35,823 research outputs found
Space-Efficient Data Structures for Lattices
A lattice is a partially-ordered set in which every pair of elements has a
unique meet (greatest lower bound) and join (least upper bound). We present new
data structures for lattices that are simple, efficient, and nearly optimal in
terms of space complexity.
Our first data structure can answer partial order queries in constant time
and find the meet or join of two elements in time, where is
the number of elements in the lattice. It occupies bits of
space, which is only a factor from the -bit
lower bound for storing lattices. The preprocessing time is . This
structure admits a simple space-time tradeoff so that, for any , the data structure supports meet and join queries in
time, occupies bits of space, and can be
constructed in time.
Our second data structure uses bits of space and supports
meet and join in time, where is the maximum
degree of any element in the transitive reduction graph of the lattice. This
structure is much faster for lattices with low-degree elements.
This paper also identifies an error in a long-standing solution to the
problem of representing lattices. We discuss the issue with this previous work.Comment: Accepted in SWAT 202
An Efficient Linear Programming Algorithm to Generate the Densest Lattice Sphere Packings
Finding the densest sphere packing in -dimensional Euclidean space
is an outstanding fundamental problem with relevance in many
fields, including the ground states of molecular systems, colloidal crystal
structures, coding theory, discrete geometry, number theory, and biological
systems. Numerically generating the densest sphere packings becomes very
challenging in high dimensions due to an exponentially increasing number of
possible sphere contacts and sphere configurations, even for the restricted
problem of finding the densest lattice sphere packings. In this paper, we apply
the Torquato-Jiao packing algorithm, which is a method based on solving a
sequence of linear programs, to robustly reproduce the densest known lattice
sphere packings for dimensions 2 through 19. We show that the TJ algorithm is
appreciably more efficient at solving these problems than previously published
methods. Indeed, in some dimensions, the former procedure can be as much as
three orders of magnitude faster at finding the optimal solutions than earlier
ones. We also study the suboptimal local density-maxima solutions (inherent
structures or "extreme" lattices) to gain insight about the nature of the
topography of the "density" landscape.Comment: 23 pages, 3 figure
Quantum walk approach to search on fractal structures
We study continuous-time quantum walks mimicking the quantum search based on
Grover's procedure. This allows us to consider structures, that is, databases,
with arbitrary topological arrangements of their entries. We show that the
topological structure of the database plays a crucial role by analyzing, both
analytically and numerically, the transition from the ground to the first
excited state of the Hamiltonian associated with different (fractal)
structures. Additionally, we use the probability of successfully finding a
specific target as another indicator of the importance of the topological
structure.Comment: 15 pages, 14 figure
Block Circulant and Toeplitz Structures in the Linearized Hartree–Fock Equation on Finite Lattices: Tensor Approach
This paper introduces and analyses the new grid-based tensor approach to
approximate solution of the elliptic eigenvalue problem for the 3D
lattice-structured systems. We consider the linearized Hartree-Fock equation
over a spatial lattice for both periodic and
non-periodic problem setting, discretized in the localized Gaussian-type
orbitals basis. In the periodic case, the Galerkin system matrix obeys a
three-level block-circulant structure that allows the FFT-based
diagonalization, while for the finite extended systems in a box (Dirichlet
boundary conditions) we arrive at the perturbed block-Toeplitz representation
providing fast matrix-vector multiplication and low storage size. The proposed
grid-based tensor techniques manifest the twofold benefits: (a) the entries of
the Fock matrix are computed by 1D operations using low-rank tensors
represented on a 3D grid, (b) in the periodic case the low-rank tensor
structure in the diagonal blocks of the Fock matrix in the Fourier space
reduces the conventional 3D FFT to the product of 1D FFTs. Lattice type systems
in a box with Dirichlet boundary conditions are treated numerically by our
previous tensor solver for single molecules, which makes possible calculations
on rather large lattices due to reduced numerical
cost for 3D problems. The numerical simulations for both box-type and periodic
lattice chain in a 3D rectangular "tube" with up to
several hundred confirm the theoretical complexity bounds for the
block-structured eigenvalue solvers in the limit of large .Comment: 30 pages, 12 figures. arXiv admin note: substantial text overlap with
arXiv:1408.383
The mechanical response of cellular materials with spinodal topologies
The mechanical response of cellular materials with spinodal topologies is
numerically and experimentally investigated. Spinodal microstructures are
generated by the numerical solution of the Cahn-Hilliard equation. Two
different topologies are investigated: "solid models," where one of the two
phases is modeled as a solid material and the remaining volume is void space;
and "shell models," where the interface between the two phases is assumed to be
a solid shell, with the rest of the volume modeled as void space. In both
cases, a wide range of relative densities and spinodal characteristic feature
sizes are investigated. The topology and morphology of all the numerically
generated models are carefully characterized to extract key geometrical
features and ensure that the distribution of curvatures and the aging law are
consistent with the physics of spinodal decomposition. Finite element meshes
are generated for each model, and the uniaxial compressive stiffness and
strength are extracted. We show that while solid spinodal models in the density
range of 30-70% are relatively inefficient (i.e., their strength and stiffness
exhibit a high-power scaling with relative density), shell spinodal models in
the density range of 0.01-1% are exceptionally stiff and strong. Spinodal shell
materials are also shown to be remarkably imperfection insensitive. These
findings are verified experimentally by in-situ uniaxial compression of
polymeric samples printed at the microscale by Direct Laser Writing (DLW). At
low relative densities, the strength and stiffness of shell spinodal models
outperform those of most lattice materials and approach theoretical bounds for
isotropic cellular materials. Most importantly, these materials can be produced
by self-assembly techniques over a range of length scales, providing unique
scalability
Multi-core computation of transfer matrices for strip lattices in the Potts model
The transfer-matrix technique is a convenient way for studying strip lattices
in the Potts model since the compu- tational costs depend just on the periodic
part of the lattice and not on the whole. However, even when the cost is
reduced, the transfer-matrix technique is still an NP-hard problem since the
time T(|V|, |E|) needed to compute the matrix grows ex- ponentially as a
function of the graph width. In this work, we present a parallel
transfer-matrix implementation that scales performance under multi-core
architectures. The construction of the matrix is based on several repetitions
of the deletion- contraction technique, allowing parallelism suitable to
multi-core machines. Our experimental results show that the multi-core
implementation achieves speedups of 3.7X with p = 4 processors and 5.7X with p
= 8. The efficiency of the implementation lies between 60% and 95%, achieving
the best balance of speedup and efficiency at p = 4 processors for actual
multi-core architectures. The algorithm also takes advantage of the lattice
symmetry, making the transfer matrix computation to run up to 2X faster than
its non-symmetric counterpart and use up to a quarter of the original space
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