2,900 research outputs found
Sparse Fault-Tolerant BFS Trees
This paper addresses the problem of designing a sparse {\em fault-tolerant}
BFS tree, or {\em FT-BFS tree} for short, namely, a sparse subgraph of the
given network such that subsequent to the failure of a single edge or
vertex, the surviving part of still contains a BFS spanning tree for
(the surviving part of) . Our main results are as follows. We present an
algorithm that for every -vertex graph and source node constructs a
(single edge failure) FT-BFS tree rooted at with O(n \cdot
\min\{\Depth(s), \sqrt{n}\}) edges, where \Depth(s) is the depth of the BFS
tree rooted at . This result is complemented by a matching lower bound,
showing that there exist -vertex graphs with a source node for which any
edge (or vertex) FT-BFS tree rooted at has edges. We then
consider {\em fault-tolerant multi-source BFS trees}, or {\em FT-MBFS trees}
for short, aiming to provide (following a failure) a BFS tree rooted at each
source for some subset of sources . Again, tight bounds
are provided, showing that there exists a poly-time algorithm that for every
-vertex graph and source set of size constructs a
(single failure) FT-MBFS tree from each source , with
edges, and on the other hand there exist
-vertex graphs with source sets of cardinality , on
which any FT-MBFS tree from has edges.
Finally, we propose an approximation algorithm for constructing
FT-BFS and FT-MBFS structures. The latter is complemented by a hardness result
stating that there exists no approximation algorithm for these
problems under standard complexity assumptions
Distributed Computing in the Asynchronous LOCAL model
The LOCAL model is among the main models for studying locality in the
framework of distributed network computing. This model is however subject to
pertinent criticisms, including the facts that all nodes wake up
simultaneously, perform in lock steps, and are failure-free. We show that
relaxing these hypotheses to some extent does not hurt local computing. In
particular, we show that, for any construction task associated to a locally
checkable labeling (LCL), if is solvable in rounds in the LOCAL model,
then remains solvable in rounds in the asynchronous LOCAL model.
This improves the result by Casta\~neda et al. [SSS 2016], which was restricted
to 3-coloring the rings. More generally, the main contribution of this paper is
to show that, perhaps surprisingly, asynchrony and failures in the computations
do not restrict the power of the LOCAL model, as long as the communications
remain synchronous and failure-free
A two-dimensional representation of four-dimensional gravitational waves
The Einstein equation in D dimensions, if restricted to the class of
space-times possessing n = D - 2 commuting hypersurface-orthogonal Killing
vectors, can be equivalently written as metric-dilaton gravity in 2 dimensions
with n scalar fields. For n = 2, this results reduces to the known reduction of
certain 4-dimensional metrics which include gravitational waves. Here, we give
such a representation which leads to a new proof of the Birkhoff theorem for
plane-symmetric space--times, and which leads to an explanation, in which sense
two (spin zero-) scalar fields in 2 dimensions may incorporate the (spin two-)
gravitational waves in 4 dimensions. (This result should not be mixed up with
well--known analogous statements where, however, the 4-dimensional space-time
is supposed to be spherically symmetric, and then, of course, the equivalent
2-dimensional picture cannot mimic any gravitational waves.) Finally, remarks
on hidden symmetries in 2 dimensions are made.Comment: 12 pages, LaTeX, no figures, Int. J. Mod. Phys. D in prin
Simulations of the angular dependence of the dipole-dipole interaction among Rydberg atoms
The dipole-dipole interaction between two Rydberg atoms depends on the
relative orientation of the atoms and on the change in the magnetic quantum
number. We simulate the effect of this anisotropy on the energy transport in an
amorphous many atom system subject to a homogeneous applied electric field. We
consider two experimentally feasible geometries and find that the effects
should be measurable in current generation imaging experiments. In both
geometries atoms of character are localized to a small region of space
which is immersed in a larger region that is filled with atoms of
character. Energy transfer due to the dipole-dipole interaction can lead to a
spread of character into the region initially occupied by atoms. Over
long timescales the energy transport is confined to the volume near the border
of the region which is suggestive of Anderson localization. We calculate a
correlation length of 6.3~m for one particular geometry.Comment: 6 pages, 5 figures, revised draf
Normal scaling in globally conserved interface-controlled coarsening of fractal clusters
Globally conserved interface-controlled coarsening of fractal clusters
exhibits dynamic scale invariance and normal scaling. This is demonstrated by a
numerical solution of the Ginzburg-Landau equation with a global conservation
law. The sharp-interface limit of this equation is volume preserving motion by
mean curvature. The scaled form of the correlation function has a power-law
tail accommodating the fractal initial condition. The coarsening length
exhibits normal scaling with time. Finally, shrinking of the fractal clusters
with time is observed. The difference between global and local conservation is
discussed.Comment: 4 pages, 3 eps figure
Node Labels in Local Decision
The role of unique node identifiers in network computing is well understood
as far as symmetry breaking is concerned. However, the unique identifiers also
leak information about the computing environment - in particular, they provide
some nodes with information related to the size of the network. It was recently
proved that in the context of local decision, there are some decision problems
such that (1) they cannot be solved without unique identifiers, and (2) unique
node identifiers leak a sufficient amount of information such that the problem
becomes solvable (PODC 2013).
In this work we give study what is the minimal amount of information that we
need to leak from the environment to the nodes in order to solve local decision
problems. Our key results are related to scalar oracles that, for any given
, provide a multiset of labels; then the adversary assigns the
labels to the nodes in the network. This is a direct generalisation of the
usual assumption of unique node identifiers. We give a complete
characterisation of the weakest oracle that leaks at least as much information
as the unique identifiers.
Our main result is the following dichotomy: we classify scalar oracles as
large and small, depending on their asymptotic behaviour, and show that (1) any
large oracle is at least as powerful as the unique identifiers in the context
of local decision problems, while (2) for any small oracle there are local
decision problems that still benefit from unique identifiers.Comment: Conference version to appear in the proceedings of SIROCCO 201
Robots with Lights: Overcoming Obstructed Visibility Without Colliding
Robots with lights is a model of autonomous mobile computational entities
operating in the plane in Look-Compute-Move cycles: each agent has an
externally visible light which can assume colors from a fixed set; the lights
are persistent (i.e., the color is not erased at the end of a cycle), but
otherwise the agents are oblivious. The investigation of computability in this
model, initially suggested by Peleg, is under way, and several results have
been recently established. In these investigations, however, an agent is
assumed to be capable to see through another agent. In this paper we start the
study of computing when visibility is obstructable, and investigate the most
basic problem for this setting, Complete Visibility: The agents must reach
within finite time a configuration where they can all see each other and
terminate. We do not make any assumption on a-priori knowledge of the number of
agents, on rigidity of movements nor on chirality. The local coordinate system
of an agent may change at each activation. Also, by definition of lights, an
agent can communicate and remember only a constant number of bits in each
cycle. In spite of these weak conditions, we prove that Complete Visibility is
always solvable, even in the asynchronous setting, without collisions and using
a small constant number of colors. The proof is constructive. We also show how
to extend our protocol for Complete Visibility so that, with the same number of
colors, the agents solve the (non-uniform) Circle Formation problem with
obstructed visibility
Supersymmetric spacetimes in 2+1 adS-supergravity models
We find a class of (2+1)-dimensional spacetimes admitting Killing spinors
appropriate to (2,0) adS-supergravity. The vacuum spacetimes include anti-de
Sitter (adS) space and charged extreme black holes, but there are many others,
including spacetimes of arbitrarily large negative energy that have only
conical singularities, and the spacetimes of fractionally charged point
particles. The non-vacuum spacetimes are those of self-gravitating solitons
obtained by coupling (2,0) adS supergravity to sigma-model matter. We show,
subject to a condition on the matter currents (satisfied by the sigma model),
and a conjecture concerning global obstructions to the existence of certain
types of spinor fields, that the mass of each supersymmetric spacetime
saturates a classical bound, in terms of the angular momentum and charge, on
the total energy of arbitrary field configurations with the same boundary
conditions, although these bounds may be violated quantum mechanically.Comment: 47 pages, phyzzx.tex, no figures
Beeping a Maximal Independent Set
We consider the problem of computing a maximal independent set (MIS) in an
extremely harsh broadcast model that relies only on carrier sensing. The model
consists of an anonymous broadcast network in which nodes have no knowledge
about the topology of the network or even an upper bound on its size.
Furthermore, it is assumed that an adversary chooses at which time slot each
node wakes up. At each time slot a node can either beep, that is, emit a
signal, or be silent. At a particular time slot, beeping nodes receive no
feedback, while silent nodes can only differentiate between none of its
neighbors beeping, or at least one of its neighbors beeping.
We start by proving a lower bound that shows that in this model, it is not
possible to locally converge to an MIS in sub-polynomial time. We then study
four different relaxations of the model which allow us to circumvent the lower
bound and find an MIS in polylogarithmic time. First, we show that if a
polynomial upper bound on the network size is known, it is possible to find an
MIS in O(log^3 n) time. Second, if we assume sleeping nodes are awoken by
neighboring beeps, then we can also find an MIS in O(log^3 n) time. Third, if
in addition to this wakeup assumption we allow sender-side collision detection,
that is, beeping nodes can distinguish whether at least one neighboring node is
beeping concurrently or not, we can find an MIS in O(log^2 n) time. Finally, if
instead we endow nodes with synchronous clocks, it is also possible to find an
MIS in O(log^2 n) time.Comment: arXiv admin note: substantial text overlap with arXiv:1108.192
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