Results 1  10
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16
Expander Codes
 IEEE TRANSACTIONS ON INFORMATION THEORY
, 1996
"... We present a new class of asymptotically good, linear errorcorrecting codes based upon expander graphs. These codes have linear time sequential decoding algorithms, logarithmic time parallel decoding algorithms with a linear number of processors, and are simple to understand. We present both random ..."
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Cited by 340 (10 self)
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We present a new class of asymptotically good, linear errorcorrecting codes based upon expander graphs. These codes have linear time sequential decoding algorithms, logarithmic time parallel decoding algorithms with a linear number of processors, and are simple to understand. We present both randomized and explicit constructions for some of these codes. Experimental results demonstrate the extremely good performance of the randomly chosen codes.
Lineartime Encodable and Decodable ErrorCorrecting Codes
, 1996
"... We present a new class of asymptotically good, linear errorcorrecting codes. These codes can be both encoded and decoded in linear time. They can also be encoded by logarithmicdepth circuits of linear size and decoded by logarithmic depth circuits of size 0 (n log n). We present both randomized an ..."
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Cited by 145 (5 self)
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We present a new class of asymptotically good, linear errorcorrecting codes. These codes can be both encoded and decoded in linear time. They can also be encoded by logarithmicdepth circuits of linear size and decoded by logarithmic depth circuits of size 0 (n log n). We present both randomized and explicit constructions of these codes.
Improved NonApproximability Results
, 1994
"... We indicate strong nonapproximability factors for central problems: N^{1/4} for Max Clique; N^{1/10} for Chromatic Number; and 66/65 for Max 3SAT. Underlying the Max Clique result is a proof system in... ..."
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Cited by 110 (13 self)
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We indicate strong nonapproximability factors for central problems: N^{1/4} for Max Clique; N^{1/10} for Chromatic Number; and 66/65 for Max 3SAT. Underlying the Max Clique result is a proof system in...
The Complexity and Approximability of Finding Maximum Feasible Subsystems of Linear Relations
 Theoretical Computer Science
, 1993
"... We study the combinatorial problem which consists, given a system of linear relations, of finding a maximum feasible subsystem, that is a solution satisfying as many relations as possible. The computational complexity of this general problem, named Max FLS, is investigated for the four types of rela ..."
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Cited by 92 (11 self)
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We study the combinatorial problem which consists, given a system of linear relations, of finding a maximum feasible subsystem, that is a solution satisfying as many relations as possible. The computational complexity of this general problem, named Max FLS, is investigated for the four types of relations =, , ? and 6=. Various constrained versions of Max FLS, where a subset of relations must be satisfied or where the variables take bounded discrete values, are also considered. We establish the complexity of solving these problems optimally and, whenever they are intractable, we determine their degree of approximability. Max FLS with =, or ? relations is NPhard even when restricted to homogeneous systems with bipolar coefficients, whereas it can be solved in polynomial time for 6= relations with real coefficients. The various NPhard versions of Max FLS belong to different approximability classes depending on the type of relations and the additional constraints. We show that the ran...
Expanders that Beat the Eigenvalue Bound: Explicit Construction and Applications
 Combinatorica
, 1993
"... For every n and 0 ! ffi ! 1, we construct graphs on n nodes such that every two sets of size n ffi share an edge, having essentially optimal maximum degree n 1\Gammaffi+o(1) . Using known and new reductions from these graphs, we explicitly construct: 1. A k round sorting algorithm using n 1+1=k ..."
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Cited by 85 (25 self)
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For every n and 0 ! ffi ! 1, we construct graphs on n nodes such that every two sets of size n ffi share an edge, having essentially optimal maximum degree n 1\Gammaffi+o(1) . Using known and new reductions from these graphs, we explicitly construct: 1. A k round sorting algorithm using n 1+1=k+o(1) comparisons. 2. A k round selection algorithm using n 1+1=(2 k \Gamma1)+o(1) comparisons. 3. A depth 2 superconcentrator of size n 1+o(1) . 4. A depth k widesense nonblocking generalized connector of size n 1+1=k+o(1) . All of these results improve on previous constructions by factors of n\Omega\Gamma37 , and are optimal to within factors of n o(1) . These results are based on an improvement to the extractor construction of Nisan & Zuckerman: our algorithm extracts an asymptotically optimal number of random bits from a defective random source using a small additional number of truly random bits. 1
Eigenvalues and Expansion of Regular Graphs
 Journal of the ACM
, 1995
"... The spectral method is the best currently known technique to prove lower bounds on expansion. Ramanujan graphs, which have asymptotically optimal second eigenvalue, are the best known explicit expanders. The spectral method yielded a lower bound of k=4 on the expansion of linear sized subsets of kr ..."
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Cited by 49 (1 self)
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The spectral method is the best currently known technique to prove lower bounds on expansion. Ramanujan graphs, which have asymptotically optimal second eigenvalue, are the best known explicit expanders. The spectral method yielded a lower bound of k=4 on the expansion of linear sized subsets of kregular Ramanujan graphs. We improve the lower bound on the expansion of Ramanujan graphs to approximately k=2. Moreover, we construct a family of kregular graphs with asymptotically optimal second eigenvalue and linear expansion equal to k=2. This shows that k=2 is the best bound one can obtain using the second eigenvalue method. We also show an upper bound of roughly 1 + p k \Gamma 1 on the average degree of linearsized induced subgraphs of Ramanujan graphs. This compares positively with the classical bound 2 p k \Gamma 1. As a byproduct, we obtain improved results on random walks on expanders and construct selection networks (resp. extrovert graphs) of smaller size (resp. degree) th...
Random Cayley Graphs and Expanders
 Random Structures Algorithms
, 1997
"... For every 1 ? ffi ? 0 there exists a c = c(ffi) ? 0 such that for every group G of order n, and for a set S of c(ffi) log n random elements in the group, the expected value of the second largest eigenvalue of the normalized adjacency matrix of the Cayley graph X(G;S) is at most (1\Gammaffi). Thi ..."
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Cited by 38 (1 self)
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For every 1 ? ffi ? 0 there exists a c = c(ffi) ? 0 such that for every group G of order n, and for a set S of c(ffi) log n random elements in the group, the expected value of the second largest eigenvalue of the normalized adjacency matrix of the Cayley graph X(G;S) is at most (1\Gammaffi). This implies that almost every such a graph is an "(ffi)expander. For Abelian groups this is essentially tight, and explicit constructions can be given in some cases. Department of Mathematics, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, Ramat Aviv, Tel Aviv, Israel. Research supported in part by a U.S.A.Israeli BSF grant. y Department of Mathematics, Hebrew University of Jerusalem, Givat Ram, Jerusalem, Israel 0 1.
Convergence to Equilibrium in Local Interaction Games
"... Abstract — We study a simple game theoretic model for the spread of an innovation in a network. The diffusion of the innovation is modeled as the dynamics of a coordination game in which the adoption of a common strategy between players has a higher payoff. Classical results in game theory provide a ..."
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Cited by 32 (2 self)
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Abstract — We study a simple game theoretic model for the spread of an innovation in a network. The diffusion of the innovation is modeled as the dynamics of a coordination game in which the adoption of a common strategy between players has a higher payoff. Classical results in game theory provide a simple condition for an innovation to become widespread in the network. The present paper characterizes the rate of convergence as a function of graph structure. In particular, we derive a dichotomy between wellconnected (e.g. random) graphs that show slow convergence and poorly connected, low dimensional graphs that show fast convergence. 1.
FROM FINDING MAXIMUM FEASIBLE SUBSYSTEMS OF LINEAR SYSTEMS TO FEEDFORWARD NEURAL NETWORK DESIGN
, 1994
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Isoperimetric Inequalities and Eigenvalues
, 1997
"... An upper bound is given on the minimum distance between i subsets of the same size of a regular graph in terms of the ith largest eigenvalue in absolute value. This yields a bound on the diameter in terms of the ith largest eigenvalue, for any integer i. Our bounds are shown to be asymptotically t ..."
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Cited by 7 (0 self)
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An upper bound is given on the minimum distance between i subsets of the same size of a regular graph in terms of the ith largest eigenvalue in absolute value. This yields a bound on the diameter in terms of the ith largest eigenvalue, for any integer i. Our bounds are shown to be asymptotically tight. A recent result by Quenell relating the diameter, the second eigenvalue, and the girth of a regular graph is obtained as a byproduct. Key words. eigenvalues, diameter, Chebychev polynomials, expanders AMS(MOS) subject classification. 05C35 1 Introduction Many combinatorial properties of a graph are related to the spectrum of its adjacency matrix [2, 3, 4, 18]. The adjacency matrix A of an undirected graph is the 0 0 1 matrix indexed by the vertices, and such that the entry (u; v) is equal to 1 if and only if (u; v) is an edge. Since the adjacency matrix of any graph H on n vertices is symmetric and real, its eigenvalues are real and will be denoted by 0 (H) 1 (H) 1 1 1 n01 (...