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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 189 (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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Property Testing in Bounded Degree Graphs
 Algorithmica
, 1997
"... We further develop the study of testing graph properties as initiated by Goldreich, Goldwasser and Ron. Whereas they view graphs as represented by their adjacency matrix and measure distance between graphs as a fraction of all possible vertex pairs, we view graphs as represented by boundedlength in ..."
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Cited by 119 (36 self)
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We further develop the study of testing graph properties as initiated by Goldreich, Goldwasser and Ron. Whereas they view graphs as represented by their adjacency matrix and measure distance between graphs as a fraction of all possible vertex pairs, we view graphs as represented by boundedlength incidence lists and measure distance between graphs as a fraction of the maximum possible number of edges. Thus, while the previous model is most appropriate for the study of dense graphs, our model is most appropriate for the study of boundeddegree graphs. In particular, we present randomized algorithms for testing whether an unknown boundeddegree graph is connected, kconnected (for k ? 1), planar, etc. Our algorithms work in time polynomial in 1=ffl, always accept the graph when it has the tested property, and reject with high probability if the graph is fflaway from having the property. For example, the 2Connectivity algorithm rejects (w.h.p.) any Nvertex ddegree graph for which more ...
Eigenvalues, geometric expanders, sorting in rounds, and Ramsey theory
 CORNBINATORICA
, 1986
"... Expanding graphs are relevant to theoretical computer science in several ways. Here we show that the points versus hyperplanes incidence graphs of finite geometries form highly (nonlinear) expanding graphs with essentially the smallest possible number of edges. The expansion properties of the graphs ..."
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Cited by 45 (12 self)
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Expanding graphs are relevant to theoretical computer science in several ways. Here we show that the points versus hyperplanes incidence graphs of finite geometries form highly (nonlinear) expanding graphs with essentially the smallest possible number of edges. The expansion properties of the graphs are proved using the eigenvalues of their adjacency matrices. These graphs enable us to improve previous results on a parallel sorting problem that arises in structural modeling, by describing an explicit algorithm to sort n elements in k time units using O(n ~k) parallel processors, where, e.g., cq=7/4, ~q8/5, 0q=26/17 and ~q=22/15. Our approach also yields several applications to Ramsey Theory and other extremal problems in
Pseudorandom Generators Hard for kDNF Resolution and Polynomial Calculus Resolution
, 2003
"... A pseudorandom generator G n : f0; 1g is hard for a propositional proof system P if (roughly speaking) P can not ef ciently prove the statement G n (x 1 ; : : : ; x n ) 6= b for any string b 2 . We present a function (m 2 ) generator which is hard for Res( log n); here Res(k) is the ..."
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Cited by 42 (4 self)
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A pseudorandom generator G n : f0; 1g is hard for a propositional proof system P if (roughly speaking) P can not ef ciently prove the statement G n (x 1 ; : : : ; x n ) 6= b for any string b 2 . We present a function (m 2 ) generator which is hard for Res( log n); here Res(k) is the propositional proof system that extends Resolution by allowing kDNFs instead of clauses.
Mixnetworks with Restricted Routes
 Proceedings of Privacy Enhancing Technologies workshop (PET 2003). SpringerVerlag, LNCS 2760
, 2003
"... We present a mix network topology that is based on sparse expander graphs, with each mix only communicating with a few neighbouring others. We analyse the anonymity such networks provide, and compare it with fully connected mix networks and mix cascades. We prove that such a topology is efficient si ..."
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Cited by 42 (9 self)
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We present a mix network topology that is based on sparse expander graphs, with each mix only communicating with a few neighbouring others. We analyse the anonymity such networks provide, and compare it with fully connected mix networks and mix cascades. We prove that such a topology is efficient since it only requires the route length of messages to be relatively small in comparison with the number of mixes to achieve maximal anonymity. Additionally mixes can resist intersection attacks while their batch size, that is directly linked to the latency of the network, remains constant. A worked example of a network is also presented to illustrate how these results can be applied to create secure mix networks in practise.
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...
Juggling Networks
, 1993
"... : Switching networks of various kinds have come to occupy a prominent position in computer science as well as communication engineering. The classical switching network technology has been spacedivisionmultiplex switching, in which each switching function is performed by a spatially separate switc ..."
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Cited by 14 (0 self)
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: Switching networks of various kinds have come to occupy a prominent position in computer science as well as communication engineering. The classical switching network technology has been spacedivisionmultiplex switching, in which each switching function is performed by a spatially separate switching component (such as a crossbar switch). A recent trend in switching network technology has been the advent of timedivisionmultiplex switching, wherein a single switching component performs the function of many switches at successive moments of time according to a periodic schedule. This technology has the advantage that nearly all of the cost of the network is in inertial memory (such as delay lines), with the cost of switching elements growing much more slowly as a function of the capacity of the network. In order for a classical spacedivisionmultiplex network to be adaptable to timedivision multiplex technology, its interconnection pattern must satisfy stringent requirements. For ...