89 research outputs found
Hypergraph expanders from Cayley graphs
We present a simple mechanism, which can be randomised, for constructing
sparse -uniform hypergraphs with strong expansion properties. These
hypergraphs are constructed using Cayley graphs over and have
vertex degree which is polylogarithmic in the number of vertices. Their
expansion properties, which are derived from the underlying Cayley graphs,
include analogues of vertex and edge expansion in graphs, rapid mixing of the
random walk on the edges of the skeleton graph, uniform distribution of edges
on large vertex subsets and the geometric overlap property.Comment: 13 page
Construction of asymptotically good low-rate error-correcting codes through pseudo-random graphs
A novel technique, based on the pseudo-random properties of certain graphs known as expanders, is used to obtain novel simple explicit constructions of asymptotically good codes. In one of the constructions, the expanders are used to enhance Justesen codes by replicating, shuffling, and then regrouping the code coordinates. For any fixed (small) rate, and for a sufficiently large alphabet, the codes thus obtained lie above the Zyablov bound. Using these codes as outer codes in a concatenated scheme, a second asymptotic good construction is obtained which applies to small alphabets (say, GF(2)) as well. Although these concatenated codes lie below the Zyablov bound, they are still superior to previously known explicit constructions in the zero-rate neighborhood
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Entropy Waves, The Zig-Zag Graph Product, and New Constant-Degree Expanders and Extractors
The main contribution of this work is a new type of graph product, which we call the zig-zag product. Taking a product of a large graph with a small graph, the resulting graph inherits (roughly) its size from the large one, its degree from the small one, and its expansion properties from both! Iteration yields simple explicit constructions of constant-degree expanders of every size, starting from one constant-size expander.
Crucial to our intuition (and simple analysis) of the properties of this graph product is the view of expanders as functions which act as "entropy wave" propagators --- they transform probability distributions in which entropy is concentrated in one area to distributions where that concentration is dissipated. In these terms, the graph product affords the constructive interference of two such waves.
A variant of this product can be applied to extractors, giving the first explicit extractors whose seed length depends (poly)logarithmically on only the entropy deficiency of the source (rather than its length) and that extract almost all the entropy of high min-entropy sources. These high min-entropy extractors have several interesting applications, including the first constant-degree explicit expanders which beat the "eigenvalue bound."Engineering and Applied Science
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Randomness Conductors and Constant-Degree Lossless Expanders [Extended Abstract]
The main concrete result of this paper is the first explicit construction of constant degree lossless expanders. In these graphs, the expansion factor is almost as large as possible: (1-[epsilon])D, where D is the degree and [epsilon] is an arbitrarily small constant. The best previous explicit constructions gave expansion factor D/2, which is too weak for many applications. The D/2 bound was obtained via the eigenvalue method, and is known that that method cannot give better bounds.
The main abstract contribution of this paper is the introduction and initial study of randomness conductors, a notion which generalizes extractors, expanders, condensers and other similar objects. In all these functions, certain guarantee on the input "entropy" is converted to a guarantee on the output "entropy". For historical reasons, specific objects used specific guarantees of different flavors. We show that the flexibility afforded by the conductor definition leads to interesting combinations of these objects, and to better constructions such as those above. The main technical tool in these constructions is a natural generalization to conductors of the zig-zag graph product, previously defined for expanders and extractors.Engineering and Applied Science
On Eigenvalues of Random Complexes
We consider higher-dimensional generalizations of the normalized Laplacian
and the adjacency matrix of graphs and study their eigenvalues for the
Linial-Meshulam model of random -dimensional simplicial complexes
on vertices. We show that for , the eigenvalues of
these matrices are a.a.s. concentrated around two values. The main tool, which
goes back to the work of Garland, are arguments that relate the eigenvalues of
these matrices to those of graphs that arise as links of -dimensional
faces. Garland's result concerns the Laplacian; we develop an analogous result
for the adjacency matrix. The same arguments apply to other models of random
complexes which allow for dependencies between the choices of -dimensional
simplices. In the second part of the paper, we apply this to the question of
possible higher-dimensional analogues of the discrete Cheeger inequality, which
in the classical case of graphs relates the eigenvalues of a graph and its edge
expansion. It is very natural to ask whether this generalizes to higher
dimensions and, in particular, whether the higher-dimensional Laplacian spectra
capture the notion of coboundary expansion - a generalization of edge expansion
that arose in recent work of Linial and Meshulam and of Gromov. We show that
this most straightforward version of a higher-dimensional discrete Cheeger
inequality fails, in quite a strong way: For every and , there is a -dimensional complex on vertices that
has strong spectral expansion properties (all nontrivial eigenvalues of the
normalised -dimensional Laplacian lie in the interval
) but whose coboundary expansion is bounded
from above by and so tends to zero as ;
moreover, can be taken to have vanishing integer homology in dimension
less than .Comment: Extended full version of an extended abstract that appeared at SoCG
2012, to appear in Israel Journal of Mathematic
Variations on Classical and Quantum Extractors
Many constructions of randomness extractors are known to work in the presence
of quantum side information, but there also exist extractors which do not
[Gavinsky {\it et al.}, STOC'07]. Here we find that spectral extractors
with a bound on the second largest eigenvalue
are quantum-proof. We then discuss fully
quantum extractors and call constructions that also work in the presence of
quantum correlations decoupling. As in the classical case we show that spectral
extractors are decoupling. The drawback of classical and quantum spectral
extractors is that they always have a long seed, whereas there exist classical
extractors with exponentially smaller seed size. For the quantum case, we show
that there exists an extractor with extremely short seed size
, where denotes the quality of the
randomness. In contrast to the classical case this is independent of the input
size and min-entropy and matches the simple lower bound
.Comment: 7 pages, slightly enhanced IEEE ISIT submission including all the
proof
Expander Graphs and Coding Theory
Expander graphs are highly connected sparse graphs which lie at the interface of many diļ¬erent ļ¬elds of study. For example, they play important roles in prime sieves, cryptography, compressive sensing, metric embedding, and coding theory to name a few. This thesis focuses on the connections between sparse graphs and coding theory. It is a major challenge to explicitly construct sparse graphs with good expansion properties, for example Ramanujan graphs. Nevertheless, explicit constructions do exist, and in this thesis, we survey many of these constructions up to this point including a new construction which slightly improves on an earlier edge expansion bound. The edge expansion of a graph is crucial in applications, and it is well-known that computing the edge expansion of an arbitrary graph is NP-hard. We present a simple algo-rithm for approximating the edge expansion of a graph using linear programming techniques. While Andersen and Lang (2008) proved similar results, our analysis attacks the problem from a diļ¬erent vantage point and was discovered independently. The main contribution in the thesis is a new result in fast decoding for expander codes. Current algorithms in the literature can decode a constant fraction of errors in linear time but require that the underlying graphs have vertex expansion at least 1/2. We present a fast decoding algorithm that can decode a constant fraction of errors in linear time given any vertex expansion (even if it is much smaller than 1/2) by using a stronger local code, and the fraction of errors corrected almost doubles that of Viderman (2013)
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