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Section: New Results

Classical computational geometry

Complexity analysis of random geometric structures made simpler

Average-case analysis of data-structures or algorithms is commonly used in computational geometry when the more classical worst-case analysis is deemed overly pessimistic. Since these analyses are often intricate, the models of random geometric data that can be handled are often simplistic and far from "realistic inputs".

In a joint work with Olivier Devillers and Marc Glisse (Inria GEOMETRICA) [20] , we presented a new simple scheme for the analysis of geometric structures. While this scheme only produces results up to a polylog factor, it is much simpler to apply than the classical techniques and therefore succeeds in analyzing new input distributions related to smoothed complexity analysis. We illustrated our method on two classical structures: convex hulls and Delaunay triangulations. Specifically, we gave short and elementary proofs of the classical results that $n$ points uniformly distributed in a ball in $R^d$ have a convex hull and a Delaunay triangulation of respective expected complexities $\tilde{\Theta }\left(n^{\left(\right(d+1)/(d-1\left)\right)}\right)$ and $\tilde{\Theta }\left(n\right)$. We then prove that if we start with $n$ points well-spread on a sphere, e.g. an $(ϵ,\kappa )$-sample of that sphere, and perturb that sample by moving each point randomly and uniformly within distance at most $\delta$ of its initial position, then the expected complexity of the convex hull of the resulting point set is $\tilde{\Theta }{\left(\sqrt{(}n\right)}^{(1-1/d)}{\delta }^{-(d-1)/\left(4d\right)})$.

On the monotonicity of the expected number of facets of a random polytope

Let $K$ be a compact convex body in $R^d$, let $K_n$ be the convex hull of $n$ points chosen uniformly and independently in $K$, and let $f_i\left(K_n\right)$ denote the number of $i$-dimensional faces of $K_n$.

In a joint work with Olivier Devillers and Marc Glisse (Inria GEOMETRICA) and Matthias Reitzner (Univ. Osnabruck) [21] , we showed that for planar convex sets, $E\left(f_0\left(K_n\right)\right)$ is increasing in $n$. In dimension $d\ge 3$ we prove that if $li{m}_{n\to \infty }\frac{E\left(f_{d-1}\left(K_n\right)\right)}{A{n}^c}=1$ for some constants $A$ and $c>0$ then the function $E\left(f_{d-1}\left(K_n\right)\right)$ is increasing for $n$ large enough. In particular, the number of facets of the convex hull of $n$ random points distributed uniformly and independently in a smooth compact convex body is asymptotically increasing. Our proof relies on a random sampling argument.

Embedding geometric structures

We continued working this year on the problem of embedding geometric objects on a grid of ${ℝ}^3$. Essentially all industrial applications take, as input, models defined with a fixed-precision floating-point arithmetic, typically doubles. As a consequence, geometric objects constructed using exact arithmetic must be embedded on a fixed-precision grid before they can be used as input in other software. More precisely, the problem is, given a geometric object, to find a similar object representable with fixed-precision floating-point arithmetic, where similar means topologically equivalent, close according to some distance function, etc. We are working on the problem of rounding polyhedral subdivisions on a grid of ${ℝ}^3$, where the only known method, due to Fortune in 1999, considers a grid whose refinement depends on the combinatorial complexity of the input, which does not solve the problem at hand. This project is joint work with Olivier Devillers (Inria Geometrica) and William Lenhart (Williams College, USA) who was in sabbatical in our team in 2012.