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Stretch Factor of Long Paths in a planar Poisson-Delaunay Triangulation

Report published in 2016 by Nicolas Chenavier, Olivier Devillers ORCID
This paper is available in a repository.
This paper is available in a repository.

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Preprint: policy unknown
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Postprint: policy unknown
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Abstract

Let $X:=X_n∪\{(0,0),(1,0)\}$, where $X_n$ is a planar Poisson point process of intensity $n$. We provide a first non-trivial 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 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 equivalent to $2.16\sqrt{n}$ and that the expected length of a particular path converges to 1.18 giving an upper bound on the stretch factor. ; Comment: 30 pages