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).