7,195 research outputs found
The random link approximation for the Euclidean traveling salesman problem
The traveling salesman problem (TSP) consists of finding the length of the
shortest closed tour visiting N ``cities''. We consider the Euclidean TSP where
the cities are distributed randomly and independently in a d-dimensional unit
hypercube. Working with periodic boundary conditions and inspired by a
remarkable universality in the kth nearest neighbor distribution, we find for
the average optimum tour length = beta_E(d) N^{1-1/d} [1+O(1/N)] with
beta_E(2) = 0.7120 +- 0.0002 and beta_E(3) = 0.6979 +- 0.0002. We then derive
analytical predictions for these quantities using the random link
approximation, where the lengths between cities are taken as independent random
variables. From the ``cavity'' equations developed by Krauth, Mezard and
Parisi, we calculate the associated random link values beta_RL(d). For d=1,2,3,
numerical results show that the random link approximation is a good one, with a
discrepancy of less than 2.1% between beta_E(d) and beta_RL(d). For large d, we
argue that the approximation is exact up to O(1/d^2) and give a conjecture for
beta_E(d), in terms of a power series in 1/d, specifying both leading and
subleading coefficients.Comment: 29 pages, 6 figures; formatting and typos correcte
Preconditioning of Improved and ``Perfect'' Fermion Actions
We construct a locally-lexicographic SSOR preconditioner to accelerate the
parallel iterative solution of linear systems of equations for two improved
discretizations of lattice fermions: the Sheikholeslami-Wohlert scheme where a
non-constant block-diagonal term is added to the Wilson fermion matrix and
renormalization group improved actions which incorporate couplings beyond
nearest neighbors of the lattice fermion fields. In case (i) we find the block
llssor-scheme to be more effective by a factor about 2 than odd-even
preconditioned solvers in terms of convergence rates, at beta=6.0. For type
(ii) actions, we show that our preconditioner accelerates the iterative
solution of a linear system of hypercube fermions by a factor of 3 to 4.Comment: 27 pages, Latex, 17 Figures include
From Proximity to Utility: A Voronoi Partition of Pareto Optima
We present an extension of Voronoi diagrams where when considering which site
a client is going to use, in addition to the site distances, other site
attributes are also considered (for example, prices or weights). A cell in this
diagram is then the locus of all clients that consider the same set of sites to
be relevant. In particular, the precise site a client might use from this
candidate set depends on parameters that might change between usages, and the
candidate set lists all of the relevant sites. The resulting diagram is
significantly more expressive than Voronoi diagrams, but naturally has the
drawback that its complexity, even in the plane, might be quite high.
Nevertheless, we show that if the attributes of the sites are drawn from the
same distribution (note that the locations are fixed), then the expected
complexity of the candidate diagram is near linear.
To this end, we derive several new technical results, which are of
independent interest. In particular, we provide a high-probability,
asymptotically optimal bound on the number of Pareto optima points in a point
set uniformly sampled from the -dimensional hypercube. To do so we revisit
the classical backward analysis technique, both simplifying and improving
relevant results in order to achieve the high-probability bounds
A directed isoperimetric inequality with application to Bregman near neighbor lower bounds
Bregman divergences are a class of divergences parametrized by a
convex function and include well known distance functions like
and the Kullback-Leibler divergence. There has been extensive
research on algorithms for problems like clustering and near neighbor search
with respect to Bregman divergences, in all cases, the algorithms depend not
just on the data size and dimensionality , but also on a structure
constant that depends solely on and can grow without bound
independently.
In this paper, we provide the first evidence that this dependence on
might be intrinsic. We focus on the problem of approximate near neighbor search
for Bregman divergences. We show that under the cell probe model, any
non-adaptive data structure (like locality-sensitive hashing) for
-approximate near-neighbor search that admits probes must use space
. In contrast, for LSH under the best
bound is .
Our new tool is a directed variant of the standard boolean noise operator. We
show that a generalization of the Bonami-Beckner hypercontractivity inequality
exists "in expectation" or upon restriction to certain subsets of the Hamming
cube, and that this is sufficient to prove the desired isoperimetric inequality
that we use in our data structure lower bound.
We also present a structural result reducing the Hamming cube to a Bregman
cube. This structure allows us to obtain lower bounds for problems under
Bregman divergences from their analog. In particular, we get a
(weaker) lower bound for approximate near neighbor search of the form
for an -query non-adaptive data structure,
and new cell probe lower bounds for a number of other near neighbor questions
in Bregman space.Comment: 27 page
Privacy sets for constrained space-filling
The paper provides typology for space filling into what we call "soft" and
"hard" methods along with introducing the central notion of privacy sets for
dealing with the latter. A heuristic algorithm based on this notion is
presented and we compare its performance on some well-known examples
An Emulator for the Lyman-alpha Forest
We present methods for interpolating between the 1-D flux power spectrum of
the Lyman- forest, as output by cosmological hydrodynamic simulations.
Interpolation is necessary for cosmological parameter estimation due to the
limited number of simulations possible. We construct an emulator for the
Lyman- forest flux power spectrum from small simulations using
Latin hypercube sampling and Gaussian process interpolation. We show that this
emulator has a typical accuracy of 1.5% and a worst-case accuracy of 4%, which
compares well to the current statistical error of 3 - 5% at from BOSS
DR9. We compare to the previous state of the art, quadratic polynomial
interpolation. The Latin hypercube samples the entire volume of parameter
space, while quadratic polynomial emulation samples only lower-dimensional
subspaces. The Gaussian process provides an estimate of the emulation error and
we show using test simulations that this estimate is reasonable. We construct a
likelihood function and use it to show that the posterior constraints generated
using the emulator are unbiased. We show that our Gaussian process emulator has
lower emulation error than quadratic polynomial interpolation and thus produces
tighter posterior confidence intervals, which will be essential for future
Lyman- surveys such as DESI.Comment: 28 pages, 10 figures, accepted to JCAP with minor change
A partitioning strategy for nonuniform problems on multiprocessors
The partitioning of a problem on a domain with unequal work estimates in different subddomains is considered in a way that balances the work load across multiple processors. Such a problem arises for example in solving partial differential equations using an adaptive method that places extra grid points in certain subregions of the domain. A binary decomposition of the domain is used to partition it into rectangles requiring equal computational effort. The communication costs of mapping this partitioning onto different microprocessors: a mesh-connected array, a tree machine and a hypercube is then studied. The communication cost expressions can be used to determine the optimal depth of the above partitioning
Geometric Measurement of Topological Susceptibility on Large Lattices
The topological susceptibility of the quenched QCD vacuum is measured on
large lattices for three values from to . Charges possibly
induced by dislocations are identified and shown to have little effect
on the measured susceptibility. As increases, fewer such questionable
charges are found. Scaling is checked by examining the ratios of the
susceptibility to previously existing values of the rho mass, string tension,
F-pi, and lambda-lattice.Comment: LaTeX article, 3 pages, uuencoded compressed tar file, 2 figures
included as tex files using axismacros, DVIPS driver required to show
figures. Talk presented by Jeffrey Grandy at Lattice 93, Dallas, Texas. Los
Alamos Preprint number pendin
- …