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The NPcompleteness column: an ongoing guide
 Journal of Algorithms
, 1985
"... This is the nineteenth edition of a (usually) quarterly column that covers new developments in the theory of NPcompleteness. The presentation is modeled on that used by M. R. Garey and myself in our book ‘‘Computers and Intractability: A Guide to the Theory of NPCompleteness,’ ’ W. H. Freeman & Co ..."
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Cited by 188 (0 self)
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This is the nineteenth edition of a (usually) quarterly column that covers new developments in the theory of NPcompleteness. The presentation is modeled on that used by M. R. Garey and myself in our book ‘‘Computers and Intractability: A Guide to the Theory of NPCompleteness,’ ’ W. H. Freeman & Co., New York, 1979 (hereinafter referred to as ‘‘[G&J]’’; previous columns will be referred to by their dates). A background equivalent to that provided by [G&J] is assumed, and, when appropriate, crossreferences will be given to that book and the list of problems (NPcomplete and harder) presented there. Readers who have results they would like mentioned (NPhardness, PSPACEhardness, polynomialtimesolvability, etc.) or open problems they would like publicized, should
Expander Graphs and their Applications
, 2003
"... Contents 1 The Magical Mystery Tour 7 1.1 Some Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.1.1 Hardness results for linear transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.1.2 Error Correcting Codes . . . . . . . ..."
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Cited by 188 (5 self)
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Contents 1 The Magical Mystery Tour 7 1.1 Some Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.1.1 Hardness results for linear transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.1.2 Error Correcting Codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.1.3 Derandomizing Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.2 Magical Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.2.1 A Super Concentrator with O(n) edges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.2.2 Error Correcting Codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.2.3 Derandomizing Random Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 1.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The Laplacian spectrum of graphs
 Graph Theory, Combinatorics, and Applications
, 1991
"... Abstract. The paper is essentially a survey of known results about the spectrum of the Laplacian matrix of graphs with special emphasis on the second smallest Laplacian eigenvalue λ2 and its relation to numerous graph invariants, including connectivity, expanding properties, isoperimetric number, m ..."
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Cited by 151 (1 self)
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Abstract. The paper is essentially a survey of known results about the spectrum of the Laplacian matrix of graphs with special emphasis on the second smallest Laplacian eigenvalue λ2 and its relation to numerous graph invariants, including connectivity, expanding properties, isoperimetric number, maximum cut, independence number, genus, diameter, mean distance, and bandwidthtype parameters of a graph. Some new results and generalizations are added. † This article appeared in “Graph Theory, Combinatorics, and Applications”, Vol. 2,
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 118 (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.
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 90 (27 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
Lossless condensers, unbalanced expanders, and extractors
 In Proceedings of the 33rd Annual ACM Symposium on Theory of Computing
, 2001
"... Abstract Trevisan showed that many pseudorandom generator constructions give rise to constructionsof explicit extractors. We show how to use such constructions to obtain explicit lossless condensers. A lossless condenser is a probabilistic map using only O(log n) additional random bitsthat maps n bi ..."
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Cited by 89 (20 self)
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Abstract Trevisan showed that many pseudorandom generator constructions give rise to constructionsof explicit extractors. We show how to use such constructions to obtain explicit lossless condensers. A lossless condenser is a probabilistic map using only O(log n) additional random bitsthat maps n bits strings to poly(log K) bit strings, such that any source with support size Kis mapped almost injectively to the smaller domain. Our construction remains the best lossless condenser to date.By composing our condenser with previous extractors, we obtain new, improved extractors. For small enough minentropies our extractors can output all of the randomness with only O(log n) bits. We also obtain a new disperser that works for every entropy loss, uses an O(log n)bit seed, and has only O(log n) entropy loss. This is the best disperser construction to date,and yields other applications. Finally, our lossless condenser can be viewed as an unbalanced
Bounds For Dispersers, Extractors, And DepthTwo Superconcentrators
 SIAM JOURNAL ON DISCRETE MATHEMATICS
, 2000
"... ..."
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 52 (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...
Tight Bounds for DepthTwo Superconcentrators
 IN PROC. OF FOCS
, 1997
"... We show that the minimum size of a depthtwo Nsuperconcentrator is \Theta(N log² N= log log N ). Before this work, optimal bounds were known for all depths except two. For the upper bound, we build superconcentrators by putting together a small number of disperser graphs; these disperser graphs are ..."
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Cited by 24 (2 self)
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We show that the minimum size of a depthtwo Nsuperconcentrator is \Theta(N log² N= log log N ). Before this work, optimal bounds were known for all depths except two. For the upper bound, we build superconcentrators by putting together a small number of disperser graphs; these disperser graphs are obtained using a probabilistic argument. We present two different methods for showing lower bounds. First, we show that superconcentrators contain several disjoint disperser graphs. When combined with the lower bound for disperser graphs due to Kovari, S'os and Tur'an, this gives an almost optimal lower bound of \Omega\Gamma N(log N= log log N )²) on the size of N  superconcentrators. The second method, based on the work of Hansel, gives the optimal lower bound. The method of the Kovari, S'os and Tur'an can be extended to give tight lower bounds for extractors, both in terms of the number of truly random bits needed to extract one additional bit and in terms of the unavoidable entr...
SelfRouting Superconcentrators
, 1996
"... : Superconcentrators are switching systems that solve the generic problem of interconnecting clients and servers during sessions, in situations where either the clients or the servers are interchangeable (so that it does not matter which client is connected to which server). Previous constructions o ..."
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Cited by 22 (0 self)
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: Superconcentrators are switching systems that solve the generic problem of interconnecting clients and servers during sessions, in situations where either the clients or the servers are interchangeable (so that it does not matter which client is connected to which server). Previous constructions of superconcentrators have required an external agent to find the interconnections appropriate in each instance. We remedy this shortcoming by constructing superconcentrators that are "selfrouting", in the sense that they compute for themselves the required interconnections. Specifically, we show how to construct, for each n, a system Sn with the following properties. (1) The system Sn has n inputs, n outputs, and O(n) components, each of which is of one of a fixed finite number of finite automata, and is connected to a fixed finite number of other components through cables, each of which carries signals from a fixed finite alphabet. (2) When some of the inputs, and an equal number of outpu...