Advertisement

Preprocessing Imprecise Points and Splitting Triangulations

  • Marc van Kreveld
  • Maarten Löffler
  • Joseph S. B. Mitchell
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5369)

Abstract

Given a triangulation of a set of n points in the plane, each colored red or blue, we show how to compute a triangulation of just the blue points in time O(n). We apply this result to show that one can preprocess a set of disjoint regions (representing “imprecise points”) in the plane having total complexity n in O(n logn) time so that if one point per region is specified with precise coordinates, a triangulation of the n points can be computed in O(n) time.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Abellanas, M., Hurtado, F., Ramos, P.A.: Structural tolerance and Delaunay triangulation. Inf. Proc. Lett. 71, 221–227 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bandyopadhyay, D., Snoeyink, J.: Almost-Delaunay simplices: Nearest neighbour relations for imprecise points. In: Proc. 15th ACM-SIAM Symposium on Discrete Algorithms, pp. 410–419 (2004)Google Scholar
  3. 3.
    Chazelle, B., Devillers, O., Hurtado, F., Mora, M., Sacristán, V., Teillaud, M.: Splitting a Delaunay triangulation in linear time. Algorithmica 34, 39–46 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Chin, F.Y.L., Wang, C.A.: Finding the constrained Delaunay triangulation and constrained Voronoi diagram of a simple polygon in linear time. SIAM J. Comput. 28(2), 471–486 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Edelsbrunner, H., Robison, A.D., Shen, X.: Covering convex sets with non-overlapping polygons. Discrete Math. 81, 153–164 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Ely, J.S., Leclerc, A.P.: Correct Delaunay triangulation in the presence of inexact inputs and arithmetic. Reliable Computing 6, 23–38 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Guibas, L.J., Salesin, D., Stolfi, J.: Constructing strongly convex approximate hulls with inaccurate primitives. Algorithmica 9, 534–560 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Held, M., Mitchell, J.S.B.: Triangulating input-constrained planar point sets. Information Processing Letters (to appear, 2008)Google Scholar
  9. 9.
    Kendall, W.S., Barndorff-Nielson, O., van Lieshout, M.C.: Current Trends in Stochastic Geometry: Likelihood and Computation. CRC Press, Boca Raton (1998)Google Scholar
  10. 10.
    Khanban, A.A.: Basic Algorithms of Computational Geometry with Imprecise Input. PhD thesis, Imperial College, London (2005)Google Scholar
  11. 11.
    Khanban, A.A., Edalat, A.: Computing Delaunay triangulation with imprecise input data. In: Proc. 15th Canad. Conf. Comput. Geom., pp. 94–97 (2003)Google Scholar
  12. 12.
    Khanban, A.A., Edalat, A., Lieutier, A.: Computability of partial Delaunay triangulation and Voronoi diagram. In: Brattka, V., Schröder, M., Weihrauch, K. (eds.) Electronic Notes in Theoretical Computer Science, vol. 66. Elsevier, Amsterdam (2002)Google Scholar
  13. 13.
    Löffler, M., Snoeyink, J.: Delaunay triangulations of imprecise points in linear time after preprocessing. In: Proc. 24th Sympoium on Computational Geometry, pp. 298–304 (2008)Google Scholar
  14. 14.
    Löffler, M., van Kreveld, M.: Largest bounding box, smallest diameter, and related problems on imprecise points. In: Dehne, F., Sack, J.-R., Zeh, N. (eds.) WADS 2007. LNCS, vol. 4619, pp. 446–457. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  15. 15.
    Löffler, M., van Kreveld, M.: Largest and smallest convex hulls for imprecise points. Algorithmica (to appear, 2008)Google Scholar
  16. 16.
    Pocchiola, M., Vegter, G.: Pseudo-triangulations: Theory and applications. In: Proc. 12th Annu. ACM Sympos. Comput. Geom., pp. 291–300 (1996)Google Scholar
  17. 17.
    Rosenfeld, A.: Fuzzy geometry: An updated overview. Inf. Sci. 110(3-4), 127–133 (1998)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Rote, G., Wang, C.A., Wang, L., Xu, Y.: On constrained minimum pseudotriangulations. In: Warnow, T.J., Zhu, B. (eds.) COCOON 2003. LNCS, vol. 2697, pp. 445–454. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  19. 19.
    Seidel, R.: A method for proving lower bounds for certain geometric problems. In: Toussaint, G.T. (ed.) Computational Geometry, pp. 319–334. North-Holland, Amsterdam (1985)CrossRefGoogle Scholar
  20. 20.
    Sember, J., Evans, W.: Guaranteed Voronoi diagrams of uncertain sites. In: Proc. 20th Canad. Conf. Comput. Geom. (2008)Google Scholar
  21. 21.
    Weller, F.: Stability of Voronoi neighborship under perturbations of the sites. In: Proc. 9th Canad. Conf. Comput. Geom., pp. 251–256 (1997)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Marc van Kreveld
    • 1
  • Maarten Löffler
    • 1
  • Joseph S. B. Mitchell
    • 2
  1. 1.Department of Information and Computing SciencesUtrecht UniversityThe Netherlands
  2. 2.Department of Applied Mathematics and StatisticsState University of New York at Stony BrookUSA

Personalised recommendations