## Section: New Results

### Probabilistic Analysis of Geometric Data Structures and Algorithms

Participants : Olivier Devillers, Charles Duménil.

#### Delaunay triangulation of a random sample of a good sample has linear size

A good sample is a point set such that any ball of radius $\u03f5$ contains a constant number of points. The Delaunay triangulation of a good sample is proved to have linear size, unfortunately this is not enough to ensure a good time complexity of the randomized incremental construction of the Delaunay triangulation. In this paper we prove that a random Bernoulli sample of a good sample has a triangulation of linear size. This result allows to prove that the randomized incremental construction needs an expected linear size and an expected $O(nlogn)$ time [8].

This work was done in collaboration with Marc Glisse (Project-team Datashape ).

#### Delaunay triangulation of a random sampling of a generic surface

The complexity of the Delaunay triangulation of $n$ points distributed on a surface ranges from linear to quadratic. We prove that when the points are evenly distributed on a smooth compact generic surface the expected size of the Delaunay triangulation is $O\left(n\right)$. This result has to be compared with a bound of $O(nlogn)$ when the points are a deterministic good sample of the surface under the same hypotheses on the surface [13].