5,022 research outputs found
Maximum Distance Separable Codes for Symbol-Pair Read Channels
We study (symbol-pair) codes for symbol-pair read channels introduced
recently by Cassuto and Blaum (2010). A Singleton-type bound on symbol-pair
codes is established and infinite families of optimal symbol-pair codes are
constructed. These codes are maximum distance separable (MDS) in the sense that
they meet the Singleton-type bound. In contrast to classical codes, where all
known q-ary MDS codes have length O(q), we show that q-ary MDS symbol-pair
codes can have length \Omega(q^2). In addition, we completely determine the
existence of MDS symbol-pair codes for certain parameters
Distributed Weight Selection in Consensus Protocols by Schatten Norm Minimization
In average consensus protocols, nodes in a network perform an iterative
weighted average of their estimates and those of their neighbors. The protocol
converges to the average of initial estimates of all nodes found in the
network. The speed of convergence of average consensus protocols depends on the
weights selected on links (to neighbors). We address in this paper how to
select the weights in a given network in order to have a fast speed of
convergence for these protocols. We approximate the problem of optimal weight
selection by the minimization of the Schatten p-norm of a matrix with some
constraints related to the connectivity of the underlying network. We then
provide a totally distributed gradient method to solve the Schatten norm
optimization problem. By tuning the parameter p in our proposed minimization,
we can simply trade-off the quality of the solution (i.e. the speed of
convergence) for communication/computation requirements (in terms of number of
messages exchanged and volume of data processed). Simulation results show that
our approach provides very good performance already for values of p that only
needs limited information exchange. The weight optimization iterative procedure
can also run in parallel with the consensus protocol and form a joint
consensus-optimization procedure.Comment: N° RR-8078 (2012
Bidimensionality and Geometric Graphs
In this paper we use several of the key ideas from Bidimensionality to give a
new generic approach to design EPTASs and subexponential time parameterized
algorithms for problems on classes of graphs which are not minor closed, but
instead exhibit a geometric structure. In particular we present EPTASs and
subexponential time parameterized algorithms for Feedback Vertex Set, Vertex
Cover, Connected Vertex Cover, Diamond Hitting Set, on map graphs and unit disk
graphs, and for Cycle Packing and Minimum-Vertex Feedback Edge Set on unit disk
graphs. Our results are based on the recent decomposition theorems proved by
Fomin et al [SODA 2011], and our algorithms work directly on the input graph.
Thus it is not necessary to compute the geometric representations of the input
graph. To the best of our knowledge, these results are previously unknown, with
the exception of the EPTAS and a subexponential time parameterized algorithm on
unit disk graphs for Vertex Cover, which were obtained by Marx [ESA 2005] and
Alber and Fiala [J. Algorithms 2004], respectively.
We proceed to show that our approach can not be extended in its full
generality to more general classes of geometric graphs, such as intersection
graphs of unit balls in R^d, d >= 3. Specifically we prove that Feedback Vertex
Set on unit-ball graphs in R^3 neither admits PTASs unless P=NP, nor
subexponential time algorithms unless the Exponential Time Hypothesis fails.
Additionally, we show that the decomposition theorems which our approach is
based on fail for disk graphs and that therefore any extension of our results
to disk graphs would require new algorithmic ideas. On the other hand, we prove
that our EPTASs and subexponential time algorithms for Vertex Cover and
Connected Vertex Cover carry over both to disk graphs and to unit-ball graphs
in R^d for every fixed d
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