Section: New Results
Probabilistic analysis of geometric data structures and algorithms
Participant : Olivier Devillers.
The worst visibility walk in a random Delaunay triangulation is
We show that the memoryless routing algorithms Greedy Walk, Compass Walk, and all variants of visibility walk based on orientation predicates are asymptotically optimal in the average case on the Delaunay triangulation. More specifically, we consider the Delaunay triangulation of an unbounded Poisson point process of unit rate and demonstrate that the worst-case path between any two vertices inside a domain of area has a number of steps that is not asymptotically more than the shortest path which exists between those two vertices with probability converging to one (as long as the vertices are sufficiently far apart.) As a corollary, it follows that the worst-case path has steps in the limiting case, under the same conditions. Our results have applications in routing in mobile networks and also settle a long-standing conjecture in point location using walking algorithms. Our proofs use techniques from percolation theory and stochastic geometry [24] .
This work was done in collaboration with Ross Hemsley (formerly in Inria Geometrica).
Smooth analysis of convex hulls
We establish an upper bound on the smoothed complexity of convex hulls in under uniform Euclidean () noise. Specifically, let be an arbitrary set of points in the unit ball in and let , where are chosen independently from the unit ball of radius . We show that the expected complexity, measured as the number of faces of all dimensions, of the convex hull of is ; the magnitude of the noise may vary with . For this bound improves to .
We also analyze the expected complexity of the convex hull of and Gaussian perturbations of a nice sample of a sphere, giving a lower-bound for the smoothed complexity. We identify the different regimes in terms of the scale, as a function of , and show that as the magnitude of the noise increases, that complexity varies monotonically for Gaussian noise but non-monotonically for noise [13] .
This work was done in collaboration with Xavier Goaoc (Univ. Marne la Vallée), Marc Glisse and Remy Thomasse (Inria Geometrica).