430 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
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
A Sparse Delaunay Filtration
We show how a filtration of Delaunay complexes can be used to approximate the persistence diagram of the distance to a point set in ?^d. Whereas the full Delaunay complex can be used to compute this persistence diagram exactly, it may have size O(n^?d/2?). In contrast, our construction uses only O(n) simplices. The central idea is to connect Delaunay complexes on progressively denser subsamples by considering the flips in an incremental construction as simplices in d+1 dimensions. This approach leads to a very simple and straightforward proof of correctness in geometric terms, because the final filtration is dual to a (d+1)-dimensional Voronoi construction similar to the standard Delaunay filtration. We also, show how this complex can be efficiently constructed
Improved Incremental Randomized Delaunay Triangulation
We 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, and small memory occupation.
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 levels below. Point location is done by marching in a
triangulation to determine the nearest neighbor of the query at that level,
then the march restarts from that neighbor at the level below. Using a small
sample (3%) allows a small memory occupation; the march and the use of the
nearest neighbor to change levels quickly locate the query.Comment: 19 pages, 7 figures Proc. 14th Annu. ACM Sympos. Comput. Geom.,
106--115, 199
Self-Improving Algorithms
We investigate ways in which an algorithm can improve its expected
performance by fine-tuning itself automatically with respect to an unknown
input distribution D. We assume here that D is of product type. More precisely,
suppose that we need to process a sequence I_1, I_2, ... of inputs I = (x_1,
x_2, ..., x_n) of some fixed length n, where each x_i is drawn independently
from some arbitrary, unknown distribution D_i. The goal is to design an
algorithm for these inputs so that eventually the expected running time will be
optimal for the input distribution D = D_1 * D_2 * ... * D_n.
We give such self-improving algorithms for two problems: (i) sorting a
sequence of numbers and (ii) computing the Delaunay triangulation of a planar
point set. Both algorithms achieve optimal expected limiting complexity. The
algorithms begin with a training phase during which they collect information
about the input distribution, followed by a stationary regime in which the
algorithms settle to their optimized incarnations.Comment: 26 pages, 8 figures, preliminary versions appeared at SODA 2006 and
SoCG 2008. Thorough revision to improve the presentation of the pape
Witness (Delaunay) Graphs
Proximity graphs are used in several areas in which a neighborliness
relationship for input data sets is a useful tool in their analysis, and have
also received substantial attention from the graph drawing community, as they
are a natural way of implicitly representing graphs. However, as a tool for
graph representation, proximity graphs have some limitations that may be
overcome with suitable generalizations. We introduce a generalization, witness
graphs, that encompasses both the goal of more power and flexibility for graph
drawing issues and a wider spectrum for neighborhood analysis. We study in
detail two concrete examples, both related to Delaunay graphs, and consider as
well some problems on stabbing geometric objects and point set discrimination,
that can be naturally described in terms of witness graphs.Comment: 27 pages. JCCGG 200
VoroCrust: Voronoi Meshing Without Clipping
Polyhedral meshes are increasingly becoming an attractive option with
particular advantages over traditional meshes for certain applications. What
has been missing is a robust polyhedral meshing algorithm that can handle broad
classes of domains exhibiting arbitrarily curved boundaries and sharp features.
In addition, the power of primal-dual mesh pairs, exemplified by
Voronoi-Delaunay meshes, has been recognized as an important ingredient in
numerous formulations. The VoroCrust algorithm is the first provably-correct
algorithm for conforming polyhedral Voronoi meshing for non-convex and
non-manifold domains with guarantees on the quality of both surface and volume
elements. A robust refinement process estimates a suitable sizing field that
enables the careful placement of Voronoi seeds across the surface circumventing
the need for clipping and avoiding its many drawbacks. The algorithm has the
flexibility of filling the interior by either structured or random samples,
while preserving all sharp features in the output mesh. We demonstrate the
capabilities of the algorithm on a variety of models and compare against
state-of-the-art polyhedral meshing methods based on clipped Voronoi cells
establishing the clear advantage of VoroCrust output.Comment: 18 pages (including appendix), 18 figures. Version without compressed
images available on https://www.dropbox.com/s/qc6sot1gaujundy/VoroCrust.pdf.
Supplemental materials available on
https://www.dropbox.com/s/6p72h1e2ivw6kj3/VoroCrust_supplemental_materials.pd
Load-Balancing for Parallel Delaunay Triangulations
Computing the Delaunay triangulation (DT) of a given point set in
is one of the fundamental operations in computational geometry.
Recently, Funke and Sanders (2017) presented a divide-and-conquer DT algorithm
that merges two partial triangulations by re-triangulating a small subset of
their vertices - the border vertices - and combining the three triangulations
efficiently via parallel hash table lookups. The input point division should
therefore yield roughly equal-sized partitions for good load-balancing and also
result in a small number of border vertices for fast merging. In this paper, we
present a novel divide-step based on partitioning the triangulation of a small
sample of the input points. In experiments on synthetic and real-world data
sets, we achieve nearly perfectly balanced partitions and small border
triangulations. This almost cuts running time in half compared to
non-data-sensitive division schemes on inputs exhibiting an exploitable
underlying structure.Comment: Short version submitted to EuroPar 201
Well-Centered Triangulation
Meshes composed of well-centered simplices have nice orthogonal dual meshes
(the dual Voronoi diagram). This is useful for certain numerical algorithms
that prefer such primal-dual mesh pairs. We prove that well-centered meshes
also have optimality properties and relationships to Delaunay and minmax angle
triangulations. We present an iterative algorithm that seeks to transform a
given triangulation in two or three dimensions into a well-centered one by
minimizing a cost function and moving the interior vertices while keeping the
mesh connectivity and boundary vertices fixed. The cost function is a direct
result of a new characterization of well-centeredness in arbitrary dimensions
that we present. Ours is the first optimization-based heuristic for
well-centeredness, and the first one that applies in both two and three
dimensions. We show the results of applying our algorithm to small and large
two-dimensional meshes, some with a complex boundary, and obtain a
well-centered tetrahedralization of the cube. We also show numerical evidence
that our algorithm preserves gradation and that it improves the maximum and
minimum angles of acute triangulations created by the best known previous
method.Comment: Content has been added to experimental results section. Significant
edits in introduction and in summary of current and previous results. Minor
edits elsewher
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