimal in the time-processor product sense.

Key words

Computational geometry Voronoi diagrams abstract Voronoi diagrams convex polygons 


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  1. 1.
    A. Aggarwal, L.J. Guibas, J. Saxe, and P.W. Shor. A Linear-Time Algorithm for Computing the Voronoi Diagram of a Convex Polygon. Discrete and Computational Geometry 2, 1987, Springer Verlag.Google Scholar
  2. 2.
    C. Clarkson and P.W. Shor. Applications of Random Sampling in Computational Geometry II. Discrete and Computational Geometry, 4:387–421, 1989.Google Scholar
  3. 3.
    R. Cole. Parallel merge sort. SIAM J. Computing, 17 (1988), pp. 770–785.CrossRefGoogle Scholar
  4. 4.
    R. Cole, M.T. Goodrich and C. Ó Dúnlaing. Merging Free Trees in Parallel for Efficient Voronoi Diagram Construction. Proc. 17th ICALP, LNCS 443, Springer Verlag, pp. 432–445.Google Scholar
  5. 5.
    H. Djidjev and A. Lingas. On Computing the Voronoi Diagram for Restricted Planar Figures. To appear in Proc. WADS'91, LNCS, Springer Verlag.Google Scholar
  6. 6.
    O. Devillers. Randomization Yields Simple O(n log* n) Algorithms for Difficult ω(n) Problems. International Journal of Computational Geometry & Applications 2(1), pp. 97–111, 1992.Google Scholar
  7. 7.
    H. Edelsbrunner and R. Seidel. Voronoi Diagrams and Arrangements. Discrete and Computational Geometry, 1:25–44, 1986.Google Scholar
  8. 8.
    S. Fortune. A sweepline algorithm for Voronoi diagrams. Algorithmica, 2 (2), pp. 153–174, 1987.CrossRefGoogle Scholar
  9. 9.
    R. M. Karp and V. Ramachandran, Parallel Algorithms for Shared-Memory Machines. Handbook of Theoretical Computer Science, Edited by J. van Leeuwen, Volume 1, Elsevier Science Publishers B.V., 1990.Google Scholar
  10. 10.
    R. Klein. Concrete and Abstract Voronoi Diagrams. LNCS 400, Springer Verlag, 1989.Google Scholar
  11. 11.
    R. Klein, K. Mehlhorn, and S. Meiser. Randomized Incremental Construction of Abstract Voronoi Diagrams. Computational Geometry: Theory and Applications 3 (1993), pp. 157–184.Google Scholar
  12. 12.
    S. Meiser. Zur Konstruktion abstrakter Voronoidiagramme. Ph.D. Thesis, Universität des Saarlandes, Saarbrücken, 1993.Google Scholar
  13. 13.
    A. Okabe, B. Boots, and K. Sugihara. Spatial Tessellations: Concepts and Applications of Voronoi Diagrams. John Wiley & Sons Ltd., Chichester, 1992.Google Scholar
  14. 14.
    G. Rote. Curves with Increasing Chords. Math. Proc. Cambridge Phil. Soc. 115, 1, 1994, pp. 1–12.Google Scholar
  15. 15.
    M.I. Shamos and D. Hoey. Closest Point Problems. In Proc. of the 16th Annual IEEE Symposium on Foundations of Computer Science, pp. 151–162, 1975.Google Scholar
  16. 16.
    C. Yap and H. Alt. Motion Planning in the CL-Environment. In F. Dehne, J.-R. Sack, and N. Santoro (eds.), Algorithms and Data Structures (WADS' 89), LNCS 382, Springer Verlag, pp.373–380.Google Scholar
  17. 17.
    C. Yap. An O(n log n) Algorithm for the Voronoi Diagram of a Set of Simple Curve Segments. Discrete and Computational Geometry 2, pp. 365–393, 1987.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1994

Authors and Affiliations

  • Rolf Klein
    • 1
  • Andrzej Lingas
    • 2
  1. 1.Praktische Informatik VIFern Universität HagenHagenGermany
  2. 2.Department of Computer ScienceLund UniversitySweden

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