Expected Length of the Voronoi Path in a High Dimensional Poisson–Delaunay Triangulation

  • Pedro Machado Manhães de Castro
  • Olivier Devillers


Let X be a d dimensional Poisson point process. We prove that the expected length of the Voronoi path between two points at distance 1 in the Delaunay triangulation associated with X is \(\sqrt{\frac{2d}{\pi }}+O(d^{-\frac{1}{2}})\) when \(d\rightarrow \infty \). In any dimension, we also provide a precise interval containing the actual value; in 3D the expected length is between 1.4977 and 1.50007.


Random distribution Walking strategies Routing Point location 

Mathematics Subject Classification

68 - Computer science 60 - Probability theory and stochastic processes 



Authors gratefully acknowledge Sylvain Lazard for his help and comments in preparing the final version of the paper.

Supplementary material

454_2017_9866_MOESM1_ESM.pdf (333 kb)
Supplementary material 1 (pdf 333 KB)
454_2017_9866_MOESM2_ESM.mw (201 kb)
Supplementary material 2 (mw 202 KB)


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Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  • Pedro Machado Manhães de Castro
    • 1
  • Olivier Devillers
    • 2
    • 3
    • 4
  1. 1.Centro de Informática da Universidade Federal de PernambucoRecifeBrasil
  2. 2.Inria, Centre de recherche Nancy-Grand EstVillers-lès-NancyFrance
  3. 3.Universté de LorraineNancy CedexFrance
  4. 4.CNRSLoriaFrance

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