Section: New Results

Data Structures and Robust Geometric Computation

The Stability of Delaunay triangulations

Participants : Jean-Daniel Boissonnat, Ramsay Dyer.

In collaboration with Arijit Ghosh (Indian Statistical Institute)

We introduce a parametrized notion of genericity for Delaunay triangulations which, in particular, implies that the Delaunay simplices of $\delta$-generic point sets are thick [45] . Equipped with this notion, we study the stability of Delaunay triangulations under perturbations of the metric and of the vertex positions. We quantify the magnitude of the perturbations under which the Delaunay triangulation remains unchanged.

Delaunay Stability via Perturbations

Participants : Jean-Daniel Boissonnat, Ramsay Dyer.

In collaboration with Arijit Ghosh (Indian Statistical Institute)

We present an algorithm that takes as input a finite point set in Euclidean space, and performs a perturbation that guarantees that the Delaunay triangulation of the resulting perturbed point set has quantifiable stability with respect to the metric and the point positions [43] . There is also a guarantee on the quality of the simplices: they cannot be too flat. The algorithm provides an alternative tool to the weighting or refinement methods to remove poorly shaped simplices in Delaunay triangulations of arbitrary dimension, but in addition it provides a guarantee of stability for the resulting triangulation.

Deletions in 3D Delaunay Triangulation

Participant : Olivier Devillers.

In collaboration with Kevin Buchin (Technical University Eindhoven, The Netherlands), Wolfgang Mulzer (Freie Universität Berlin, Germany), Okke Schrijvers, (Stanford University, USA) and Jonathan Shewchuk (University of California at Berkeley, USA)

Deleting a vertex in a Delaunay triangulation is much more difficult than inserting a new vertex because the information present in the triangulation before the deletion is difficult to exploit to speed up the computation of the new triangulation.

The removal of the tetrahedra incident to the deleted vertex creates a hole in the triangulation that need to be retriangulated. First we propose a technically sound framework to compute incrementally a triangulation of the hole vertices: the conflict Delaunay triangulation. The conflict Delaunay triangulation matches the hole boundary and avoid to compute extra tetrahedra outside the hole. Second, we propose a method that uses guided randomized reinsertion to speed up the point location during the computation of the conflict triangulation. The hole boundary is a polyhedron, this polyhedron is simplified by deleting its vertices one by one in a random order maintaining a polyhedron called link Delaunay triangulation, then the points are inserted in reverse order into the conflict Delaunay triangulation using the information from the link Delaunay triangulation to avoid point location [30] .

A Convex Body with a Chaotic Random Polytope

Participants : Olivier Devillers, Marc Glisse, Rémy Thomasse.

Consider a sequence of points in a convex body in dimension $d$ whose convex hull is dynamically maintained when the points are inserted one by one, the convex hull size may increase, decrease, or being constant when a new point is added. Studying the expected size of the convex hull when the points are evenly distributed in the convex is a classical problem of probabilistic geometry that yields to some surprising facts. For example, although it seems quite natural to think that the expected size of the convex hull is increasing with $n$ the number of points, this fact is only formally proven for $n$ big enough [16] . The asymptotic behavior of the expected size is known to be logarithmic for a polyhedral body and polynomial for a smooth one. If for a polyhedral or a smooth body, the asymptotic behavior is somehow “nice" it is possible to construct strange convex objects that have no such nice behaviors and we exhibit a convex body, such that the behavior of the expected size of a random polytope oscillates between the polyhedral and smooth behaviors when $n$ increases [51] .

Delaunay Triangulations and Cycles on Closed Hyperbolic surfaces

Participants : Mikhail Bogdanov, Monique Teillaud.

This work [40] is motivated by applications of periodic Delaunay triangulations in the Poincaré disk conformal model of the hyperbolic plane ${ℍ}^2$. A periodic triangulation is defined by an infinite point set that is the image of a finite point set by a (non commutative) discrete group $G$ generated by hyperbolic translations, such that the hyperbolic area of a Dirichlet region is finite (i.e., a cocompact Fuchsian group acting on ${ℍ}^2$ without fixed points).

We consider the projection of such a Delaunay triangulation onto the closed orientable hyperbolic surface $M={ℍ}^2/G$. The graph of its edges may have cycles of length one or two. We prove that there always exists a finite-sheeted covering space of $M$ in which there is no cycle of length $\le$ 2. We then focus on the group defining the Bolza surface (homeomorphic to a torus having two handles), and we explicitly construct a sequence of subgroups of finite index allowing us to exhibit a covering space of the Bolza surface in which, for any input point set, there is no cycle of length one, and another covering space in which there is no cycle of length two. We also exhibit a small point set such that the projection of the Delaunay triangulation on the Bolza surface for any superset has no cycle of length $\le 2$.

The work uses mathematical proofs, algorithmic constructions, and implementation.

Universal Point Sets for Planar Graph Drawings with Circular Arcs

Participant : Monique Teillaud.

In collaboration with Patrizio Angelini (Roma Tre University), David Eppstein (University of California, Irvine), Fabrizio Frati (The University of Sydney), Michael Kaufmann (MPI, Tübingen), Sylvain Lazard (EPI Vegas ), Tamara Mchedlidze (Karlsruhe Institute of Technology), and Alexander Wolff (Universität Würzburg).

We prove that there exists a set $S$ of $n$ points in the plane such that every $n$-vertex planar graph $G$ admits a plane drawing in which every vertex of $G$ is placed on a distinct point of $S$ and every edge of $G$ is drawn as a circular arc. [25]

A Generic Implementation of $d$D Combinatorial Maps in CGAL

Participant : Monique Teillaud.

In collaboration with Guillaume Damiand (Université de Lyon, LIRIS, UMR 5205 CNRS)

We present a generic implementation of $d$D combinatorial maps and linear cell complexes in cgal . A combinatorial map describes an object subdivided into cells; a linear cell complex describes the linear geometry embedding of such a subdivision. In this paper [49] , we show how generic programming and new techniques recently introduced in the C++11 standard allow a fully generic and customizable implementation of these two data structures, while maintaining optimal memory footprint and direct access to all information. To the best of our knowledge, the cgal software packages presented here [59] , [60] offer the only available generic implementation of combinatorial maps in any dimension.

Silhouette of a Random Polytope

Participant : Marc Glisse.

In collaboration with Sylvain Lazard and Marc Pouget (EPI vegas ) and Julien Michel (LMA-Poitiers).

We consider random polytopes defined as the convex hull of a Poisson point process on a sphere in ${ℝ}^3$ such that its average number of points is $n$. We show [52] that the expectation over all such random polytopes of the maximum size of their silhouettes viewed from infinity is $\Theta \left(\sqrt{n}\right)$.

A New Approach to Output-Sensitive Voronoi Diagrams and Delaunay Triangulations

Participant : Donald Sheehy.

In collaboration with Gary Miller (Carnegie Mellon University)

We describe [35] a new algorithm for computing the Voronoi diagram of a set of n points in constant-dimensional Euclidean space. The running time of our algorithm is $O(flognlog\Delta )$ where $f$ is the output complexity of the Voronoi diagram and $\Delta$ is the spread of the input, the ratio of largest to smallest pairwise distances. Despite the simplicity of the algorithm and its analysis, it improves on the state of the art for all inputs with polynomial spread and near-linear output size. The key idea is to first build the Voronoi diagram of a superset of the input points using ideas from Voronoi refinement mesh generation. Then, the extra points are removed in a straightforward way that allows the total work to be bounded in terms of the output complexity, yielding the output sensitive bound. The removal only involves local flips and is inspired by kinetic data structures.

A Fast Algorithm for Well-Spaced Points and Approximate Delaunay Graphs

Participant : Donald Sheehy.

In collaboration with Gary Miller and Ameya Velingker (Carnegie Mellon University)

We present [32] a new algorithm that produces a well-spaced superset of points conforming to a given input set in any dimension with guaranteed optimal output size. We also provide an approximate Delaunay graph on the output points. Our algorithm runs in expected time $O\left(2^{O\left(d\right)}(nlogn+m)\right)$, where $n$ is the input size, $m$ is the output point set size, and $d$ is the ambient dimension. The constants only depend on the desired element quality bounds.

To gain this new efficiency, the algorithm approximately maintains the Voronoi diagram of the current set of points by storing a superset of the Delaunay neighbors of each point. By retaining quality of the Voronoi diagram and avoiding the storage of the full Voronoi diagram, a simple exponential dependence on $d$ is obtained in the running time. Thus, if one only wants the approximate neighbors structure of a refined Delaunay mesh conforming to a set of input points, the algorithm will return a size $2^{O\left(d\right)}m$ graph in $2^{O\left(d\right)}(nlogn+m)$ expected time. If $m$ is superlinear in $n$, then we can produce a hierarchically well-spaced superset of size $2^{O\left(d\right)}n$ in $2^{O\left(d\right)}nlogn$ expected time.

Geometric Separators and the Parabolic Lift

Participant : Donald Sheehy.

A geometric separator for a set $U$ of $n$ geometric objects (usually balls) is a small (sublinear in $n$) subset whose removal disconnects the intersection graph of $U$ into roughly equal sized parts. These separators provide a natural way to do divide and conquer in geometric settings. A particularly nice geometric separator algorithm originally introduced by Miller and Thurston has three steps: compute a centerpoint in a space of one dimension higher than the input, compute a conformal transformation that “centers” the centerpoint, and finally, use the computed transformation to sample a sphere in the original space. The output separator is the subset of S intersecting this sphere. It is both simple and elegant. We show [36] that a change of perspective (literally) can make this algorithm even simpler by eliminating the entire middle step. By computing the centerpoint of the points lifted onto a paraboloid rather than using the stereographic map as in the original method, one can sample the desired sphere directly, without computing the conformal transformation.