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)


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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  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