Accessibility setting

Section: New Results

Probabilistic analysis of geometric data structures and algorithms

Participant : Olivier Devillers.

The worst visibility walk in a random Delaunay triangulation is $O\left(\sqrt{n}\right)$

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 $n$ 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 $O\left(\sqrt{n}\right)$ 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 ${ℝ}^d$ under uniform Euclidean (${\ell }^2$) noise. Specifically, let $\{p_1^{*},p_2^{*},...,p_n^{*}\}$ be an arbitrary set of $n$ points in the unit ball in ${ℝ}^d$ and let $p_i=p_i^{*}+x_i$, where $x_1,x_2,...,x_n$ are chosen independently from the unit ball of radius $\delta$. We show that the expected complexity, measured as the number of faces of all dimensions, of the convex hull of $\{p_1,p_2,...,p_n\}$ is $O\left(n^{2-\frac{4}{d+1}}{\left(1+1/\delta \right)}^{d-1}\right)$; the magnitude $\delta$ of the noise may vary with $n$. For $d=2$ this bound improves to $O\left(n^{\frac{2}{3}}(1+{\delta }^{-\frac{2}{3}})\right)$.

We also analyze the expected complexity of the convex hull of ${\ell }^2$ 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 $n$, and show that as the magnitude of the noise increases, that complexity varies monotonically for Gaussian noise but non-monotonically for ${\ell }^2$ noise [13] .

This work was done in collaboration with Xavier Goaoc (Univ. Marne la Vallée), Marc Glisse and Remy Thomasse (Inria Geometrica).