1,314 research outputs found
Colouring random graphs and maximising local diversity
We study a variation of the graph colouring problem on random graphs of
finite average connectivity. Given the number of colours, we aim to maximise
the number of different colours at neighbouring vertices (i.e. one edge
distance) of any vertex. Two efficient algorithms, belief propagation and
Walksat are adapted to carry out this task. We present experimental results
based on two types of random graphs for different system sizes and identify the
critical value of the connectivity for the algorithms to find a perfect
solution. The problem and the suggested algorithms have practical relevance
since various applications, such as distributed storage, can be mapped onto
this problem.Comment: 10 pages, 10 figure
Focused Local Search for Random 3-Satisfiability
A local search algorithm solving an NP-complete optimisation problem can be
viewed as a stochastic process moving in an 'energy landscape' towards
eventually finding an optimal solution. For the random 3-satisfiability
problem, the heuristic of focusing the local moves on the presently
unsatisfiedclauses is known to be very effective: the time to solution has been
observed to grow only linearly in the number of variables, for a given
clauses-to-variables ratio sufficiently far below the critical
satisfiability threshold . We present numerical results
on the behaviour of three focused local search algorithms for this problem,
considering in particular the characteristics of a focused variant of the
simple Metropolis dynamics. We estimate the optimal value for the
``temperature'' parameter for this algorithm, such that its linear-time
regime extends as close to as possible. Similar parameter
optimisation is performed also for the well-known WalkSAT algorithm and for the
less studied, but very well performing Focused Record-to-Record Travel method.
We observe that with an appropriate choice of parameters, the linear time
regime for each of these algorithms seems to extend well into ratios -- much further than has so far been generally assumed. We discuss the
statistics of solution times for the algorithms, relate their performance to
the process of ``whitening'', and present some conjectures on the shape of
their computational phase diagrams.Comment: 20 pages, lots of figure
Exponentially hard problems are sometimes polynomial, a large deviation analysis of search algorithms for the random Satisfiability problem, and its application to stop-and-restart resolutions
A large deviation analysis of the solving complexity of random
3-Satisfiability instances slightly below threshold is presented. While finding
a solution for such instances demands an exponential effort with high
probability, we show that an exponentially small fraction of resolutions
require a computation scaling linearly in the size of the instance only. This
exponentially small probability of easy resolutions is analytically calculated,
and the corresponding exponent shown to be smaller (in absolute value) than the
growth exponent of the typical resolution time. Our study therefore gives some
theoretical basis to heuristic stop-and-restart solving procedures, and
suggests a natural cut-off (the size of the instance) for the restart.Comment: Revtex file, 4 figure
Modeling a high mass turn down in the stellar initial mass function
Statistical sampling from the stellar initial mass function (IMF) for all
star-forming regions in the Galaxy would lead to the prediction of ~1000 Msun
stars unless there is a rapid turn-down in the IMF beyond several hundred solar
masses. Such a turndown is not necessary for dense clusters because the number
of stars sampled is always too small. Here we explore several mechanisms for an
upper mass cutoff, including an exponential decline of the star formation
probability after a turbulent crossing time. The results are in good agreement
with the observed IMF over the entire stellar mass range, and they give a
gradual turn down compared to the Salpeter function above ~100 Msun for normal
thermal Jeans mass, M_J. The upper mass turn down should scale with M_J in
different environments. A problem with the models is that they cannot give both
the observed power-law IMF out to the high-mass sampling limit in dense
clusters, as well as the observed lack of supermassive stars in whole galaxy
disks. Either there is a sharper upper-mass cutoff in the IMF, perhaps from
self-limitation, or the IMF is different for dense clusters than for the
majority of star formation that occurs at lower density. Dense clusters seem to
have an overabundance of massive stars relative to the average IMF in a galaxy.Comment: 19 pages, 2 figures, Astrophysical Journal, Vol 539, August 10, 200
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
A Random Matrix Model of Adiabatic Quantum Computing
We present an analysis of the quantum adiabatic algorithm for solving hard
instances of 3-SAT (an NP-complete problem) in terms of Random Matrix Theory
(RMT). We determine the global regularity of the spectral fluctuations of the
instantaneous Hamiltonians encountered during the interpolation between the
starting Hamiltonians and the ones whose ground states encode the solutions to
the computational problems of interest. At each interpolation point, we
quantify the degree of regularity of the average spectral distribution via its
Brody parameter, a measure that distinguishes regular (i.e., Poissonian) from
chaotic (i.e., Wigner-type) distributions of normalized nearest-neighbor
spacings. We find that for hard problem instances, i.e., those having a
critical ratio of clauses to variables, the spectral fluctuations typically
become irregular across a contiguous region of the interpolation parameter,
while the spectrum is regular for easy instances. Within the hard region, RMT
may be applied to obtain a mathematical model of the probability of avoided
level crossings and concomitant failure rate of the adiabatic algorithm due to
non-adiabatic Landau-Zener type transitions. Our model predicts that if the
interpolation is performed at a uniform rate, the average failure rate of the
quantum adiabatic algorithm, when averaged over hard problem instances, scales
exponentially with increasing problem size.Comment: 9 pages, 7 figure
An Optical and X-ray Examination of Two Radio Supernova Remnant Candidates in 30 Doradus
The giant HII region 30 Doradus is known for its violent internal motions and
bright diffuse X-ray emission, suggesting the existence of supernova remnants
(SNRs), but no nonthermal radio emission has been detected. Recently, Lazendic
et al. compared the H-alpha/H-beta and radio/H-alpha ratios and suggested two
small radio sources to be nonthermal and thus SNR candidates; however, no
optical or X-ray counterparts were detected. We have used high-resolution
optical images and high-dispersion spectra to examine the morphological,
spectral, and kinematic properties of these two SNR candidates, and still find
no optical evidence supporting their identification as SNRs. We have also
determined the X-ray luminosities of these SNR candidates, and find them 1-3
orders of magnitude lower than those commonly seen in young SNRs. High
extinction can obscure optical and X-ray signatures of an SNR, but would
prohibit the use of a high radio/H-alpha ratio to identify nonthermal radio
emission. We suggest that the SNR candidate MCRX J053831.8-690620 is associated
with a young star forming region; while the radio emission originates from the
obscured star forming region, the observed optical emission is dominated by the
foreground. We suggest that the SNR candidate MCRX J053838.8-690730 is
associated with a dust/molecular cloud, which obscures some optical emission
but not the radio emission.Comment: 13 pages, 2 figures, accepted for publication in the ApJ, Nov 10,
200
The scale-free character of the cluster mass function and the universality of the stellar IMF
Our recent determination of a Salpeter slope for the IMF in the field of 30
Doradus (Selman and Melnick 2005) appears to be in conflict with simple
probabilistic counting arguments advanced in the past to support observational
claims of a steeper IMF in the LMC field. In this paper we re-examine these
arguments and show by explicit construction that, contrary to these claims, the
field IMF is expected to be exactly the same as the stellar IMF of the clusters
out of which the field was presumably formed. We show that the current data on
the mass distribution of clusters themselves is in excellent agreement with our
model, and is consistent with a single spectrum {\it by number of stars} of the
type with beta between -1.8 and -2.2 down to the smallest clusters
without any preferred mass scale for cluster formation. We also use the random
sampling model to estimate the statistics of the maximal mass star in clusters,
and confirm the discrepancy with observations found by Weidner and Kroupa
(2006). We argue that rather than signaling the violation of the random
sampling model these observations reflect the gravitationally unstable nature
of systems with one very large mass star. We stress the importance of the
random sampling model as a \emph{null hypothesis} whose violation would signal
the presence of interesting physics.Comment: 9 pages emulateap
The Phase Diagram of 1-in-3 Satisfiability Problem
We study the typical case properties of the 1-in-3 satisfiability problem,
the boolean satisfaction problem where a clause is satisfied by exactly one
literal, in an enlarged random ensemble parametrized by average connectivity
and probability of negation of a variable in a clause. Random 1-in-3
Satisfiability and Exact 3-Cover are special cases of this ensemble. We
interpolate between these cases from a region where satisfiability can be
typically decided for all connectivities in polynomial time to a region where
deciding satisfiability is hard, in some interval of connectivities. We derive
several rigorous results in the first region, and develop the
one-step--replica-symmetry-breaking cavity analysis in the second one. We
discuss the prediction for the transition between the almost surely satisfiable
and the almost surely unsatisfiable phase, and other structural properties of
the phase diagram, in light of cavity method results.Comment: 30 pages, 12 figure
Minimizing energy below the glass thresholds
Focusing on the optimization version of the random K-satisfiability problem,
the MAX-K-SAT problem, we study the performance of the finite energy version of
the Survey Propagation (SP) algorithm. We show that a simple (linear time)
backtrack decimation strategy is sufficient to reach configurations well below
the lower bound for the dynamic threshold energy and very close to the analytic
prediction for the optimal ground states. A comparative numerical study on one
of the most efficient local search procedures is also given.Comment: 12 pages, submitted to Phys. Rev. E, accepted for publicatio
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