The visible perimeter of an arrangement of disks

Given a collection of n opaque unit disks in the plane, we want to find a stacking order for them that maximizes their visible perimeter---the total length of all pieces of their boundaries visible from above. We prove that if the centers of the disks form a dense point set, i.e., the ratio of their maximum to their minimum distance is O(n^1/2), then there is a stacking order for which the visible perimeter is Omega(n^2/3). We also show that this bound cannot be improved in the case of a sufficiently small n^1/2 by n^1/2 uniform grid. On the other hand, if the set of centers is dense and the maximum distance between them is small, then the visible perimeter is O(n^3/4) with respect to any stacking order. This latter bound cannot be improved either. Finally, we address the case where no more than c disks can have a point in common. These results partially answer some questions of Cabello, Haverkort, van Kreveld, and Speckmann.Comment: 12 pages, 5 figure

Neighborly inscribed polytopes and Delaunay triangulations

We construct a large family of neighborly polytopes that can be realized with all the vertices on the boundary of any smooth strictly convex body. In particular, we show that there are superexponentially many combinatorially distinct neighborly polytopes that admit realizations inscribed on the sphere. These are the first examples of inscribable neighborly polytopes that are not cyclic polytopes, and provide the current best lower bound for the number of combinatorial types of inscribable polytopes (which coincides with the current best lower bound for the number of combinatorial types of polytopes). Via stereographic projections, this translates into a superexponential lower bound for the number of combinatorial types of (neighborly) Delaunay triangulations.Comment: 15 pages, 2 figures. We extended our results to arbitrary smooth strictly convex bodie

Meshes Preserving Minimum Feature Size

The minimum feature size of a planar straight-line graph is the minimum distance between a vertex and a nonincident edge. When such a graph is partitioned into a mesh, the degradation is the ratio of original to final minimum feature size. For an n-vertex input, we give a triangulation (meshing) algorithm that limits degradation to only a constant factor, as long as Steiner points are allowed on the sides of triangles. If such Steiner points are not allowed, our algorithm realizes \ensuremathO(lgn) degradation. This addresses a 14-year-old open problem by Bern, Dobkin, and Eppstein

Complexity analysis of random geometric structures made simpler

Average-case analysis of data-structures or algorithms is commonly used in compu- tational 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". We present a new simple scheme for the analy- sis 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. Abstract: We illustrate our method on two classical structures: convex hulls and Delaunay triangulations. Specifically, we give short and elementary proofs of the classical results that n points uniformly distributed in a ball in Rd have a convex hull and a Delaunay triangulation of respective expected complexities Θ~(n^((d+1)/(d-1)) ) and Θ~(n). We then prove that if we start with n points well-spread on a sphere, e.g. an (ε,κ)-sample of that sphere, and perturb that sample by moving each point ran- domly and uniformly within distance at most δ of its initial position, then the expected complexity of the convex hull of the resulting point set is Θ~( sqrt(n)^(1−1/d) δ^(-(d-1/d)/4)). .L'analyse en moyenne de structure de données et d'algorithmes géométriques est fréquemment utilisée en géométrie algorithmique, un domaine ou' l'analyse dans le cas le pire est souvent très pessimiste. La difficulté de ce type d'analyse fait que les modèles probabilistes utilisés restent simples et relativement éloignées de données réalistes. Nous présentons une nouvelle approche pour l'analyse des structures géométriques. Nos résultats sont seulement 'a des facteurs logarithmiques près, mais notre méthode est plus simple que les classiques du domaine et nous réussissons 'a analyser de nouveau type de distribution liée à la smooth analysis. Nous illustrons notre méthode sur deux structures classiques: l'enveloppe convexe et la triangulation de Delaunay. Plus précisément, nous démontrons simplement le fait, classique, que n points uniformément distribués dans une boule de Rd ont une enveloppe convexe et une triangulation de Delaunay dont l'espérance de la taille est respectivement Θ~(n^((d+1)/(d-1)) ) et Θ~(n). Nous démontrons ensuite que si on se donne ensemble de n points bien distribu ́es sur une sphère, par exemple un (ε, κ)-échantillon de la sphère, et qu'on le perturbe ensuite en déplaçant chaque point uniformément d'une distance δ à partir de sa position initiale, alors l'espérance de la taille de l'enveloppe convexe de ces points est Θ~( sqrt(n)^(1−1/d) δ^(-(d-1/d)/4)).

The Bane of Low-Dimensionality Clustering

In this paper, we give a conditional lower bound of $n^{\Omega(k)}$ on running time for the classic k-median and k-means clustering objectives (where n is the size of the input), even in low-dimensional Euclidean space of dimension four, assuming the Exponential Time Hypothesis (ETH). We also consider k-median (and k-means) with penalties where each point need not be assigned to a center, in which case it must pay a penalty, and extend our lower bound to at least three-dimensional Euclidean space. This stands in stark contrast to many other geometric problems such as the traveling salesman problem, or computing an independent set of unit spheres. While these problems benefit from the so-called (limited) blessing of dimensionality, as they can be solved in time $n^{O(k^{1-1/d})}$ or $2^{n^{1-1/d}}$ in d dimensions, our work shows that widely-used clustering objectives have a lower bound of $n^{\Omega(k)}$, even in dimension four. We complete the picture by considering the two-dimensional case: we show that there is no algorithm that solves the penalized version in time less than $n^{o(\sqrt{k})}$, and provide a matching upper bound of $n^{O(\sqrt{k})}$. The main tool we use to establish these lower bounds is the placement of points on the moment curve, which takes its inspiration from constructions of point sets yielding Delaunay complexes of high complexity

A Linear Bound on the Complexity of the Delaunay triangulation of points on polyhedral surfaces

Delaunay triangulations and Voronoi diagrams have found numerous applications in surface modeling, surface mesh generation, deformable surface modeling and surface reconstruction. Many algorithms in these applications begin by constructing the three-dimensional Delaunay triangulation of a finite set of points scattered over a surface. Their running-time therefore depends on the complexity of the Delaunay triangulation of such point sets. Although the Delaunay triangulation of points in ^3 can be quadratic in the worst-case, we show that, under some mild sampling condition, the complexity of the 3D Delaunay triangulation of points distributed on a fixed number of facets of ^3 (e.g. the facets of a polyhedron) is linear. Our bound is deterministic and the constants are explicitly given

On the Number of Facets of Three-Dimensional Dirichlet Stereohedra III: Full Cubic Groups

We are interested in the maximum possible number of facets that Dirichlet stereohedra for three-dimensional crystallographic groups can have. The problem for non-cubic groups was studied in previous papers by D. Bochis and the second author (Discrete Comput. Geom. 25:3 (2001), 419-444, and Beitr. Algebra Geom., 47:1 (2006), 89-120). This paper deals with ''full'' cubic groups, while ''quarter'' cubic groups are left for a subsequent paper. Here, ''full'' and ''quarter'' refers to the recent classification of three-dimensional crystallographic groups by Conway, Delgado-Friedrichs, Huson and Thurston (math.MG/9911185, Beitr. Algebra Geom. 42.2 (2001), 475-507). Our main result in this paper is that Dirichlet stereohedra for any of the 27 full groups cannot have more than 25 facets. We also find stereohedra with 17 facets for one of these groups.Comment: 28 pages, 12 figures. Changes from v1: apart of some editing (mostly at the end of the introduction) and addition of references, an appendix has been added, which analyzes the case where the base point does not have trivial stabilize

Dense point sets have sparse Delaunay triangulations

The spread of a finite set of points is the ratio between the longest and shortest pairwise distances. We prove that the Delaunay triangulation of any set of n points in R^3 with spread D has complexity O(D^3). This bound is tight in the worst case for all D = O(sqrt{n}). In particular, the Delaunay triangulation of any dense point set has linear complexity. We also generalize this upper bound to regular triangulations of k-ply systems of balls, unions of several dense point sets, and uniform samples of smooth surfaces. On the other hand, for any n and D=O(n), we construct a regular triangulation of complexity Omega(nD) whose n vertices have spread D.Comment: 31 pages, 11 figures. Full version of SODA 2002 paper. Also available at http://www.cs.uiuc.edu/~jeffe/pubs/screw.htm