5,574 research outputs found
Finding long cycles in graphs
We analyze the problem of discovering long cycles inside a graph. We propose
and test two algorithms for this task. The first one is based on recent
advances in statistical mechanics and relies on a message passing procedure.
The second follows a more standard Monte Carlo Markov Chain strategy. Special
attention is devoted to Hamiltonian cycles of (non-regular) random graphs of
minimal connectivity equal to three
Complexity Analysis and Efficient Measurement Selection Primitives for High-Rate Graph SLAM
Sparsity has been widely recognized as crucial for efficient optimization in
graph-based SLAM. Because the sparsity and structure of the SLAM graph reflect
the set of incorporated measurements, many methods for sparsification have been
proposed in hopes of reducing computation. These methods often focus narrowly
on reducing edge count without regard for structure at a global level. Such
structurally-naive techniques can fail to produce significant computational
savings, even after aggressive pruning. In contrast, simple heuristics such as
measurement decimation and keyframing are known empirically to produce
significant computation reductions. To demonstrate why, we propose a
quantitative metric called elimination complexity (EC) that bridges the
existing analytic gap between graph structure and computation. EC quantifies
the complexity of the primary computational bottleneck: the factorization step
of a Gauss-Newton iteration. Using this metric, we show rigorously that
decimation and keyframing impose favorable global structures and therefore
achieve computation reductions on the order of and , respectively,
where is the pruning rate. We additionally present numerical results
showing EC provides a good approximation of computation in both batch and
incremental (iSAM2) optimization and demonstrate that pruning methods promoting
globally-efficient structure outperform those that do not.Comment: Pre-print accepted to ICRA 201
Infinite Randomness Phases and Entanglement Entropy of the Disordered Golden Chain
Topological insulators supporting non-abelian anyonic excitations are at the
center of attention as candidates for topological quantum computation. In this
paper, we analyze the ground-state properties of disordered non-abelian anyonic
chains. The resemblance of fusion rules of non-abelian anyons and real space
decimation strongly suggests that disordered chains of such anyons generically
exhibit infinite-randomness phases. Concentrating on the disordered golden
chain model with nearest-neighbor coupling, we show that Fibonacci anyons with
the fusion rule exhibit two
infinite-randomness phases: a random-singlet phase when all bonds prefer the
trivial fusion channel, and a mixed phase which occurs whenever a finite
density of bonds prefers the fusion channel. Real space RG analysis
shows that the random-singlet fixed point is unstable to the mixed fixed point.
By analyzing the entanglement entropy of the mixed phase, we find its effective
central charge, and find that it increases along the RG flow from the random
singlet point, thus ruling out a c-theorem for the effective central charge.Comment: 16 page
Exactly Solvable Lattice Models with Crossing Symmetry
We show how to compute the exact partition function for lattice
statistical-mechanical models whose Boltzmann weights obey a special "crossing"
symmetry. The crossing symmetry equates partition functions on different
trivalent graphs, allowing a transformation to a graph where the partition
function is easily computed. The simplest example is counting the number of
nets without ends on the honeycomb lattice, including a weight per branching.
Other examples include an Ising model on the Kagome' lattice with three-spin
interactions, dimers on any graph of corner-sharing triangles, and non-crossing
loops on the honeycomb lattice, where multiple loops on each edge are allowed.
We give several methods for obtaining models with this crossing symmetry, one
utilizing discrete groups and another anyon fusion rules. We also present
results indicating that for models which deviate slightly from having crossing
symmetry, a real-space decimation (renormalization-group-like) procedure
restores the crossing symmetry
Survey-propagation decimation through distributed local computations
We discuss the implementation of two distributed solvers of the random K-SAT
problem, based on some development of the recently introduced
survey-propagation (SP) algorithm. The first solver, called the "SP diffusion
algorithm", diffuses as dynamical information the maximum bias over the system,
so that variable nodes can decide to freeze in a self-organized way, each
variable making its decision on the basis of purely local information. The
second solver, called the "SP reinforcement algorithm", makes use of
time-dependent external forcing messages on each variable, which let the
variables get completely polarized in the direction of a solution at the end of
a single convergence. Both methods allow us to find a solution of the random
3-SAT problem in a range of parameters comparable with the best previously
described serialized solvers. The simulated time of convergence towards a
solution (if these solvers were implemented on a distributed device) grows as
log(N).Comment: 18 pages, 10 figure
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