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
A generator of random convex polygons in a disc
Participants : Olivier Devillers, Rémy Thomasse.
In collaboration with Philippe Duchon (LABRI)
Let
where
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
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