1,898 research outputs found
Decomposition of multiple packings with subquadratic union complexity
Suppose is a positive integer and is a -fold packing of
the plane by infinitely many arc-connected compact sets, which means that every
point of the plane belongs to at most sets. Suppose there is a function
with the property that any members of determine
at most holes, which means that the complement of their union has at
most bounded connected components. We use tools from extremal graph
theory and the topological Helly theorem to prove that can be
decomposed into at most (-fold) packings, where is a constant
depending only on and .Comment: Small generalization of the main result, improvements in the proofs,
minor correction
The Galois Complexity of Graph Drawing: Why Numerical Solutions are Ubiquitous for Force-Directed, Spectral, and Circle Packing Drawings
Many well-known graph drawing techniques, including force directed drawings,
spectral graph layouts, multidimensional scaling, and circle packings, have
algebraic formulations. However, practical methods for producing such drawings
ubiquitously use iterative numerical approximations rather than constructing
and then solving algebraic expressions representing their exact solutions. To
explain this phenomenon, we use Galois theory to show that many variants of
these problems have solutions that cannot be expressed by nested radicals or
nested roots of low-degree polynomials. Hence, such solutions cannot be
computed exactly even in extended computational models that include such
operations.Comment: Graph Drawing 201
Defragmenting the Module Layout of a Partially Reconfigurable Device
Modern generations of field-programmable gate arrays (FPGAs) allow for
partial reconfiguration. In an online context, where the sequence of modules to
be loaded on the FPGA is unknown beforehand, repeated insertion and deletion of
modules leads to progressive fragmentation of the available space, making
defragmentation an important issue. We address this problem by propose an
online and an offline component for the defragmentation of the available space.
We consider defragmenting the module layout on a reconfigurable device. This
corresponds to solving a two-dimensional strip packing problem. Problems of
this type are NP-hard in the strong sense, and previous algorithmic results are
rather limited. Based on a graph-theoretic characterization of feasible
packings, we develop a method that can solve two-dimensional defragmentation
instances of practical size to optimality. Our approach is validated for a set
of benchmark instances.Comment: 10 pages, 11 figures, 1 table, Latex, to appear in "Engineering of
Reconfigurable Systems and Algorithms" as a "Distinguished Paper
TS2PACK: A Two-Level Tabu Search for the Three-dimensional Bin Packing Problem
Three-dimensional orthogonal bin packing is a problem NP-hard in the strong sense where a set of boxes must be orthogonally packed into the minimum number of three-dimensional bins. We present a two-level tabu search for this problem. The first-level aims to reduce the number of bins. The second optimizes the packing of the bins. This latter procedure is based on the Interval Graph representation of the packing, proposed by Fekete and Schepers, which reduces the size of the search space. We also introduce a general method to increase the size of the associated neighborhoods, and thus the quality of the search, without increasing the overall complexity of the algorithm. Extensive computational results on benchmark problem instances show the effectiveness of the proposed approach, obtaining better results compared to the existing one
Complexity in surfaces of densest packings for families of polyhedra
Packings of hard polyhedra have been studied for centuries due to their
mathematical aesthetic and more recently for their applications in fields such
as nanoscience, granular and colloidal matter, and biology. In all these
fields, particle shape is important for structure and properties, especially
upon crowding. Here, we explore packing as a function of shape. By combining
simulations and analytic calculations, we study three 2-parameter families of
hard polyhedra and report an extensive and systematic analysis of the densest
packings of more than 55,000 convex shapes. The three families have the
symmetries of triangle groups (icosahedral, octahedral, tetrahedral) and
interpolate between various symmetric solids (Platonic, Archimedean, Catalan).
We find that optimal (maximum) packing density surfaces that reveal unexpected
richness and complexity, containing as many as 130 different structures within
a single family. Our results demonstrate the utility of thinking of shape not
as a static property of an object in the context of packings, but rather as but
one point in a higher dimensional shape space whose neighbors in that space may
have identical or markedly different packings. Finally, we present and
interpret our packing results in a consistent and generally applicable way by
proposing a method to distinguish regions of packings and classify types of
transitions between them.Comment: 16 pages, 8 figure
Pore-scale Modeling of Viscous Flow and Induced Forces in Dense Sphere Packings
We propose a method for effectively upscaling incompressible viscous flow in
large random polydispersed sphere packings: the emphasis of this method is on
the determination of the forces applied on the solid particles by the fluid.
Pore bodies and their connections are defined locally through a regular
Delaunay triangulation of the packings. Viscous flow equations are upscaled at
the pore level, and approximated with a finite volume numerical scheme. We
compare numerical simulations of the proposed method to detailed finite element
(FEM) simulations of the Stokes equations for assemblies of 8 to 200 spheres. A
good agreement is found both in terms of forces exerted on the solid particles
and effective permeability coefficients
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