## Section: New Results

Keywords : Voronoi diagrams, compact data structures.

### Combinatorics, data structures and algorithms

#### Curved Voronoi diagrams

Participants : Jean-Daniel Boissonnat, Camille Wormser, Mariette Yvinec.

Voronoi diagrams are fundamental data structures that have been extensively studied in Computational Geometry. A Voronoi diagram can be defined as the minimization diagram of a finite set of continuous functions. Usually, each of those functions is interpreted as the distance function to an object. The associated Voronoi diagram subdivides the embedding space into regions, each region consisting of the points that are closer to a given object than to the others. We may define many variants of Voronoi diagrams depending on the class of objects, the distance functions and the embedding space. Affine diagrams, i.e. diagrams whose cells are convex polytopes, are well understood. Their properties can be deduced from the properties of polytopes and they can be constructed efficiently. The situation is very different for Voronoi diagrams with curved regions. Curved Voronoi diagrams arise in various contexts where the objects are not punctual or the distance is not the Euclidean distance. This work [16] is a survey of the main results on curved Voronoi diagrams. We describe in some detail two general mechanisms to obtain effective algorithms for some classes of curved Voronoi diagrams. The first one consists in linearizing the diagram and applies, in particular, to diagrams whose bisectors are algebraic hypersurfaces. The second one is a randomized incremental paradigm that can construct affine and several planar non-affine diagrams.

#### Complexity of Delaunay triangulation for points on lower-dimensional polyhedra

Participant : Olivier Devillers.

In collaboration with N. Amenta (University of California at Davis) and D. Attali (LIS-Grenoble)

Following previous 3D results of Attali and Boissonnat,
we show that the Delaunay triangulation of a set of points
distributed nearly uniformly on a polyhedron (not necessarily convex) of
dimension p in d-dimensional space is O(n^{(d-1)/p}) .
For all 2pd-1 , this improves on the well-known
worst-case bound of [50] .

#### Bregman Voronoi diagrams of probabilistic distributions

Participant : Jean-Daniel Boissonnat.

In collaboration with F. Nielsen (Sony Computer Science Laboratories) and R. Nock (University of Antilles-Guyane)

We investigate a framework for defining and building Voronoi diagrams for a broad class of distortion measures called Bregman divergences, that includes not only the traditional (squared) Euclidean distance, but also various divergence measures based on entropic functions. As a by-product, Bregman Voronoi diagrams allow one to define information-theoretic Voronoi diagrams in statistical parametric spaces based on the relative entropy of distributions. We show that for a given Bregman divergence, one can define several types of Voronoi diagrams related to each other by convex duality or embedding. Moreover, we can always compute them indirectly as power diagrams in primal or dual spaces, or directly after linearization in an extra-dimensional space as the projection of a Euclidean polytope. Finally, our paper proposes to generalize Bregman divergences to higher-order terms, called k-jet Bregman divergences, and touch upon their Voronoi diagrams [43] .

#### Compact representation of geometric data structures

Participants : Luca Castelli-Aleardi, Olivier Devillers, Abdelkrim Mebarki.

In collaboration with G. Schaeffer (LIX-Palaiseau)

We address the problem of representing the connectivity information of geometric objects using as little memory as possible. As opposed to raw compression issues, the focus is here on designing data structures that preserve the possibility of answering incidence queries in constant time. We propose in particular the first optimal representations for 3-connected planar graphs and triangulations, which are the most standard classes of graphs underlying meshes with spherical topology. Optimal means that these representations asymptotically match the respective entropy of the two classes, namely 2 bits per edge for 3-c planar graphs, and 1.62 bits per triangle or equivalently 3.24 bits per vertex for triangulations [38] . These results remain rather theoretical because the asymptotic behavior is far from the practical one, mainly due to a sub-linear term in which is actually not negligible. We are investigating some simplified framework inspired by the theoretical method but taking care of practical issues [37] .

#### Exact circle arrangements on a sphere, with applications in structural biology

Participants : Frédéric Cazals, Sébastien Loriot.

Given a collection of circles in a sphere, we adapt the
Bentley-Ottmann algorithm to the spherical setting to compute the *exact* arrangement of the circles [58] . Assuming the
circles are induced by balls, we also extend the algorithm to report
the balls covering each face of the arrangement.
The algorithm consists of sweeping the sphere by a meridian, which is
non trivial because of the degenerate cases and the algebraic
specification of event points.
The paper focuses on the construction of the arrangement, the
predicates and algebraic questions being developed in a companion
paper.

This construction is motivated by the calculation of parameters describing multi-body contacts in structural biology, so as to go beyond the classical exposed and buried surface areas. As an illustration, statistics featuring spectacular changes wrt traditional observations on protein - protein complexes are provided.