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SmallBias Probability Spaces: Efficient Constructions and Applications
 SIAM J. Comput
, 1993
"... We show how to efficiently construct a small probability space on n binary random variables such that for every subset, its parity is either zero or one with "almost" equal probability. They are called fflbiased random variables. The number of random bits needed to generate the random variables is ..."
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Cited by 258 (15 self)
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We show how to efficiently construct a small probability space on n binary random variables such that for every subset, its parity is either zero or one with "almost" equal probability. They are called fflbiased random variables. The number of random bits needed to generate the random variables is O(log n + log 1 ffl ). Thus, if ffl is polynomially small, then the size of the sample space is also polynomial. Random variables that are fflbiased can be used to construct "almost" kwise independent random variables where ffl is a function of k. These probability spaces have various applications: 1. Derandomization of algorithms: many randomized algorithms that require only k wise independence of their random bits (where k is bounded by O(log n)), can be derandomized by using fflbiased random variables. 2. Reducing the number of random bits required by certain randomized algorithms, e.g., verification of matrix multiplication. 3. Exhaustive testing of combinatorial circui...
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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Dispersers, Deterministic Amplification, and Weak Random Sources.
, 1989
"... We use a certain type of expanding bipartite graphs, called disperser graphs, to design procedures for picking highly correlated samples from a finite set, with the property that the probability of hitting any sufficiently large subset is high. These procedures require a relatively small number of r ..."
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Cited by 93 (11 self)
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We use a certain type of expanding bipartite graphs, called disperser graphs, to design procedures for picking highly correlated samples from a finite set, with the property that the probability of hitting any sufficiently large subset is high. These procedures require a relatively small number of random bits and are robust with respect to the quality of the random bits. Using these sampling procedures to sample random inputs of polynomial time probabilistic algorithms, we can simulate the performance of some probabilistic algorithms with less random bits or with low quality random bits. We obtain the following results: 1. The error probability of an RP or BPP algorithm that operates with a constant error bound and requires n random bits, can be made exponentially small (i.e. 2 \Gamman ), with only (3 + ffl)n random bits, as opposed to standard amplification techniques that require \Omega\Gamma n 2 ) random bits for the same task. This result is nearly optimal, since the informati...
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...
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...
On approximate majority and probabilistic time
 in Proceedings of the 22nd IEEE Conference on Computational Complexity
, 2007
"... We prove new results on the circuit complexity of Approximate Majority, which is the problem of computing Majority of a given bit string whose fraction of 1’s is bounded away from 1/2 (by a constant). We then apply these results to obtain new relationships between probabilistic time, BPTime (t), and ..."
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Cited by 18 (6 self)
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We prove new results on the circuit complexity of Approximate Majority, which is the problem of computing Majority of a given bit string whose fraction of 1’s is bounded away from 1/2 (by a constant). We then apply these results to obtain new relationships between probabilistic time, BPTime (t), and alternating time, Σ O(1)Time (t). Our main results are the following: 1. We prove that 2 n0.1�size depth3 circuits for Approximate Majority on n bits have bottom fanin Ω(log n). As a corollary we obtain that BPTime (t) �⊆ Σ2Time � o(t 2) � with respect to some oracle. This complements the result that BPTime (t) ⊆ Σ2Time � t 2 · poly log t � with respect to every oracle (Sipser and Gács, STOC ’83; Lautemann, IPL ’83). 2. We prove that Approximate Majority is computable by uniform polynomialsize circuits of depth 3. Prior to our work, the only known polynomialsize depth3 circuits for Approximate Majority were nonuniform (Ajtai, Ann. Pure Appl. Logic ’83). We also prove that BPTime (t) ⊆ Σ3Time (t · poly log t). This complements our results in (1). 3. We prove new lower bounds for solving QSAT 3 ∈ Σ3Time (n · poly log n) on probabilistic computational models. In particular, we prove that solving QSAT 3 requires time n 1+Ω(1) on Turing machines with a randomaccess input tape and a sequentialaccess work tape that is initialized with random bits. No lower bound was previously known on this model (for a function computable in linear space). ∗ Author supported by NSF grant CCR0324906. Any opinions, findings and conclusions or recommendations expressed in this material are those of the author and do not necessarily reflect the views of the
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 ...
A new family of Cayley expanders
 Proc. 36th STOC, 2004
"... We assume that for some fixed large enough integer d, the symmetric group Sd can be generated as an expander using d 1/30 generators. Under this assumption, we explicitly construct an infinite family of groups Gn, and explicit sets of generators Yn ⊂ Gn, such that all generating sets have bounded si ..."
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Cited by 8 (3 self)
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We assume that for some fixed large enough integer d, the symmetric group Sd can be generated as an expander using d 1/30 generators. Under this assumption, we explicitly construct an infinite family of groups Gn, and explicit sets of generators Yn ⊂ Gn, such that all generating sets have bounded size (at most d 1/7), and the associated Cayley graphs are all expanders. The groups Gn above are very simple, and completely different from previous known examples of expanding groups. Indeed, Gn is (essentially) all symmetries of the dregular tree of depth n. The proof is completely elementary, using only simple combinatorics and linear algebra. The recursive structure of the groups Gn (iterated wreath products of the alternating group Ad) allows for an inductive proof of expansion, using the group theoretic analogue [4] of the zigzag graph product of [37]. The explicit construction of the generating sets Yn uses an efficient algorithm for solving certain equations over these groups, which relies on the work of [32] on the commutator width of perfect groups. We stress that our assumption above on weak expansion in the symmetric group is an open problem. We conjecture that it holds for all d. We discuss known results related to its likelihood in the paper.
Deterministic Amplification of SpaceBounded Probabilistic Algorithms
 In Proceedings of the 14th Annual IEEE Conference on Computational Complexity
, 1999
"... This paper initiates the study of deterministic amplification of spacebounded probabilistic algorithms. The straightforward implementations of known amplification methods cannot be used for such algorithms, since they consume too much space. We present a new implementation of the AjtaiKomlosSzeme ..."
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Cited by 7 (2 self)
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This paper initiates the study of deterministic amplification of spacebounded probabilistic algorithms. The straightforward implementations of known amplification methods cannot be used for such algorithms, since they consume too much space. We present a new implementation of the AjtaiKomlosSzemeredi method, that enables to amplify an Sspace algorithm that uses r random bits and errs with probability ffl to an O(kS)space algorithm that uses r +O(k) random bits and errs with probability ffl\Omega\Gamma k) . This method can be used to reduce the error probability of BPL algorithms below any constant, with only a constant addition of new random bits. This is weaker than the exponential reduction that can be achieved for BPP algorithms by methods that use only O(r) random bits. However, we prove that any blackbox amplification method that uses O(r) random bits and makes at most p parallel simulations reduces the error to at most ffl O(p) . Hence, in BPL, where p should be a con...