Skip to main content

Stretch factor in a planar Poisson–Delaunay triangulation with a large intensity

  • Nicolas Chenavier (a1) and Olivier Devillers (a2)

Let X := X n ∪ {(0, 0), (1, 0)}, where X n is a planar Poisson point process of intensity n. We provide a first nontrivial lower bound for the distance between the expected length of the shortest path between (0, 0) and (1, 0) in the Delaunay triangulation associated with X when the intensity of X n goes to ∞. Simulations indicate that the correct value is about 1.04. We also prove that the expected length of the so-called upper path converges to 35 / 3π2, yielding an upper bound for the expected length of the smallest path.

Corresponding author
* Postal address: LMPA Joseph Liouville, Université du Littoral Côte d'Opale, 50 rue Ferdinand Buisson, BP 699, 62228 Calais Cedex, France. Email address:
** Postal address: Université de Lorraine, 615 rue du Jardin Botanique, B.P. 101, 54602 Villers-lès-Nancy Cedex, France. Email address:
Hide All
[1] Aurenhammer, F., Klein, R. and Lee, D.-T. (2013). Voronoi Diagrams and Delaunay Triangulations. World Scientific, Hackensack, NJ.
[2] Baccelli, F., Tchoumatchenko, K. and Zuyev, S. (2000). Markov paths on the Poisson-Delaunay graph with applications to routing in mobile networks. Adv. Appl. Prob. 32, 118.
[3] Bose, P. and Devroye, L. (2007). On the stabbing number of a random Delaunay triangulation. Comput. Geom. 36, 89105.
[4] Bose, P. and Morin, P. (2004). Online routing in triangulations. SIAM J. Comput. 33, 937951.
[5] Calka, P. (2002). The distributions of the smallest disks containing the Poisson-Voronoi typical cell and the Crofton cell in the plane. Adv. Appl. Prob. 34, 702717.
[6] Cazals, F. and Giesen, J. (2006). Delaunay triangulation based surface reconstruction. In Effective Computational Geometry for Curves and Surfaces, Springer, Berlin, pp. 231276.
[7] Chenavier, N. and Devillers, O. (2016). Stretch factor of long paths in a planar Poisson-Delaunay triangulation. Res. Rep. 8935, INRIA.
[8] Cheng, S.-W., Dey, T. K. and Shewchuk, J. R. (2013). Delaunay Mesh Generation. Chapman & Hall/CRC, Boca Raton, FL.
[9] Cox, J. T., Gandolfi, A., Griffin, P. S. and Kesten, H. (1993). Greedy lattice animals. I. Upper bounds. Ann. Appl. Prob. 3, 11511169.
[10] De Castro, P. M. M. and Devillers, O. (2018). Expected length of the Voronoi path in a high dimensional Poisson–Delaunay triangulation. Discrete Comput. Geom. 120. Available at
[11] Devillers, O. and Hemsley, R. (2016). The worst visibility walk in random Delaunay triangulation is O(√n). J. Comput. Geom. 7, 332359.
[12] Devillers, O. and Noizet, L. (2016). Walking in a planar Poisson-Delaunay triangulation: shortcuts in the Voronoi path. Res. Rep. 8946, INRIA.
[13] Devillers, O., Pion, S. and Teillaud, M. (2002). Walking in a triangulation. Internat. J. Found. Comput. Sci. 13, 181199.
[14] Devroye, L., Lemaire, C. and Moreau, J.-M. (2004). Expected time analysis for Delaunay point location. Comput. Geom. 29, 6189.
[15] Dobkin, D. P., Friedman, S. J. and Supowit, K. J. (1990). Delaunay graphs are almost as good as complete graphs. Discrete Comput. Geom. 5, 399407.
[16] Gerard, Y., Vacavant, A. and Favreau, J.-M. (2016). Tight bounds in the quadtree complexity theorem and the maximal number of pixels crossed by a curve of given length. Theoret. Comput. Sci. 624, 4155.
[17] Hirsch, C., Neuhä;user, D. and Schmidt, V. (2016). Moderate deviations for shortest-path lengths on random segment processes. ESAIM Prob. Statist. 20, 261292.
[18] Keil, J. M. and Gutwin, C. A. (1989). The Delaunay triangulation closely approximates the complete Euclidean graph. In Algorithms and Data Structures (Lecture Notes Comput. Sci. 382), Springer, Berlin, pp. 4756.
[19] Penrose, M. (2003). Random Geometric Graphs. Oxford University Press.
[20] Schneider, R. and Weil, W. (2008). Stochastic and Integral Geometry. Springer, Berlin.
[21] Thä;le, C. and Yukich, J. E. (2016). Asymptotic theory for statistics of the Poisson-Voronoi approximation. Bernoulli 22, 23722400.
[22] Xia, G. (2013). The stretch factor of the Delaunay triangulation is less than 1.998. SIAM J. Comput. 42, 16201659.
[23] Xia, G. and Zhang, L. (2011). Toward the tight bound of the stretch factor of Delaunay triangulations. In Proceedings 23th Canadian Conference on Computational Geometry, 6 pp.
[24] Yukich, J. E. (2015). Surface order scaling in stochastic geometry. Ann. Appl. Prob. 25, 177210.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Advances in Applied Probability
  • ISSN: 0001-8678
  • EISSN: 1475-6064
  • URL: /core/journals/advances-in-applied-probability
Please enter your name
Please enter a valid email address
Who would you like to send this to? *


MSC classification


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed