215,050 research outputs found
Continuous Average Straightness in Spatial Graphs
The Straightness is a measure designed to characterize a pair of vertices in
a spatial graph. It is defined as the ratio of the Euclidean distance to the
graph distance between these vertices. It is often used as an average, for
instance to describe the accessibility of a single vertex relatively to all the
other vertices in the graph, or even to summarize the graph as a whole. In some
cases, one needs to process the Straightness between not only vertices, but
also any other points constituting the graph of interest. Suppose for instance
that our graph represents a road network and we do not want to limit ourselves
to crossroad-to-crossroad itineraries, but allow any street number to be a
starting point or destination. In this situation, the standard approach
consists in: 1) discretizing the graph edges, 2) processing the
vertex-to-vertex Straightness considering the additional vertices resulting
from this discretization, and 3) performing the appropriate average on the
obtained values. However, this discrete approximation can be computationally
expensive on large graphs, and its precision has not been clearly assessed. In
this article, we adopt a continuous approach to average the Straightness over
the edges of spatial graphs. This allows us to derive 5 distinct measures able
to characterize precisely the accessibility of the whole graph, as well as
individual vertices and edges. Our method is generic and could be applied to
other measures designed for spatial graphs. We perform an experimental
evaluation of our continuous average Straightness measures, and show how they
behave differently from the traditional vertex-to-vertex ones. Moreover, we
also study their discrete approximations, and show that our approach is
globally less demanding in terms of both processing time and memory usage. Our
R source code is publicly available under an open source license
Chaotic Orbits in Thermal-Equilibrium Beams: Existence and Dynamical Implications
Phase mixing of chaotic orbits exponentially distributes these orbits through
their accessible phase space. This phenomenon, commonly called ``chaotic
mixing'', stands in marked contrast to phase mixing of regular orbits which
proceeds as a power law in time. It is operationally irreversible; hence, its
associated e-folding time scale sets a condition on any process envisioned for
emittance compensation. A key question is whether beams can support chaotic
orbits, and if so, under what conditions? We numerically investigate the
parameter space of three-dimensional thermal-equilibrium beams with space
charge, confined by linear external focusing forces, to determine whether the
associated potentials support chaotic orbits. We find that a large subset of
the parameter space does support chaos and, in turn, chaotic mixing. Details
and implications are enumerated.Comment: 39 pages, including 14 figure
The Densest k-Subhypergraph Problem
The Densest -Subgraph (DS) problem, and its corresponding minimization
problem Smallest -Edge Subgraph (SES), have come to play a central role
in approximation algorithms. This is due both to their practical importance,
and their usefulness as a tool for solving and establishing approximation
bounds for other problems. These two problems are not well understood, and it
is widely believed that they do not an admit a subpolynomial approximation
ratio (although the best known hardness results do not rule this out).
In this paper we generalize both DS and SES from graphs to hypergraphs.
We consider the Densest -Subhypergraph problem (given a hypergraph ,
find a subset of vertices so as to maximize the number of
hyperedges contained in ) and define the Minimum -Union problem (given a
hypergraph, choose of the hyperedges so as to minimize the number of
vertices in their union). We focus in particular on the case where all
hyperedges have size 3, as this is the simplest non-graph setting. For this
case we provide an -approximation (for arbitrary constant )
for Densest -Subhypergraph and an -approximation for
Minimum -Union. We also give an -approximation for Minimum
-Union in general hypergraphs. Finally, we examine the interesting special
case of interval hypergraphs (instances where the vertices are a subset of the
natural numbers and the hyperedges are intervals of the line) and prove that
both problems admit an exact polynomial time solution on these instances.Comment: 21 page
Characterization of chaos in random maps
We discuss the characterization of chaotic behaviours in random maps both in
terms of the Lyapunov exponent and of the spectral properties of the
Perron-Frobenius operator. In particular, we study a logistic map where the
control parameter is extracted at random at each time step by considering
finite dimensional approximation of the Perron-Frobenius operatorComment: Plane TeX file, 15 pages, and 5 figures available under request to
[email protected]
Polynomial iterative algorithms for coloring and analyzing random graphs
We study the graph coloring problem over random graphs of finite average
connectivity . Given a number of available colors, we find that graphs
with low connectivity admit almost always a proper coloring whereas graphs with
high connectivity are uncolorable. Depending on , we find the precise value
of the critical average connectivity . Moreover, we show that below
there exist a clustering phase in which ground states
spontaneously divide into an exponential number of clusters. Furthermore, we
extended our considerations to the case of single instances showing consistent
results. This lead us to propose a new algorithm able to color in polynomial
time random graphs in the hard but colorable region, i.e when .Comment: 23 pages, 10 eps figure
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