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 points
uniformly distributed in a ball in have a convex hull and a
Delaunay triangulation of respective expected complexities
and .
We then prove that if we start with points well-spread on a
sphere, e.g. an -sample of that sphere, and perturb
that sample by moving each point randomly and uniformly within
distance at most of its initial position, then the expected
complexity of the convex hull of the resulting point set is
.
On the monotonicity of the expected number of facets of a random polytope
Let be a compact convex body in , let be the convex
hull of points chosen uniformly and independently in , and
let denote the number of -dimensional faces of .
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, is increasing in . In dimension
we prove that if for some constants and
then the function is increasing for large
enough. In particular, the number of facets of the convex hull of
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 . 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 , 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.