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How to compute the Voronoi diagram of line segments: Theoretical and experimental results

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Algorithms — ESA '94 (ESA 1994)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 855))

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Abstract

Given a set of non-intersecting (except at endpoints) line segments in the plane we want to compute their Voronoi diagram. Although there are several algorithms for this problem in the literature [Yap87, For87, CS89, BDS+92, KMM90] nobody claims to have a correct implementation. This is due to the fact that the algorithms presuppose exact arithmetic and that the Voronoi diagram of segments requires to compute with non-rational algebraic numbers. We report about a detailed study of the numerical precision required for evaluating the geometric test exactly and about first experimental experiences. More specifically, we improve the precision bound implied by classical root separation results by more than two orders of magnitude and we compare the implementation strategies suggested by our theoretical results.

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References

  1. F. Aurenhammer. Voronoi diagrams: a survey of a fundamental geometric data structure. ACM Comput. Surv., 23:345–405, 1991.

    Article  Google Scholar 

  2. J.D. Boissonnat, O. Devillers, R. Schott, M. Teillaud, and M. Yvinec. Applications of random sampling to on-line algorithms in computational geometry. Technical report, INRIA, 1990.

    Google Scholar 

  3. J.-D. Boissonnat, O. Devillers, R. Schott, M. Teillaud, and M. Yvinec. Applications of random sampling to on-line algorithms in computational geometry. Discrete and Computational Geometry, 8:51–71, 1992.

    MathSciNet  Google Scholar 

  4. M.O. Benouamer, P. Jaillon, D. Michelucci, and J-M. Moreau. A “lazy” solution to imprecision in computational geometry. In 5th Canadian Conf. on Computational Geometry, pages 73–78, 1993.

    Google Scholar 

  5. J. Blömer. Computing sums of radicals in polynomial time. In FOCS91, pages 670–677, 1991.

    Google Scholar 

  6. J. Blömer. Computing sums of radicals in polynomial time. Technical Report B93-13, Freie Universität Berlin, 1993.

    Google Scholar 

  7. B. Butz. Robuste Implementierung eines Algorithmus zur Berechung eines Voronoi-Diagrams für Polygone. Diplomarbeit, 1994.

    Google Scholar 

  8. K. L. Clarkson and P. W. Shor. Applications of random sampling in computational geometry, II. Discrete and Computational Geometry, pages 387–421, 1989.

    Google Scholar 

  9. T. Dube and C.K. Yap. A basis for implementing exact computational geometry. extended abstract, 1994.

    Google Scholar 

  10. S. J. Fortune. A sweepline algorithm for Voronoi diagrams. Algorithmica, 2:153–174, 1987.

    Article  MATH  MathSciNet  Google Scholar 

  11. S. Fortune. Numerical stability of algorithms for 2d Delaunay triangulations and Voronoi diagrams. In ACM Symposium on Computational Geometry, volume 8, pages 83–92, 1992.

    Google Scholar 

  12. S. Fortune and C. van Wyk. Efficient exact arithmetic for computational geometry. Proc. of the 9th Symp. on Computational Geometry, pages 163–171, 1993.

    Google Scholar 

  13. S. Fortune and C. van Wyk. Efficient exact arithmetic for computational geometry. In Proc. of the 9th ACM Symp. on Computational Geometry, pages 163–172, 1993.

    Google Scholar 

  14. M. Karasick, D. Lieber, and L.R. Nackman. Efficient Delaunay triangulation using rational arithmetic. ACM Transactions on Graphics, 10(1):71–91, 1991.

    Article  Google Scholar 

  15. R. Klein, K. Mehlhorn, and S. Meiser. On the construction of abstract Voronoi diagrams ii. In Proc. SIGAL Symp. on Algorithms, Tokyo, 1990. Springer Verlag. LNCS 450.

    Google Scholar 

  16. Z. Li and V. Milenkovic. Constructing strongly convex hulls using exact or rounded arithmetic. Algorithmica, 8:345–364, 1992.

    Article  MathSciNet  Google Scholar 

  17. R. Loos. Computing in algebraic extensions. In B. Buchberger, G.E. Collins, and R. Loos, editors, Computer Algebra, pages 173–187. Springer Verlag, 1982.

    Google Scholar 

  18. M. Mignotte. Mathematics for Computer Algebra. Springer Verlag, 1992.

    Google Scholar 

  19. K. Mehlhorn and S. Näher. Implementation of a sweep line algorithm for the segment intersection problem. manuscript.

    Google Scholar 

  20. A. Okabe, B. Boots, and Sugihara, K. Spatial tessellations: concepts and applications of Voronoi diagrams. Wiley, New York, 1992.

    Google Scholar 

  21. T. Ottmann, G. Thiemt, and C. Ullrich. Numerical stability of geometric algorithms. In Proc. of the 3rd ACM Symp. on Computational Geometry, pages 119–125, 1987.

    Google Scholar 

  22. C. O'Dúnlaing and C. Yap. A “retraction” method for planning the motion of a disk. Journal of Algorithms, 6:104–111, 1985.

    MathSciNet  Google Scholar 

  23. M. Seel. Ausarbeitung und Implementierung eines Algorithmus zur Konstruktion abstrakter Voronoi-Diagramme. Diplomarbeit, 1994.

    Google Scholar 

  24. K. Sugihara, Y. Ooishi, and T. Imai. Topology-oriented approach to robustness and its applications to several Voronoi-diagram algorithms. In Proc. 2nd Canad. Conf. Comput. Geom., pages 36–39, 1990.

    Google Scholar 

  25. C. Yap. An O(n log n) algorithm for the Voronoi diagram of a set of simple curve segments. Discrete and Computational Geometry, 2:365–393, 1987.

    MATH  MathSciNet  Google Scholar 

  26. C.K. Yap. Towards exact geometric computation. In 5th Canadian Conf. on Computational Geometry, pages 405–419, 1993.

    Google Scholar 

  27. C.K. Yap. Fundamental Problems in Algorithmic Algebra. Princeton University Press, 1994.

    Google Scholar 

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Author information

Authors and Affiliations

  1. Max-Planck-Institut für Informatik, 66123, Saarbrücken, Germany

    Christoph Burnikel, Kurt Mehlhorn & Stefan Schirra

Authors
  1. Christoph Burnikel
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  2. Kurt Mehlhorn
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  3. Stefan Schirra
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Jan van Leeuwen

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© 1994 Springer-Verlag Berlin Heidelberg

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Burnikel, C., Mehlhorn, K., Schirra, S. (1994). How to compute the Voronoi diagram of line segments: Theoretical and experimental results. In: van Leeuwen, J. (eds) Algorithms — ESA '94. ESA 1994. Lecture Notes in Computer Science, vol 855. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0049411

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  • DOI: https://doi.org/10.1007/BFb0049411

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-58434-6

  • Online ISBN: 978-3-540-48794-4

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