Section: New Results

Classical and probabilistic computational geometry

Participants : Xavier Goaoc, Guillaume Moroz, Sylvain Lazard, Marc Pouget.

Probabilistic complexity analysis of random geometric structures

Average-case analysis of data-structures or algorithms is commonly used in computational geometry when the more classical worst-case analysis is deemed overly pessimistic. Since these analyses are often intricate, the models of random geometric data that can be handled are often simplistic and far from "realistic inputs".

Complexity analysis of random geometric structures made simpler.   In a joint work with Olivier Devillers and Marc Glisse (Inria Geometrica), we presented a new simple scheme for the analysis of geometric structures. While this scheme only produces results up to a polylog factor, it is much simpler to apply than the classical techniques and therefore succeeds in analyzing new input distributions related to smoothed complexity analysis. We illustrated our method on two classical structures: convex hulls and Delaunay triangulations. Specifically, we gave short and elementary proofs of the classical results that $n$ points uniformly distributed in a ball in $R^d$ have a convex hull and a Delaunay triangulation of respective expected complexities $\tilde{\Theta }\left(n^{\left(\right(d+1)/(d-1\left)\right)}\right)$ and $\tilde{\Theta }\left(n\right)$. We then prove that if we start with $n$ points well-spread on a sphere, e.g. an $(ϵ,\kappa )$-sample of that sphere, and perturb that sample by moving each point randomly and uniformly within distance at most $\delta$ of its initial position, then the expected complexity of the convex hull of the resulting point set is $\tilde{\Theta }{\left(\sqrt{(}n\right)}^{(1-1/d)}{\delta }^{-(d-1)/\left(4d\right)})$. We presented these results in the Symposium on Computational Geometry 2013 [20] .

Monotonicity of the number of facets of random polytopes.   We also proved a result on the size of the convex hull $K_n$ of $n$ points sampled uniformly in a convex set $K$. More precisely, let $u_n^{K,i}$ be the expected number of facets of dimension $i$ of the convex hull. We proved that, in the plane, $u_n^{K,0}$ is an increasing sequence. In higher dimension, if $K$ is a convex, smooth, compact body, then we showed that the sequence $u_n^{K,d-1}$ is asymptotically increasing. This result, published in the Electronic Communications in Probability [13] , was obtained in collaboration with Olivier Devillers and Marc Glisse (Inria Geometrica) and Matthias Reitzner (Osnabruck Univ.).

Worst-case silhouette size of random polytopes.   Finally, we studied from a probabilistic point of view the size of the silhouette of a polyhedron. While the silhouette size of a polyhedron with $n$ vertices may be linear for some view points, several experimental and theoretical studies show a sublinear behavior for a wide range of constraints. The latest result on the subject proves a bound in $\Theta \left(\sqrt{n}\right)$ on the size of the silhouette from a random view point of polyhedra of size $n$ approximating non-convex surfaces in a reasonable way [9] . This result considers the polyhedron given and average the sizes of the silhouettes over all view points. This year, we addressed the problem of bounding the worst-case size of the silhouette where the average is taken over a set of polyhedra. Namely, 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 that the expectation over all such random polytopes of the maximum size of their silhouettes viewed from infinity is $\Theta \left(\sqrt{n}\right)$. This work was done in collaboration with Marc Glisse (Inria Geometrica) and Julien Michel (Université de Poitiers) [24] .

Embedding geometric structures

We continued working this year on the problem of embedding geometric objects on a grid of ${ℝ}^3$. Essentially all industrial applications take, as input, models defined with a fixed-precision floating-point arithmetic, typically doubles. As a consequence, geometric objects constructed using exact arithmetic must be embedded on a fixed-precision grid before they can be used as input in other software. More precisely, the problem is, given a geometric object, to find a similar object representable with fixed-precision floating-point arithmetic, where similar means topologically equivalent, close according to some distance function, etc. We are working on the problem of rounding polyhedral subdivisions on a grid of ${ℝ}^3$, where the only known method, due to Fortune in 1999, considers a grid whose refinement depends on the combinatorial complexity of the input, which does not solve the problem at hand. This project is joint work with Olivier Devillers (Inria Geometrica) and William Lenhart (Williams College, USA).

Bounded-Curvature Shortest Paths

We considered the problem of computing shortest paths having curvature at most one almost everywhere and visiting a sequence of $n$ points in the plane in a given order. This problem is a sub-problem of the Dubins Traveling Salesman Problem and also arises naturally in path planning for point car-like robots in the presence of polygonal obstacles. We showed that when consecutive waypoints are distance at least four apart, this question reduces to a family of convex optimization problems over polyhedra in ${ℝ}^n$. This result, done in collaboration with Hyo-Sil Kim (KAIST) was published in the SIAM Journal on Computing [15] .

Approximating Geodesics in Meshes

A standard way to approximate the distance between any two vertices $p$ and $q$ on a mesh is to compute, in the associated graph, 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 of the graph yields an efficient computation of approximate distances between any two vertices. One standard method for choosing $k$ sources, which has been used extensively and successfully for isometry-invariant surface processing, 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 ${ℱ}_{FPS}$ 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 show that ${ℱ}_{FPS}$ can be bounded in terms of the minimal value ${ℱ}^{*}$ of the stretch factor obtained using an optimal placement of $k$ sources as ${ℱ}_{FPS}\le 2r_e^2{ℱ}^{*}+2r_e^2+8r_e+1$, where $r_e$ is the ratio of the lengths of the longest and the shortest edges of the graph. This provides some evidence explaining why farthest point sampling has been used successfully for isometry-invariant shape processing. Furthermore, we showed that it is NP-complete to find $k$ sources that minimize the stretch factor [25] .

On Point-sets that Support Planar Graphs

A set of points is said universal if it supports a crossing-free drawing of any planar graph. For a planar graph with $n$ vertices, if bends on edges of the drawing are permitted, universal point-sets of size $n$ are known, but only if the bend-points are in arbitrary positions. If the locations of the bend-points must also be specified as part of the point-set, no result was known, and we prove that any planar graph with $n$ vertices can be drawn on a universal set $𝒮$ of $O(n^2/logn)$ points with at most one bend per edge and with the vertices and the bend points in $𝒮$. If two bends per edge are allowed, we show that $O(nlogn)$ points are sufficient, and if three bends per edge are allowed, $\Theta \left(n\right)$ points are sufficient. When no bends on edges are permitted, no universal point-set of size $o\left(n^2\right)$ is known for the class of planar graphs. We show that a set of $n$ points in balanced biconvex position supports the class of maximum-degree-3 series-parallel lattices. These results were published this year in the journal Computational Geometry: Theory and Application [14] .

We also considered the setting in which graphs are drawn with curved edges. We proved that, surprisingly, there exists a universal set of $n$ points in the plane for which every $n$-vertex planar graph admits a planar drawing in which the edges are drawn as a circular arc. This result was presented in the Canadian Conference on Computational Geometry [17] .