2,474 research outputs found
Electrical networks and Stephenson's conjecture
In this paper, we consider a planar annulus, i.e., a bounded, two-connected,
Jordan domain, endowed with a sequence of triangulations exhausting it. We then
construct a corresponding sequence of maps which converge uniformly on compact
subsets of the domain, to a conformal homeomorphism onto the interior of a
Euclidean annulus bounded by two concentric circles. As an application, we will
affirm a conjecture raised by Ken Stephenson in the 90's which predicts that
the Riemann mapping can be approximated by a sequence of electrical networks.Comment: Comments are welcome
Relaxed spanners for directed disk graphs
Let be a finite metric space, where is a set of points
and is a distance function defined for these points. Assume that
has a constant doubling dimension and assume that each point
has a disk of radius around it. The disk graph that corresponds
to and is a \emph{directed} graph , whose vertices are
the points of and whose edge set includes a directed edge from to
if . In \cite{PeRo08} we presented an algorithm for
constructing a (1+\eps)-spanner of size O(n/\eps^d \log M), where is
the maximal radius . The current paper presents two results. The first
shows that the spanner of \cite{PeRo08} is essentially optimal, i.e., for
metrics of constant doubling dimension it is not possible to guarantee a
spanner whose size is independent of . The second result shows that by
slightly relaxing the requirements and allowing a small perturbation of the
radius assignment, considerably better spanners can be constructed. In
particular, we show that if it is allowed to use edges of the disk graph
I(V,E,r_{1+\eps}), where r_{1+\eps}(p) = (1+\eps)\cdot r(p) for every , then it is possible to get a (1+\eps)-spanner of size O(n/\eps^d) for
. Our algorithm is simple and can be implemented efficiently
On the solution of a `solvable' model of an ideal glass of hard spheres displaying a jamming transition
We discuss the analytical solution through the cavity method of a mean field
model that displays at the same time an ideal glass transition and a set of
jamming points. We establish the equations describing this system, and we
discuss some approximate analytical solutions and a numerical strategy to solve
them exactly. We compare these methods and we get insight into the reliability
of the theory for the description of finite dimensional hard spheres.Comment: 31 pages, 8 figure
Wireless Scheduling with Power Control
We consider the scheduling of arbitrary wireless links in the physical model
of interference to minimize the time for satisfying all requests. We study here
the combined problem of scheduling and power control, where we seek both an
assignment of power settings and a partition of the links so that each set
satisfies the signal-to-interference-plus-noise (SINR) constraints.
We give an algorithm that attains an approximation ratio of , where is the number of links and is the ratio
between the longest and the shortest link length. Under the natural assumption
that lengths are represented in binary, this gives the first approximation
ratio that is polylogarithmic in the size of the input. The algorithm has the
desirable property of using an oblivious power assignment, where the power
assigned to a sender depends only on the length of the link. We give evidence
that this dependence on is unavoidable, showing that any
reasonably-behaving oblivious power assignment results in a -approximation.
These results hold also for the (weighted) capacity problem of finding a
maximum (weighted) subset of links that can be scheduled in a single time slot.
In addition, we obtain improved approximation for a bidirectional variant of
the scheduling problem, give partial answers to questions about the utility of
graphs for modeling physical interference, and generalize the setting from the
standard 2-dimensional Euclidean plane to doubling metrics. Finally, we explore
the utility of graph models in capturing wireless interference.Comment: Revised full versio
Recognizing Weighted Disk Contact Graphs
Disk contact representations realize graphs by mapping vertices bijectively
to interior-disjoint disks in the plane such that two disks touch each other if
and only if the corresponding vertices are adjacent in the graph. Deciding
whether a vertex-weighted planar graph can be realized such that the disks'
radii coincide with the vertex weights is known to be NP-hard. In this work, we
reduce the gap between hardness and tractability by analyzing the problem for
special graph classes. We show that it remains NP-hard for outerplanar graphs
with unit weights and for stars with arbitrary weights, strengthening the
previous hardness results. On the positive side, we present constructive
linear-time recognition algorithms for caterpillars with unit weights and for
embedded stars with arbitrary weights.Comment: 24 pages, 21 figures, extended version of a paper to appear at the
International Symposium on Graph Drawing and Network Visualization (GD) 201
Beyond Geometry : Towards Fully Realistic Wireless Models
Signal-strength models of wireless communications capture the gradual fading
of signals and the additivity of interference. As such, they are closer to
reality than other models. However, nearly all theoretic work in the SINR model
depends on the assumption of smooth geometric decay, one that is true in free
space but is far off in actual environments. The challenge is to model
realistic environments, including walls, obstacles, reflections and anisotropic
antennas, without making the models algorithmically impractical or analytically
intractable.
We present a simple solution that allows the modeling of arbitrary static
situations by moving from geometry to arbitrary decay spaces. The complexity of
a setting is captured by a metricity parameter Z that indicates how far the
decay space is from satisfying the triangular inequality. All results that hold
in the SINR model in general metrics carry over to decay spaces, with the
resulting time complexity and approximation depending on Z in the same way that
the original results depends on the path loss term alpha. For distributed
algorithms, that to date have appeared to necessarily depend on the planarity,
we indicate how they can be adapted to arbitrary decay spaces.
Finally, we explore the dependence on Z in the approximability of core
problems. In particular, we observe that the capacity maximization problem has
exponential upper and lower bounds in terms of Z in general decay spaces. In
Euclidean metrics and related growth-bounded decay spaces, the performance
depends on the exact metricity definition, with a polynomial upper bound in
terms of Z, but an exponential lower bound in terms of a variant parameter phi.
On the plane, the upper bound result actually yields the first approximation of
a capacity-type SINR problem that is subexponential in alpha
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