Section: New Results

Data Structures and Robust Geometric Computation

Efficiently Navigating a Random Delaunay Triangulation

Participants : Olivier Devillers, Ross Hemsley.

In collaboration with Nicolas Broutin (EPI rap )

Planar graph navigation is an important problem with significant implications to both point location in geometric data structures and routing in networks. Whilst many algorithms have been proposed, very little theoretical analysis is available for the properties of the paths generated or the computational resources required to generate them. In this work, we propose and analyse a new planar navigation algorithm for the Delaunay triangulation. We then demonstrate a number of strong theoretical guarantees for the algorithm when it is applied to a random set of points in a convex region [33] . In a side result, we give a new polylogarithmic bound on the maximum degree of a random Delaunay triangulation in a smooth convex, that holds with probability one as the number of points goes to infinity. In particular, our new bound holds even for points arbitrarily close to the boundary of the domain.  [56]

A chaotic random convex hull

Participants : Olivier Devillers, Marc Glisse, Rémy Thomasse.

The asymptotic behavior of the expected size of the convex hull of uniformly random points in a convex body in ${ℛ}^d$ is polynomial for a smooth body and polylogarithmic for a polytope. We construct a body whose expected size of the convex hull oscillates between these two behaviors when the number of points increases [62]

A generator of random convex polygons in a disc

Participants : Olivier Devillers, Rémy Thomasse.

In collaboration with Philippe Duchon (LABRI)

Let $𝒟$ a disc in ${ℝ}^2$ with radius 1 centered at $𝔬$, and $(x_1,\cdots ,x_n)$ a sample of $n$ points uniformly and independently distributed in $𝒟$. Let's define the polygon $P_n$ as the convex hull of $(x_1,\cdots ,x_n)$, and $f_0\left(P_n\right)$ its number of vertices. This kind of polygon has been well studied, and it is known, see [65] , that

where $c>0$ is constant. To generate such a polygon, one can explicitly generate $n$ points uniformly in $𝒟$ and compute the convex hull. For a very large quantity of points, it could be interesting to generate less points to get the same polygon, for example to have some estimations on asymptotic properties, such as the distribution of the size of the edges.We propose an algorithm that generate far less points at random in order to get $P_n$, so that the time and the memory needed is reduced for $n$ large. Namely [61] , we generate a number of points of the same order of magnitude than the final hull, up to a polylogarithmic factor

On the complexity of the representation of simplicial complexes by trees

Participants : Jean-Daniel Boissonnat, Dorian Mazauric.

In [46] , we investigate the problem of the representation of simplicial complexes by trees. We introduce and analyze local and global tree representations. We prove that the global tree representation is more efficient in terms of time complexity for searching a given simplex and we show that the local tree representation is more ecient in terms of size of the structure. The simplicial complexes are modeled by hypergraphs. We then prove that the associated combinatorial optimization problems are very dicult to solve and to approximate even if the set of maximal simplices induces a cubic graph, a planar graph, or a bounded degree hypergraph. However, we prove polynomial time algorithms that compute constant factor approximations and optimal solutions for some classes of instances.

Building Efficient and Compact Data Structures for Simplicial Complexes

Participant : Jean-Daniel Boissonnat.

In collaboration with Karthik C.S (Weizmann Institute of Science, Israël) and Sébastien Tavenas (Max-Planck-Institut für Informatik, Saarbrücken, Germany).

The Simplex Tree is a recently introduced data structure that can represent abstract simplicial complexes of any dimension and allows to efficiently implement a large range of basic operations on simplicial complexes. In this paper, we show how to optimally compress the simplex tree while retaining its functionalities. In addition, we propose two new data structures called Maximal Simplex Tree and Compact Simplex Tree. We analyze the Compressed Simplex Tree, the Maximal Simplex Tree and the Compact Simplex Tree under various settings.

Delaunay triangulations over finite universes

Participant : Jean-Daniel Boissonnat.

In collaboration with Ramsay Dyer (Johann Bernouilli Institute, University of Groningen, Pays Bas) and Arijit Ghosh (Max-Planck-Institut für Informatik, Saarbrücken, Germany).

The witness complex was introduced by Carlsson and de Silva as a weak form of the Delaunay complex that is suitable for finite metric spaces and is computed using only distance comparisons. The witness complex $\mathrm{Wit}(L,W)$ is defined from two sets $L$ and $W$ in some metric space $X$: a finite set of points $L$ on which the complex is built, and a set $W$ of witnesses that serves as an approximation of $X$. A fundamental result of de Silva states that $\mathrm{Wit}(L,W)=\mathrm{Del}\left(L\right)$ if $W=X={ℝ}^d$. In this paper we give conditions on $L$ that ensure that the witness complex and the Delaunay triangulation coincide when $W\subset {ℝ}^d$ is a finite set, and we introduce a new perturbation scheme to compute a perturbed set $L^{\text{'}}$ close to $L$ such that $\mathrm{Del}\left(L^{\text{'}}\right)=\mathrm{Wit}(L^{\text{'}},W)$. The algorithm constructs $\mathrm{Wit}(L^{\text{'}},W)$ in time sublinear in $\left|W\right|$.

The only numerical operations used by our algorithms are (squared) distance comparisons (i.e., predicates of degree 2). In particular, we do not use orientation or in-sphere predicates, whose degree depends on the dimension $d$, and are difficult to implement robustly in higher dimensions. Although the algorithm does not compute any measure of simplex quality, a lower bound on the thickness of the output simplices can be guaranteed. Another novelty in the analysis is the use of the Moser-Tardos constructive proof of the general Lovász local lemma.