5,199 research outputs found
Interleaving schemes for multidimensional cluster errors
We present two-dimensional and three-dimensional interleaving techniques for correcting two- and three-dimensional bursts (or clusters) of errors, where a cluster of errors is characterized by its area or volume. Correction of multidimensional error clusters is required in holographic storage, an emerging application of considerable importance. Our main contribution is the construction of efficient two-dimensional and three-dimensional interleaving schemes. The proposed schemes are based on t-interleaved arrays of integers, defined by the property that every connected component of area or volume t consists of distinct integers. In the two-dimensional case, our constructions are optimal: they have the lowest possible interleaving degree. That is, the resulting t-interleaved arrays contain the smallest possible number of distinct integers, hence minimizing the number of codewords required in an interleaving scheme. In general, we observe that the interleaving problem can be interpreted as a graph-coloring problem, and introduce the useful special class of lattice interleavers. We employ a result of Minkowski, dating back to 1904, to establish both upper and lower bounds on the interleaving degree of lattice interleavers in three dimensions. For the case t≡0 mod 6, the upper and lower bounds coincide, and the Minkowski lattice directly yields an optimal lattice interleaver. For t≠0 mod 6, we construct efficient lattice interleavers using approximations of the Minkowski lattice
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
Improved approximation bounds for Vector Bin Packing
In this paper we propose an improved approximation scheme for the Vector Bin
Packing problem (VBP), based on the combination of (near-)optimal solution of
the Linear Programming (LP) relaxation and a greedy (modified first-fit)
heuristic. The Vector Bin Packing problem of higher dimension (d \geq 2) is not
known to have asymptotic polynomial-time approximation schemes (unless P = NP).
Our algorithm improves over the previously-known guarantee of (ln d + 1 +
epsilon) by Bansal et al. [1] for higher dimensions (d > 2). We provide a
{\theta}(1) approximation scheme for certain set of inputs for any dimension d.
More precisely, we provide a 2-OPT algorithm, a result which is irrespective of
the number of dimensions d.Comment: 15 pages, 3 algorithm
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