195,548 research outputs found
Lower bounds for local approximation
In the study of deterministic distributed algorithms it is commonly assumed that each node has a unique O(log n)-bit identifier. We prove that for a general class of graph problems, local algorithms (constant-time distributed algorithms) do not need such identifiers: a port numbering and orientation is sufficient. Our result holds for so-called simple PO-checkable graph optimisation problems; this includes many classical packing and covering problems such as vertex covers, edge covers, matchings, independent sets, dominating sets, and edge dominating sets. We focus on the case of bounded-degree graphs and show that if a local algorithm finds a constant-factor approximation of a simple PO-checkable graph problem with the help of unique identifiers, then the same approximation ratio can be achieved on anonymous networks. As a corollary of our result, we derive a tight lower bound on the local approximability of the minimum edge dominating set problem. By prior work, there is a deterministic local algorithm that achieves the approximation factor of 4 − 1/⌊Δ/2⌋ in graphs of maximum degree Δ. This approximation ratio is known to be optimal in the port-numbering model—our main theorem implies that it is optimal also in the standard model in which each node has a unique identifier. Our main technical tool is an algebraic construction of homogeneously ordered graphs: We say that a graph is (α,r)-homogeneous if its nodes are linearly ordered so that an α fraction of nodes have pairwise isomorphic radius-r neighbourhoods. We show that there exists a finite (α,r)-homogeneous 2k-regular graph of girth at least g for any α < 1 and any r, k, and g.Peer reviewe
Local approximation of a metapopulation's equilibrium
We consider the approximation of the equilibrium of a metapopulation model,
in which a finite number of patches are randomly distributed over a bounded
subset of Euclidean space. The approximation is good when a large
number of patches contribute to the colonization pressure on any given
unoccupied patch, and when the quality of the patches varies little over the
length scale determined by the colonization radius. If this is the case, the
equilibrium probability of a patch at being occupied is shown to be close
to , the equilibrium occupation probability in Levins's model, at any
point not too close to the boundary, if the local colonization
pressure and extinction rates appropriate to are assumed. The approximation
is justified by giving explicit upper and lower bounds for the occupation
probabilities, expressed in terms of the model parameters. Since the patches
are distributed randomly, the occupation probabilities are also random, and we
complement our bounds with explicit bounds on the probability that they are
satisfied at all patches simultaneously
Compound Poisson and signed compound Poisson approximations to the Markov binomial law
Compound Poisson distributions and signed compound Poisson measures are used
for approximation of the Markov binomial distribution. The upper and lower
bound estimates are obtained for the total variation, local and Wasserstein
norms. In a special case, asymptotically sharp constants are calculated. For
the upper bounds, the smoothing properties of compound Poisson distributions
are applied. For the lower bound estimates, the characteristic function method
is used.Comment: Published in at http://dx.doi.org/10.3150/09-BEJ246 the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm
Fully computable a posteriori error bounds for hybridizable discontinuous Galerkin finite element approximations
We derive a posteriori error estimates for the hybridizable discontinuous
Galerkin (HDG) methods, including both the primal and mixed formulations, for
the approximation of a linear second-order elliptic problem on conforming
simplicial meshes in two and three dimensions.
We obtain fully computable, constant free, a posteriori error bounds on the
broken energy seminorm and the HDG energy (semi)norm of the error. The
estimators are also shown to provide local lower bounds for the HDG energy
(semi)norm of the error up to a constant and a higher-order data oscillation
term. For the primal HDG methods and mixed HDG methods with an appropriate
choice of stabilization parameter, the estimators are also shown to provide a
lower bound for the broken energy seminorm of the error up to a constant and a
higher-order data oscillation term. Numerical examples are given illustrating
the theoretical results
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