5,371 research outputs found
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
Large-n conditional facedness m_n of 3D Poisson-Voronoi cells
We consider the three-dimensional Poisson-Voronoi tessellation and study the
average facedness m_n of a cell known to neighbor an n-faced cell. Whereas
Aboav's law states that m_n=A+B/n, theoretical arguments indicate an asymptotic
expansion m_n = 8 + k_1 n^{-1/6} +.... Recent new Monte Carlo data due to Lazar
et al., based on a very large data set, now clearly rule out Aboav's law. In
this work we determine the numerical value of k_1 and compare the expansion to
the Monte Carlo data. The calculation of k_1 involves an auxiliary planar
cellular structure composed of circular arcs, that we will call the
Poisson-Moebius diagram. It is a special case of more general Moebius diagrams
(or multiplicatively weighted power diagrams) and is of interest for its own
sake. We obtain exact results for the total edge length per unit area, which is
a prerequisite for the coefficient k_1, and a few other quantities in this
diagram.Comment: 18 pages, 5 figure
The projector algorithm: a simple parallel algorithm for computing Voronoi diagrams and Delaunay graphs
The Voronoi diagram is a certain geometric data structure which has numerous
applications in various scientific and technological fields. The theory of
algorithms for computing 2D Euclidean Voronoi diagrams of point sites is rich
and useful, with several different and important algorithms. However, this
theory has been quite steady during the last few decades in the sense that no
essentially new algorithms have entered the game. In addition, most of the
known algorithms are serial in nature and hence cast inherent difficulties on
the possibility to compute the diagram in parallel. In this paper we present
the projector algorithm: a new and simple algorithm which enables the
(combinatorial) computation of 2D Voronoi diagrams. The algorithm is
significantly different from previous ones and some of the involved concepts in
it are in the spirit of linear programming and optics. Parallel implementation
is naturally supported since each Voronoi cell can be computed independently of
the other cells. A new combinatorial structure for representing the cells (and
any convex polytope) is described along the way and the computation of the
induced Delaunay graph is obtained almost automatically.Comment: This is a major revision; re-organization and better presentation of
some parts; correction of several inaccuracies; improvement of some proofs
and figures; added references; modification of the title; the paper is long
but more than half of it is composed of proofs and references: it is
sufficient to look at pages 5, 7--11 in order to understand the algorith
Kinetic and Dynamic Delaunay tetrahedralizations in three dimensions
We describe the implementation of algorithms to construct and maintain
three-dimensional dynamic Delaunay triangulations with kinetic vertices using a
three-simplex data structure. The code is capable of constructing the geometric
dual, the Voronoi or Dirichlet tessellation. Initially, a given list of points
is triangulated. Time evolution of the triangulation is not only governed by
kinetic vertices but also by a changing number of vertices. We use
three-dimensional simplex flip algorithms, a stochastic visibility walk
algorithm for point location and in addition, we propose a new simple method of
deleting vertices from an existing three-dimensional Delaunay triangulation
while maintaining the Delaunay property. The dual Dirichlet tessellation can be
used to solve differential equations on an irregular grid, to define partitions
in cell tissue simulations, for collision detection etc.Comment: 29 pg (preprint), 12 figures, 1 table Title changed (mainly
nomenclature), referee suggestions included, typos corrected, bibliography
update
Alpha, Betti and the Megaparsec Universe: on the Topology of the Cosmic Web
We study the topology of the Megaparsec Cosmic Web in terms of the
scale-dependent Betti numbers, which formalize the topological information
content of the cosmic mass distribution. While the Betti numbers do not fully
quantify topology, they extend the information beyond conventional cosmological
studies of topology in terms of genus and Euler characteristic. The richer
information content of Betti numbers goes along the availability of fast
algorithms to compute them.
For continuous density fields, we determine the scale-dependence of Betti
numbers by invoking the cosmologically familiar filtration of sublevel or
superlevel sets defined by density thresholds. For the discrete galaxy
distribution, however, the analysis is based on the alpha shapes of the
particles. These simplicial complexes constitute an ordered sequence of nested
subsets of the Delaunay tessellation, a filtration defined by the scale
parameter, . As they are homotopy equivalent to the sublevel sets of
the distance field, they are an excellent tool for assessing the topological
structure of a discrete point distribution. In order to develop an intuitive
understanding for the behavior of Betti numbers as a function of , and
their relation to the morphological patterns in the Cosmic Web, we first study
them within the context of simple heuristic Voronoi clustering models.
Subsequently, we address the topology of structures emerging in the standard
LCDM scenario and in cosmological scenarios with alternative dark energy
content. The evolution and scale-dependence of the Betti numbers is shown to
reflect the hierarchical evolution of the Cosmic Web and yields a promising
measure of cosmological parameters. We also discuss the expected Betti numbers
as a function of the density threshold for superlevel sets of a Gaussian random
field.Comment: 42 pages, 14 figure
On the Complexity of Randomly Weighted Voronoi Diagrams
In this paper, we provide an bound on the expected
complexity of the randomly weighted Voronoi diagram of a set of sites in
the plane, where the sites can be either points, interior-disjoint convex sets,
or other more general objects. Here the randomness is on the weight of the
sites, not their location. This compares favorably with the worst case
complexity of these diagrams, which is quadratic. As a consequence we get an
alternative proof to that of Agarwal etal [AHKS13] of the near linear
complexity of the union of randomly expanded disjoint segments or convex sets
(with an improved bound on the latter). The technique we develop is elegant and
should be applicable to other problems
Bregman Voronoi Diagrams: Properties, Algorithms and Applications
The Voronoi diagram of a finite set of objects is a fundamental geometric
structure that subdivides the embedding space into regions, each region
consisting of the points that are closer to a given object than to the others.
We may define many variants of Voronoi diagrams depending on the class of
objects, the distance functions and the embedding space. In this paper, we
investigate a framework for defining and building Voronoi diagrams for a broad
class of distance functions called Bregman divergences. Bregman divergences
include not only the traditional (squared) Euclidean distance but also various
divergence measures based on entropic functions. Accordingly, Bregman Voronoi
diagrams allow to define information-theoretic Voronoi diagrams in statistical
parametric spaces based on the relative entropy of distributions. We define
several types of Bregman diagrams, establish correspondences between those
diagrams (using the Legendre transformation), and show how to compute them
efficiently. We also introduce extensions of these diagrams, e.g. k-order and
k-bag Bregman Voronoi diagrams, and introduce Bregman triangulations of a set
of points and their connexion with Bregman Voronoi diagrams. We show that these
triangulations capture many of the properties of the celebrated Delaunay
triangulation. Finally, we give some applications of Bregman Voronoi diagrams
which are of interest in the context of computational geometry and machine
learning.Comment: Extend the proceedings abstract of SODA 2007 (46 pages, 15 figures
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