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 , where is a planar Poisson point process of intensity . Our paper  provides a first non-trivial lower bound for the expected length of the shortest path between and in the Delaunay triangulation associated with when the intensity of 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 , 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 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  the complexity is controlled. Namely, good sample means that a sphere of size centered on the surface contains between 1 and points. Under this hypothesis, the complexity of the Delaunay triangulation is . 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 and . We can deduce  that the expected size of the Delaunay triangulation of random points of a surface is .
On Order Types of Random Point Sets
Let be a set of random points chosen uniformly in the unit square. In our paper , we examine the typical resolution of the order type of . First, we showed that with high probability, can be rounded to the grid of step 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 bits to determine the order type of , and showed that any algorithm requires at least bits. Both results extend to more general models of random point sets.
In collaboration with Philippe Duchon (LABRI) and Marc Glisse (project team Datashape ).