A Probabilistic Approach to Reducing Algebraic Complexity of Delaunay Triangulations

  • Jean-Daniel Boissonnat
  • Ramsay Dyer
  • Arijit Ghosh
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9294)


We propose algorithms to compute the Delaunay triangulation of a point set L using only (squared) distance comparisons (i.e., predicates of degree 2). Our approach is based on the witness complex, a weak form of the Delaunay complex introduced by Carlsson and de Silva. We give conditions that ensure that the witness complex and the Delaunay triangulation coincide and we introduce a new perturbation scheme to compute a perturbed set L′ close to L such that the Delaunay triangulation and the witness complex coincide. Our perturbation algorithm is a geometric application of the Moser-Tardos constructive proof of the Lovász local lemma.


Simplicial Complex Delaunay Triangulation Full Cell Good Link Distance Comparison 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Alon, N., Spencer, J.H.: The Probabilistic Method, 3rd edn. Wiley-Interscience, New York (2008)CrossRefMATHGoogle Scholar
  2. 2.
    Attali, D., Edelsbrunner, H., Mileyko, Y.: Weak witnesses for Delaunay triangulations of submanifolds. In: Proc. ACM Sympos. Solid and Physical Modeling, pp. 143–150 (2007)Google Scholar
  3. 3.
    Boissonnat, J.D., Dyer, R., Ghosh, A.: The Stability of Delaunay Triangulations. Int. J. on Comp. Geom (IJCGA) 23(4&5), 303–333 (2013)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Boissonnat, J.D., Dyer, R., Ghosh, A., Oudot, S.Y.: Only distances are required to reconstruct submanifolds. ArXiv e-prints (October 2014)Google Scholar
  5. 5.
    Boissonnat, J.D., Maria, C.: The Simplex Tree: An Efficient Data Structure for General Simplicial Complexes. Algorithmica 70(3), 406–427 (2014)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Boissonnat, J., Dyer, R., Ghosh, A.: A probabilistic approach to reducing the algebraic complexity of computing Delaunay triangulations. CoRR abs/1505.05454 (2015). http://arxiv.org/abs/1505.05454
  7. 7.
    Delaunay, B.: Sur la sphère vide. Izv. Akad. Nauk SSSR, Otdelenie Matematicheskii i Estestvennyka Nauk 7, 793–800 (1934)MATHGoogle Scholar
  8. 8.
    Funke, S., Klein, C., Mehlhorn, K., Schmitt, S.: Controlled perturbation for Delaunay triangulations. In: Proc. 16th ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 1047–1056 (2005)Google Scholar
  9. 9.
    Halperin, D.: Controlled perturbation for certified geometric computing with fixed-precision arithmetic. In: Fukuda, K., van der Hoeven, J., Joswig, M., Takayama, N. (eds.) ICMS 2010. LNCS, vol. 6327, pp. 92–95. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  10. 10.
    Har-Peled, S.: Geometric Approximation Algorithms. American Mathematical Society (2011)Google Scholar
  11. 11.
    Millman, D.L., Snoeyink, J.: Computing Planar Voronoi Diagrams in Double Precision: A Further Example of Degree-driven Algorithm Design. In: Proc. 26th ACM Symp. on Computational Geometry, pp. 386–392 (2010)Google Scholar
  12. 12.
    Moser, R.A., Tardos, G.: A constructive proof of the generalized Lovász Local Lemma. Journal of the ACM 57(2) (2010)Google Scholar
  13. 13.
    de Silva, V.: A weak characterisation of the Delaunay triangulation. Geometriae Dedicata 135(1), 39–64 (2008)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    de Silva, V., Carlsson, G.: Topological estimation using witness complexes. In: Proc. Sympos. Point-Based Graphics, pp. 157–166 (2004)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  • Jean-Daniel Boissonnat
    • 1
  • Ramsay Dyer
    • 2
  • Arijit Ghosh
    • 3
  1. 1.INRIASophia AntipolisFrance
  2. 2.University of GroningenGroningenThe Netherlands
  3. 3.Max-Planck-Institut für InformatikSaarbrückenGermany

Personalised recommendations