1,026 research outputs found
Continuum Percolation in the Relative Neighborhood Graph
In the present study, we establish the existence of nontrivial site
percolation threshold in the Relative Neighborhood Graph (RNG) for Poisson
stationary point process with unit intensity in the plane
Spatially-distributed coverage optimization and control with limited-range interactions
This paper presents coordination algorithms for groups of mobile agents
performing deployment and coverage tasks. As an important modeling constraint,
we assume that each mobile agent has a limited sensing/communication radius.
Based on the geometry of Voronoi partitions and proximity graphs, we analyze a
class of aggregate objective functions and propose coverage algorithms in
continuous and discrete time. These algorithms have convergence guarantees and
are spatially distributed with respect to appropriate proximity graphs.
Numerical simulations illustrate the results.Comment: 31 pages, some figures left out because of size limits. Complete
preprint version available at http://motion.csl.uiuc.ed
Connected Spatial Networks over Random Points and a Route-Length Statistic
We review mathematically tractable models for connected networks on random
points in the plane, emphasizing the class of proximity graphs which deserves
to be better known to applied probabilists and statisticians. We introduce and
motivate a particular statistic measuring shortness of routes in a network.
We illustrate, via Monte Carlo in part, the trade-off between normalized
network length and in a one-parameter family of proximity graphs. How close
this family comes to the optimal trade-off over all possible networks remains
an intriguing open question. The paper is a write-up of a talk developed by the
first author during 2007--2009.Comment: Published in at http://dx.doi.org/10.1214/10-STS335 the Statistical
Science (http://www.imstat.org/sts/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Beta-Skeletons have Unbounded Dilation
A fractal construction shows that, for any beta>0, the beta-skeleton of a
point set can have arbitrarily large dilation. In particular this applies to
the Gabriel graph.Comment: 8 pages, 9 figure
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