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 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 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 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 . This result has to be compared with a bound of when the points are a deterministic good sample of the surface under the same hypotheses on the surface [13].