4,112 research outputs found
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
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
On Deletion in Delaunay Triangulation
This paper presents how the space of spheres and shelling may be used to
delete a point from a -dimensional triangulation efficiently. In dimension
two, if k is the degree of the deleted vertex, the complexity is O(k log k),
but we notice that this number only applies to low cost operations, while time
consuming computations are only done a linear number of times.
This algorithm may be viewed as a variation of Heller's algorithm, which is
popular in the geographic information system community. Unfortunately, Heller
algorithm is false, as explained in this paper.Comment: 15 pages 5 figures. in Proc. 15th Annu. ACM Sympos. Comput. Geom.,
181--188, 199
Tessellations and Pattern Formation in Plant Growth and Development
The shoot apical meristem (SAM) is a dome-shaped collection of cells at the
apex of growing plants from which all above-ground tissue ultimately derives.
In Arabidopsis thaliana (thale cress), a small flowering weed of the
Brassicaceae family (related to mustard and cabbage), the SAM typically
contains some three to five hundred cells that range from five to ten microns
in diameter. These cells are organized into several distinct zones that
maintain their topological and functional relationships throughout the life of
the plant. As the plant grows, organs (primordia) form on its surface flanks in
a phyllotactic pattern that develop into new shoots, leaves, and flowers.
Cross-sections through the meristem reveal a pattern of polygonal tessellation
that is suggestive of Voronoi diagrams derived from the centroids of cellular
nuclei. In this chapter we explore some of the properties of these patterns
within the meristem and explore the applicability of simple, standard
mathematical models of their geometry.Comment: Originally presented at: "The World is a Jigsaw: Tessellations in the
Sciences," Lorentz Center, Leiden, The Netherlands, March 200
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
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