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Section: New Results

Probabilistic Analysis of Geometric Data Structures and Algorithms

Participants : Olivier Devillers, Charles Duménil, Fernand Kuiebove Pefireko.

Stretch Factor in a Planar Poisson-Delaunay Triangulation with a Large Intensity

Let $X:=X_n\cup \{(0,0),(1,0)\}$, where $X_n$ is a planar Poisson point process of intensity $n$. Our paper [4] provides a first non-trivial lower bound for 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. Simulations indicate that the correct value is about 1.04. We also prove that the expected length of the so-called upper path converges to $\frac{35}{3{\pi }^2}$, giving an upper bound for the expected length of the smallest path.

In collaboration with Nicolas Chenavier (Université du Littoral Côte d'Opale).

Delaunay triangulation of a Poisson Point Process on a Surface

The complexity of the Delaunay triangulation of $n$ points distributed on a surface ranges from linear to quadratic. We proved that when the points are evenly distributed on a smooth compact generic surface the expected size of the Delaunay triangulation is can be controlled. If the point set is a good sample of a smooth compact generic surface  [22] the complexity is controlled. Namely, good sample means that a sphere of size $ϵ$ centered on the surface contains between 1 and $\eta$ points. Under this hypothesis, the complexity of the Delaunay triangulation is $O\left(\frac{{\eta }^2}{{ϵ}^2}log\frac{1}{ϵ}\right)$. We proved that when the points are evenly distributed on a smooth compact generic surface they form a good sample with high probability for relevant values of $ϵ$ and $\eta$. We can deduce [15] that the expected size of the Delaunay triangulation of $n$ random points of a surface is $O\left(n{log}^{2}n\right)$.

On Order Types of Random Point Sets

Let $P$ be a set of $n$ random points chosen uniformly in the unit square. In our paper [19], we examine the typical resolution of the order type of $P$. First, we showed that with high probability, $P$ can be rounded to the grid of step $\frac{1}{n^{3+ϵ}}$ without changing its order type. Second, we studied algorithms for determining the order type of a point set in terms of the the number of coordinate bits they require to know. We gave an algorithm that requires on average $4n{log}_{2}n+O\left(n\right)$ bits to determine the order type of $P$, and showed that any algorithm requires at least $4n{log}_{2}n-O(nloglogn)$ bits. Both results extend to more general models of random point sets.

In collaboration with Philippe Duchon (LABRI) and Marc Glisse (project team Datashape ).