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

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)).

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

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

A Hierarchical Approach for Regular Centroidal Voronoi Tessellations

International audienceIn this paper we consider Centroidal Voronoi Tessellations (CVTs) and study their regularity. CVTs are geometric structures that enable regular tessellations of geometric objects and are widely used in shape modeling and analysis. While several efficient iterative schemes, with defined local convergence properties, have been proposed to compute CVTs, little attention has been paid to the evaluation of the resulting cell decompositions. In this paper, we propose a regularity criterion that allows us to evaluate and compare CVTs independently of their sizes and of their cell numbers. This criterion allows us to compare CVTs on a common basis. It builds on earlier theoretical work showing that second moments of cells converge to a lower bound when optimising CVTs. In addition to proposing a regularity criterion, this paper also considers computational strategies to determine regular CVTs. We introduce a hierarchical framework that propagates regularity over decomposition levels and hence provides CVTs with provably better regularities than existing methods. We illustrate these principles with a wide range of experiments on synthetic and real models

The Delaunay Hierarchy

International audienceWe propose a new data structure to compute the Delaunay triangulation of a set of points in the plane. It combines good worst case complexity, fast behavior on real data, small memory occupation and the possibility of fully dynamic insertions and deletions. The location structure is organized into several levels. The lowest level just consists of the triangulation, then each level contains the triangulation of a small sample of the level below. Point location is done by walking in a triangulation to determine the nearest neighbor of the query at that level, then the walk restarts from that neighbor at the level below. Using a small subset (3%) to sample a level allows a small memory occupation; the walk and the use of the nearest neighbor to change levels quickly locate the query