Advertisement

Fast skeleton construction

  • Rolf Klein
  • Andrzej Lingas
Session 10. Chair: Paul Spirakis
Part of the Lecture Notes in Computer Science book series (LNCS, volume 979)

Abstract

For a polygon P, the skeleton of P is a partition of P into regions assigned to the edges of P. A point p inside P belongs to the region of an edge e if and only if e is the closest edge of P. We present a randomized algorithm that builds the skeleton of a simple polygon in linear expected time. We also observe that the Delaunay triangulation (equivalently, the Voronoi diagram) of a planar point set can be computed from its connected spanning subgraph in linear expected time.

Keywords

Voronoi Diagram Computational Geometry Delaunay Triangulation Simple Polygon Left Neighbor 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  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 4, 1987.Google Scholar
  2. [2]
    F. Aurenhammer. Voronoi Diagrams—A Survey of a Fundamental Geometric Data Structure. ACM Computing Surveys 23, 1991, pp. 345–405.CrossRefGoogle Scholar
  3. [3]
    B. Chazelle. Triangulating a Simple Polygon in Linear Time. Discrete and Computational Geometry 6, 1991, pp. 485–524.Google Scholar
  4. [4]
    P. Chew. Building Voronoi Diagrams for Convex Polygons in Linear Expected Time. Manuscript, 1986.Google Scholar
  5. [5]
    P. Chew. Constrained Delaunay Triangulations Proc. 3rd ACM Symposium on Computational Geometry, 1987, pp. 215–222.Google Scholar
  6. [6]
    K.L. Clarkson, K. Mehlhorn and R. Seidel. Four results on randomized incremental constructions. Computational Geometry: Theory and Applications 3 (1993), pp. 185–212.Google Scholar
  7. [7]
    O. Devillers. Randomization yields simple O(n log* n) algorithms for difficult Ω(n) problems. International Journal of Computational Geometry and Applications, Vol2, No1 (1992), pp. 97–111Google Scholar
  8. [8]
    H. Djidjev and A. Lingas. On Computing the Voronoi Diagram for Restricted Planar Figures. Proc. WADS'91, LNCS 519, pp. 54–64, Springer Verlag. To appear in International Journal of Computational Geometry and Applications.Google Scholar
  9. [9]
    R.L. Graham and F.F. Yao. Finding the convex hull of a simple polygon. J. Algorithms 4(4), pp. 324–331, 1983.Google Scholar
  10. [10]
    M. Held. On the Computational Geometry of Pocket Machining. LNCS 500, Springer-Verlag, 1991.Google Scholar
  11. [11]
    D.G. Kirkpatrick. Efficient computation of continuous skeletons. Proceedings of the 20th FOCS, pp. 18–27.Google Scholar
  12. [12]
    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
  13. [13]
    R. Klein and A. Lingas. A Linear-Time Randomized Algorithm for the Bounded Voronoi Diagram of a Simple Polygon. 9th ACM Symposium on Computational Geometry, San Diego, U.S.A., 1993, pp. 124–132.Google Scholar
  14. [14]
    R. Klein and A. Lingas. A Note on Generalizations of Chew's Algorithm for the Voronoi Diagram of a Convex Polygon. Proc. 5CCCG, pp. 370–374, 1993.Google Scholar
  15. [15]
    R. Klein and A. Lingas. Hamiltonian Abstract Voronoi Diagrams in Linear Time. Proc. ISAAC'94, LNCS, Springer Verlag.Google Scholar
  16. [16]
    D.T. Lee. On finding the convex hull of a simple polygon. Int'l J. Comput. and Infor. Sci. 12(2), 87–98, 1983.Google Scholar
  17. [17]
    D.T. Lee and A. Lin. Generalized Delaunay Triangulations for Planar Graphs. Discrete and Computational Geometry 1, 1986, pp. 201–217.Google Scholar
  18. [18]
    A. Lingas. Voronoi Diagrams with Barriers and the Shortest Diagonal Problem. Information Processing Letters 32, 1989, pp. 191–198.Google Scholar
  19. [19]
    U. Montanari. Continuous Skeletons from Digitized Images. J. ACM 16 (1969) pp. 564–549.Google Scholar
  20. [20]
    A. Okabe, B. Boots, and K. Sugihara. Spatial Tessellations, Concepts and Applications of Voronoi Diagrams. John Wiley & Sons, West Sussex, 1992.Google Scholar
  21. [21]
    F.P. Preparata and M.I. Shamos. Computational Geometry: An Introduction. Texts and Monographs in Theoretical Computer Science, Springer Verlag, New York, 1985.Google Scholar
  22. [22]
    R. Seidel. Constrained Delaunay triangulations and Voronoi diagrams with obstacles. In Rep. 260, IIG-TU Graz, Austria, pp. 178–191.Google Scholar
  23. [23]
    R. Seidel. Backwards Analysis of Randomized Geometric Algorithms. New Trends in Discrete and Computational Geometry, Janos Pach (ed.), Springer-Verlag, 1993.Google Scholar
  24. [24]
    R. Seidel. A simple and fast incremental and randomized algorithms for computing trapezoidal decompositions and for triangulating polygons. Computational Geometry: Theory and Applications, vol. 1, no. 1, 1991.Google Scholar
  25. [25]
    C. Wang and L. Schubert. An Optimal Algorithm for Constructing the Delaunay Triangulation of a Set of Line Segments. Proc. 3rd ACM Symposium on Computational Geometry, Waterloo, pp. 223–232, 1987.Google Scholar
  26. [26]
    C. Yap. An O(n log n) algorithm for the Voronoi diagram of a set of simple curve segments. Discrete Computational Geometry 2 (1987), pp. 365–393.Google Scholar
  27. [27]
    C. Yap and H. Alt Motion Planning in the CL-Environment. Proc. WADS'89, Ottawa, Canada, LNCS 382, pp. 373–380.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1995

Authors and Affiliations

  • Rolf Klein
    • 1
  • Andrzej Lingas
    • 2
  1. 1.FernUniversität HagenGermany
  2. 2.Lund UniversitySweden

Personalised recommendations