5,618 research outputs found
The edge-disjoint path problem on random graphs by message-passing
We present a message-passing algorithm to solve the edge disjoint path
problem (EDP) on graphs incorporating under a unique framework both traffic
optimization and path length minimization. The min-sum equations for this
problem present an exponential computational cost in the number of paths. To
overcome this obstacle we propose an efficient implementation by mapping the
equations onto a weighted combinatorial matching problem over an auxiliary
graph. We perform extensive numerical simulations on random graphs of various
types to test the performance both in terms of path length minimization and
maximization of the number of accommodated paths. In addition, we test the
performance on benchmark instances on various graphs by comparison with
state-of-the-art algorithms and results found in the literature. Our
message-passing algorithm always outperforms the others in terms of the number
of accommodated paths when considering non trivial instances (otherwise it
gives the same trivial results). Remarkably, the largest improvement in
performance with respect to the other methods employed is found in the case of
benchmarks with meshes, where the validity hypothesis behind message-passing is
expected to worsen. In these cases, even though the exact message-passing
equations do not converge, by introducing a reinforcement parameter to force
convergence towards a sub optimal solution, we were able to always outperform
the other algorithms with a peak of 27% performance improvement in terms of
accommodated paths. On random graphs, we numerically observe two separated
regimes: one in which all paths can be accommodated and one in which this is
not possible. We also investigate the behaviour of both the number of paths to
be accommodated and their minimum total length.Comment: 14 pages, 8 figure
Upper bounds on quantum query complexity inspired by the Elitzur-Vaidman bomb tester
Inspired by the Elitzur-Vaidman bomb testing problem [arXiv:hep-th/9305002],
we introduce a new query complexity model, which we call bomb query complexity
. We investigate its relationship with the usual quantum query complexity
, and show that .
This result gives a new method to upper bound the quantum query complexity:
we give a method of finding bomb query algorithms from classical algorithms,
which then provide nonconstructive upper bounds on .
We subsequently were able to give explicit quantum algorithms matching our
upper bound method. We apply this method on the single-source shortest paths
problem on unweighted graphs, obtaining an algorithm with quantum
query complexity, improving the best known algorithm of [arXiv:quant-ph/0606127]. Applying this method to the maximum bipartite
matching problem gives an algorithm, improving the best known
trivial upper bound.Comment: 32 pages. Minor revisions and corrections. Regev and Schiff's proof
that P(OR) = \Omega(N) remove
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