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Section: New Results

Classical Computational Geometry and Graph Drawing

Participants : Vincent Despré, Olivier Devillers, Sylvain Lazard.

Delaunay Triangulations of Points on Circles

Delaunay triangulations of a point set in the Euclidean plane are ubiquitous in a number of computational sciences, including computational geometry. Delaunay triangulations are not well defined as soon as 4 or more points are concyclic but since it is not a generic situation, this difficulty is usually handled by using a (symbolic or explicit) perturbation. As an alternative, we proposed to define a canonical triangulation for a set of concyclic points by using a max-min angle characterization of Delaunay triangulations. This point of view leads to a well defined and unique triangulation as long as there are no symmetric quadruples of points. This unique triangulation can be computed in quasi-linear time by a very simple algorithm [18].

In collaboration with Hugo Parlier and Jean-Marc Schlenker (University of Luxembourg).

Improved Routing on the Delaunay Triangulation

A geometric graph $G=(P,E)$ is a set of points in the plane and edges between pairs of points, where the weight of each edge is equal to the Euclidean distance between the corresponding points. In $k$-local routing we find a path through $G$ from a source vertex $s$ to a destination vertex $t$, using only knowledge of the present location, the locations of $s$ and $t$, and the $k$-neighbourhood of the current vertex. We presented [11] an algorithm for 1-local routing on the Delaunay triangulation, and show that it finds a path between a source vertex $s$ and a target vertex $t$ that is not longer than $3.56\left|st\right|$, improving the previous bound of $5.9$.

In collaboration with Nicolas Bonichon (Labri), Prosenjit Bose, Jean-Lou De Carufel, Michiel Smid and Daryl Hill (Carleton University)

Limits of Order Types

We completed an extended version of a work published at SoCG 2015, in which we apply ideas from the theory of limits of dense combinatorial structures to study order types, which are combinatorial encodings of finite point sets. Using flag algebras we obtain new numerical results on the Erdös problem of finding the minimal density of 5-or 6-tuples in convex position in an arbitrary point set, and also an inequality expressing the difficulty of sampling order types uniformly. Next we establish results on the analytic representation of limits of order types by planar measures. Our main result is a rigidity theorem: we show that if sampling two measures induce the same probability distribution on order types, then these measures are projectively equivalent provided the support of at least one of them has non-empty interior. We also show that some condition on the Hausdorff dimension of the support is necessary to obtain projective rigidity and we construct limits of order types that cannot be represented by a planar measure. Returning to combinatorial geometry we relate the regularity of this analytic representation to the aforementioned problem of Erdös on the density of $k$-tuples in convex position, for large $k$ [20].

In collaboration with Alfredo Hubard (Laboratoire d'Informatique Gaspard-Monge) Rémi De Joannis de Verclos (Radboud university, Nijmegen) Jean-Sébastien Sereni (CNRS) Jan Volec (Department of Mathematics and Computer Science, Emory University)

Snap rounding polyhedral subdivisions

Let $𝒫$ be a set of $n$ polygons in ${ℝ}^3$, each of constant complexity and with pairwise disjoint interiors. We propose a rounding algorithm that maps $𝒫$ to a simplicial complex $𝒬$ whose vertices have integer coordinates. Every face of $𝒫$ is mapped to a set of faces (or edges or vertices) of $𝒬$ and the mapping from $𝒫$ to $𝒬$ can be build through a continuous motion of the faces such that (i) the $L_{\infty }$ Hausdorff distance between a face and its image during the motion is at most 3/2 and (ii) if two points become equal during the motion they remain equal through the rest of the motion. In the worse case, the size of $𝒬$ is $O\left(n^{15}\right)$, but we conjecture a good complexity of $O\left(n\sqrt{n}\right)$ in practice on non-pathological data [12].

In collaboration with William J. Lenhart (Williams College, USA).

On the Edge-length Ratio of Outerplanar Graphs

We show that any outerplanar graph admits a planar straight-line drawing such that the length ratio of the longest to the shortest edges is strictly less than 2. This result is tight in the sense that for any $ϵ>0$ there are outerplanar graphs that cannot be drawn with an edge-length ratio smaller than $2-ϵ$. We also show that this ratio cannot be bounded if the embeddings of the outerplanar graphs are given [9].

In collaboration with William J. Lenhart (Williams College, USA) and Giuseppe Liotta (Università di Perugia, Italy).