7,476 research outputs found
A Voronoi poset
Given a set S of n points in general position, we consider all k-th order
Voronoi diagrams on S, for k=1,...,n, simultaneously. We deduce symmetry
relations for the number of faces, number of vertices and number of circles of
certain orders. These symmetry relations are independent of the position of the
sites in S. As a consequence we show that the reduced Euler characteristic of
the poset of faces equals zero whenever n odd.Comment: 14 pages 4 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
On the Structure of Higher Order Voronoi Cells
The classic Voronoi cells can be generalized to a higher-order version by
considering the cells of points for which a given -element subset of the set
of sites consists of the closest sites. We study the structure of the
-order Voronoi cells and illustrate our theoretical findings with a case
study of two-dimensional higher-order Voronoi cells for four points.Comment: Minor correction
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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