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Section: New Results

Classical Computational Geometry and Graph Drawing

Participants : Olivier Devillers, Sylvain Lazard.

Monotone Simultaneous Path Embeddings in ${ℝ}^d$

We study the following problem: Given $k$ paths that share the same vertex set, is there a simultaneous geometric embedding of these paths such that each individual drawing is monotone in some direction? We prove that for any dimension $d\ge 2$, there is a set of $d+1$ paths that does not admit a monotone simultaneous geometric embedding [21].

This work was done in collaboration with David Bremner (U. New Brunswick), Marc Glisse (Inria Datashape), Giuseppe Liotta (U. Perugia), Tamara Mchedlidze (Karlsruhe Institute for Technology), Sue Whitesides (U. Victoria), and Stephen Wismath (U. Lethbridge).

Analysis of Farthest Point Sampling for Approximating Geodesics in a Graph

A standard way to approximate the distance between two vertices $p$ and $q$ in a graph is to compute a shortest path from $p$ to $q$ that goes through one of $k$ sources, which are well-chosen vertices. Precomputing the distance between each of the $k$ sources to all vertices yields an efficient computation of approximate distances between any two vertices. One standard method for choosing $k$ sources is the so-called Farthest Point Sampling (FPS), which starts with a random vertex as the first source, and iteratively selects the farthest vertex from the already selected sources.

We analyzed the stretch factor ${ℱ}_{\text{FPS}}$ of approximate geodesics computed using FPS, which is the maximum, over all pairs of distinct vertices, of their approximated distance over their geodesic distance in the graph. We showed that ${ℱ}_{\text{FPS}}$ can be bounded in terms of the minimal value ${ℱ}^{*}$ of the stretch factor obtained using an optimal placement of $k$ sources as ${ℱ}_{\text{FPS}}\le 2r_e^2{ℱ}^{*}+2r_e^2+8r_e+1$, where $r_e$ is the length ratio of longest edge over the shortest edge in the graph. We further showed that the factor $r_e$ is not an artefact of the analysis by providing a class of graphs for which ${ℱ}_{\text{FPS}}\ge \frac{1}{2}{r}_e{ℱ}^{*}$ [18].

This work was done in collaboration with Pegah Kamousi (Université Libre de Bruxelles), Anil Maheshwari (Carleton University), and Stefanie Wuhrer (Inria Grenoble Rhône-Alpes).

Recognizing Shrinkable Complexes is NP-complete

We say that a simplicial complex is shrinkable if there exists a sequence of admissible edge contractions that reduces the complex to a single vertex. We prove that it is NP-complete to decide whether a (three-dimensional) simplicial complex is shrinkable. Along the way, we describe examples of contractible complexes which are not shrinkable [10].

This work was done in collaboration with Dominique Attali (CNRS, Grenoble) and Marc Glisse (Inria Datashape).