2,072 research outputs found
Vesicle computers: Approximating Voronoi diagram on Voronoi automata
Irregular arrangements of vesicles filled with excitable and precipitating
chemical systems are imitated by Voronoi automata --- finite-state machines
defined on a planar Voronoi diagram. Every Voronoi cell takes four states:
resting, excited, refractory and precipitate. A resting cell excites if it has
at least one excited neighbour; the cell precipitates if a ratio of excited
cells in its neighbourhood to its number of neighbours exceed certain
threshold. To approximate a Voronoi diagram on Voronoi automata we project a
planar set onto automaton lattice, thus cells corresponding to data-points are
excited. Excitation waves propagate across the Voronoi automaton, interact with
each other and form precipitate in result of the interaction. Configuration of
precipitate represents edges of approximated Voronoi diagram. We discover
relation between quality of Voronoi diagram approximation and precipitation
threshold, and demonstrate feasibility of our model in approximation Voronoi
diagram of arbitrary-shaped objects and a skeleton of a planar shape.Comment: Chaos, Solitons & Fractals (2011), in pres
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
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New algorithms for minimum-measure simplices and one-dimensional weighted Voronoi diagrams
We present two new algorithms for finding the minimum-measure simplex determined by a set of n points in R^d for arbitrary d >/= 2. The first algorithm runs in time O(n^d log n) using O(n) space. The only data structure used by this algorithms a stack. The second algorithm runs in time O(n^d) using O(n^2) space, which matches the best known time bounds for this problem in all dimensions and exceeds the previous best space bounds for all d > 3. We also present a new optimal algorithm for building one-dimensional multiplicatively weighted Voronoi diagrams that runs in linear time if the points are already sorted
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
Applications of random sampling to on-line algorithms in computational geometry
This paper presents a general framework for the design and randomized analysis of geometric algorithms. These algorithms are on-line and the framework provides general bounds for their expected space and time complexities when averaging over all permutations of the input data. The method is general and can be applied to various geometric problems. The power of the technique is illustrated by new efficient on-line algorithms for constructing convex hulls and Voronoi diagrams in any dimension, Voronoi diagrams of line segments in the plane, arrangements of curves in the plane, and others
MGOS: A library for molecular geometry and its operating system
The geometry of atomic arrangement underpins the structural understanding of molecules in many fields. However, no general framework of mathematical/computational theory for the geometry of atomic arrangement exists. Here we present "Molecular Geometry (MG)'' as a theoretical framework accompanied by "MG Operating System (MGOS)'' which consists of callable functions implementing the MG theory. MG allows researchers to model complicated molecular structure problems in terms of elementary yet standard notions of volume, area, etc. and MGOS frees them from the hard and tedious task of developing/implementing geometric algorithms so that they can focus more on their primary research issues. MG facilitates simpler modeling of molecular structure problems; MGOS functions can be conveniently embedded in application programs for the efficient and accurate solution of geometric queries involving atomic arrangements. The use of MGOS in problems involving spherical entities is akin to the use of math libraries in general purpose programming languages in science and engineering. (C) 2019 The Author(s). Published by Elsevier B.V
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