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Discrete & Computational Geometry

, Volume 4, Issue 6, pp 591–604 | Cite as

A linear-time algorithm for computing the voronoi diagram of a convex polygon

  • Alok Aggarwal
  • Leonidas J. Guibas
  • James Saxe
  • Peter W. Shor
Article

Abstract

We present an algorithm for computing certain kinds of three-dimensional convex hulls in linear time. Using this algorithm, we show that the Voronoi diagram ofn sites in the plane can be computed in Θ(n) time when these sites form the vertices of a convex polygon in, say, counterclockwise order. This settles an open problem in computational geometry. Our techniques can also be used to obtain linear-time algorithms for computing the furthest-site Voronoi diagram and the medial axis of a convex polygon and for deleting a site from a general planar Voronoi diagram.

Keywords

Convex Hull Voronoi Diagram Delaunay Triangulation Convex Polygon Medial Axis 
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.

References

  1. 1.
    L. P. Chew, Constrained Delaunay triangulations,Proc. 3rd Annual ACM Symposium on Computational Geometry, 1987, pp. 223–232.Google Scholar
  2. 2.
    L. J. Guibas and J. Stolfi, Primitives for the manipulation of general subdivisions and the computation of Voronoi diagrams,ACM Trans. Graphics 4 (1985), 74–123.zbMATHCrossRefGoogle Scholar
  3. 3.
    D. G. Kirkpatrick, Efficient computation of continuous skeletons,Proc. 20th IEEE Annual Symposium on Foundations of Computer Science, 1979, pp. 18–27.Google Scholar
  4. 4.
    D. G. Kirkpatrick, Optimal search in planar subdivisions,SIAM J. Comput. 12 (1983), 28–35.zbMATHMathSciNetCrossRefGoogle Scholar
  5. 5.
    D. T. Lee, Onk-nearest neighbor Voronoi diagrams in the plane,IEEE Trans. Comput. 31 (1982), 478–487.zbMATHMathSciNetGoogle Scholar
  6. 6.
    D. T. Lee and A. K. Lin, Generalized Delaunay triangulations of planar graphs,Discrete Comput. Geom. 1 (1986), 201–217.zbMATHMathSciNetCrossRefGoogle Scholar
  7. 7.
    D. McCallum and D. Avis, A linear-time algorithm for finding the convex hull of a simple polygon,Inform. Process. Lett. 9 (1979), 201–206.zbMATHMathSciNetCrossRefGoogle Scholar
  8. 8.
    F. P. Preparata, The medial axis of a simple polygon, inMathematical Foundations of Computer Science 1977 (Proc. 6th Symp.), 443–450, Lecture Notes in Computer Science, Vol. 53, Springer-Verlag, Berlin, 1977.CrossRefGoogle Scholar
  9. 9.
    F. P. Preparata and M. I. Shamos,Computational Geometry: An Introduction, Springer-Verlag, New York, 1985.CrossRefGoogle Scholar
  10. 10.
    M. I. Shamos, Computational Geometry, Ph.D. thesis, Yale University, New Haven, CT, 1978.Google Scholar
  11. 11.
    C. A. Wang and L. Schubert, An optimal algorithm for constructing the Delaunay triangulation of a set of segments,Proc. 3rd Annual ACM Symposium on Computational Geometry, 1987, pp. 223–232.Google Scholar

Copyright information

© Springer-Verlag New York Inc. 1989

Authors and Affiliations

  • Alok Aggarwal
    • 1
  • Leonidas J. Guibas
    • 2
    • 3
  • James Saxe
    • 3
  • Peter W. Shor
    • 4
  1. 1.IBM T. J. Watson Research CenterYorktown HeightsUSA
  2. 2.Department of Computer ScienceStanford UniversityStanfordUSA
  3. 3.DEC Systems Research CenterPalo AltoUSA
  4. 4.AT&T Bell LaboratoriesMurray HillUSA

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