## 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 $X:={X}_{n}\cup \{(0,0),(1,0)\}$, 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 $\left[\right(0,0),(1,0\left)\right]$ 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 $(0,0)$ and $(1,0)$ 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 $O(1/\sqrt{n})$.
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 $n\in \mathbb{N}$ 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).