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Stretch Factor of Long Paths in a planar Poisson-Delaunay Triangulation
Stretch Factor of Long Paths in a planar Poisson-Delaunay Triangulation
Abstract : Let $X:=X_n\cup\{(0,0),(1,0)\}$, where $X_n$ is a planar Poisson point process of intensity $n$. We prove that the distance between the expected length of the shortest path between $(0,0)$ and $(1,0)$ in the Delaunay triangulation associated with $X$ belongs to $[1+2.47\cdot 10^{-11}, 1.182]$ when the intensity of $X_n$ goes to infinity.
Experimental values indicate that the correct value is about 1.04.
We also prove that the expected number of Delaunay edges crossed by the line segment $[(0,0),(1,0)]$ is $\frac{64}{3\pi^2}\sqrt{n}+O(1)\simeq2.16\sqrt{n}$.
Simulation code and Maple sheet are available with the research report.
Cited literature [17 references]
https://hal.inria.fr/hal-01346203
Contributor : Olivier Devillers <>
Submitted on : Monday, July 18, 2016 - 2:44:57 PM
Last modification on : Monday, December 14, 2020 - 5:16:20 PM
Contributor : Olivier Devillers <>
Submitted on : Monday, July 18, 2016 - 2:44:57 PM
Last modification on : Monday, December 14, 2020 - 5:16:20 PM