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##### VEGAS - 2016
Application Domains
New Software and Platforms
Bilateral Contracts and Grants with Industry
Partnerships and Cooperations
Bibliography
Application Domains
New Software and Platforms
Bilateral Contracts and Grants with Industry
Partnerships and Cooperations
Bibliography

## Section: New Results

### Probabilistic Analysis of Geometric Data Structures and Algorithms

Participants : Olivier Devillers, Louis Noizet.

#### Stretch Factor of Long Paths in a Planar Poisson-Delaunay Triangulation

Let $X:={X}_{n}\cup \left\{\left(0,0\right),\left(1,0\right)\right\}$, 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 $\left(0,0\right)$ and $\left(1,0\right)$ 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 $\left[\left(0,0\right),\left(1,0\right)\right]$ 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 [26].

This work was done in collaboration with Nicolas Chenavier (Université Littoral Côte d'Opale ).

#### Walking in a Planar Poisson-Delaunay Triangulation: Shortcuts in the Voronoi Path

Let ${X}_{n}$ be a planar Poisson point process of intensity $n$. We give a new proof that the expected length of the Voronoi path between $\left(0,0\right)$ and $\left(1,0\right)$ in the Delaunay triangulation associated with ${X}_{n}$ is $\frac{4}{\pi }\simeq 1.27$ when $n$ goes to infinity; and we also prove that the variance of this length is $O\left(1/\sqrt{n}\right)$. We investigate the length of possible shortcuts in this path, and defined a shortened Voronoi path whose expected length can be expressed as an integral that is numerically evaluated to $\simeq 1.16$. The shortened Voronoi path has the property to be locally defined; and is shorter than the previously known locally defined path in Delaunay triangulation such as the upper path whose expected length is $35/3{\pi }^{2}\simeq 1.18$ [27].

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

Let ${X}_{n}$ be a $d$ dimensional Poisson point process of intensity $n$. We prove that the expected length of the Voronoi path between two points at distance 1 in the Delaunay triangulation associated with ${X}_{n}$ is $\sqrt{\frac{2d}{\pi }}+O\left({d}^{-\frac{1}{2}}\right)$ for all $n\in ℕ$ and $d\to \infty$. In any dimension, we provide a precise interval containing the exact value, in 3D the expected length is between 1.4977 and 1.50007 [31].

This work was done in collaboration with Pedro Machado Manhães De Castro (Centro de Informática da Universidade Federal de Pernambuco).