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20
Pseudorandom generators for regular branching programs
 In Proceedings of the 51st IEEE Symposium on Foundations of Computer Science
, 2010
"... We give new pseudorandom generators for regular readonce branching programs of small width. A branching program is regular if the indegree of every vertex in it is (0 or) 2. For every width d and length n, our pseudorandom generator uses a seed of length O((log d + log log n + log(1/ɛ)) log n) to ..."
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Cited by 23 (3 self)
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We give new pseudorandom generators for regular readonce branching programs of small width. A branching program is regular if the indegree of every vertex in it is (0 or) 2. For every width d and length n, our pseudorandom generator uses a seed of length O((log d + log log n + log(1/ɛ)) log n) to produce n bits that cannot be distinguished from a uniformly random string by any regular width d length n readonce branching program, except with probability ɛ. We also give a result for general readonce branching programs, in the case that there are no vertices that are reached with small probability. We show that if a (possibly nonregular) branching program of length n and width d has the property that every vertex in the program is traversed with probability at least γ on a uniformly random input, then the error of the generator above is at most 2ɛ/γ 2. 1
NONLINEAR SPECTRAL CALCULUS AND SUPEREXPANDERS
"... Nonlinear spectral gaps with respect to uniformly convex normed spaces are shown to satisfy a spectral calculus inequality that establishes their decay along Cesàro averages. Nonlinear spectral gaps of graphs are also shown to behave submultiplicatively under zigzag products. These results yield a ..."
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Cited by 15 (4 self)
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Nonlinear spectral gaps with respect to uniformly convex normed spaces are shown to satisfy a spectral calculus inequality that establishes their decay along Cesàro averages. Nonlinear spectral gaps of graphs are also shown to behave submultiplicatively under zigzag products. These results yield a combinatorial construction of superexpanders, i.e., a sequence of 3regular graphs that does not admit a coarse embedding into any uniformly convex normed space.
Towards a Calculus for NonLinear Spectral Gaps [Extended Abstract]
"... Given a finite regular graph ..."
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Pseudorandomness for regular branching programs via fourier analysis
 In APPROXRANDOM
, 2013
"... ar ..."
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Pseudorandomness for Width 2 Branching Programs
"... Recently Bogdanov and Viola (FOCS 2007) and Lovett (ECCC07) constructed pseudorandom generators that fool degree k polynomials over F2 for an arbitrary constant k. We show that such generators can also be used to fool branching programs of width 2 and polynomial length that read k bits of inputs at ..."
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Cited by 5 (1 self)
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Recently Bogdanov and Viola (FOCS 2007) and Lovett (ECCC07) constructed pseudorandom generators that fool degree k polynomials over F2 for an arbitrary constant k. We show that such generators can also be used to fool branching programs of width 2 and polynomial length that read k bits of inputs at a time. This model generalizes polynomials of degree k over F2 and includes some other interesting classes of functions, for instance kDNF. The constructions of Bogdanov and Viola and Lovett consist of adding a constant number of independent copies of a generator that fools linear functions (an ɛbiased set). It is natural to ask, in light of our first result, whether such generators can fool branching programs of width larger than 2. Our second result is a lower bound showing that a sum of o ( √ n / log n) independent copies of any n −O(1)biased set does not fool branching programs of width 5. To the best of our knowledge this is the first lower bound for such constructions.
SpaceEfficient Algorithms for Reachability in SurfaceEmbedded Graphs
, 2010
"... We consider the reachability problem for a certain class of directed acyclic graphs embedded on surfaces. Let G(m, g) be the class of directed acyclic graphs with m = m(n) source vertices embedded on a surface (orientable or nonorientable) of genus g = g(n). We give a logspace reduction that on in ..."
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We consider the reachability problem for a certain class of directed acyclic graphs embedded on surfaces. Let G(m, g) be the class of directed acyclic graphs with m = m(n) source vertices embedded on a surface (orientable or nonorientable) of genus g = g(n). We give a logspace reduction that on input G, u, v where G ∈G(m, g) and u and v are two vertices of G, outputs G,u,v where G is directed graph, and u,v are vertices of G, so that (a) there is a directed path from u to v in G if and only if there is a directed path from u to v in G and (b) G has O(m + g) vertices. By a direct application of Savitch’s theorem on the reduced instance we get a deterministic O(log n + log 2 (m + g))space algorithm for the reachability problem for graphs in G(m, g). By setting m and g to be 2O(√log n) we get that the reachability problem for directed acyclic graphs with 2O(√log n) O( sources embedded on surfaces of genus 2 √ log n) is in L (deterministic logarithmic
Pseudorandom Generators for Combinatorial Checkerboards
, 2011
"... We define a combinatorial checkerboard to be a function f: {1,...,m} d → {1, −1} of the form f(u1,...,ud) = ∏d i=1 fi(ui) for some functions fi: {1,...,m} → {1, −1}. This is a variant of combinatorial rectangles, which can be defined in the same way but using {0, 1} instead of {1, −1}. We consider ..."
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Cited by 3 (1 self)
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We define a combinatorial checkerboard to be a function f: {1,...,m} d → {1, −1} of the form f(u1,...,ud) = ∏d i=1 fi(ui) for some functions fi: {1,...,m} → {1, −1}. This is a variant of combinatorial rectangles, which can be defined in the same way but using {0, 1} instead of {1, −1}. We consider the problem of constructing explicit pseudorandom generators for combinatorial checkerboards. This is a generalization of smallbias generators, which correspond to the case m = 2. We construct a pseudorandom generator that ǫfools all combinatorial checkerboards with seed length O () 3/2 1 log m + log d · log log d + log ǫ. Previous work by Impagliazzo, Nisan, and
St connectivity on digraphs with a known stationary distribution
 IN IEEE CONFERENCE ON COMPUTATIONAL COMPLEXITY
, 2007
"... We present a deterministic logspace algorithm for solving ST CONNECTIVITY on directed graphs if (i) we are given a stationary distribution for random walk on the graph and (ii) the random walk which starts at the source vertex s has polynomial mixing time. This result generalizes the recent determi ..."
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Cited by 1 (0 self)
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We present a deterministic logspace algorithm for solving ST CONNECTIVITY on directed graphs if (i) we are given a stationary distribution for random walk on the graph and (ii) the random walk which starts at the source vertex s has polynomial mixing time. This result generalizes the recent deterministic logspace algorithm for ST CONNECTIVITY on undirected graphs [15]. It identifies knowledge of the stationary distribution as the gap between the ST CONNECTIVITY problems we know how to solve in logspace (L) and those that capture all of randomized logspace (RL).
The Computational Complexity of Randomness
, 2013
"... This dissertation explores the multifaceted interplay between efficient computation andprobability distributions. We organize the aspects of this interplay according to whether the randomness occurs primarily at the level of the problem or the level of the algorithm, and orthogonally according to wh ..."
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This dissertation explores the multifaceted interplay between efficient computation andprobability distributions. We organize the aspects of this interplay according to whether the randomness occurs primarily at the level of the problem or the level of the algorithm, and orthogonally according to whether the output is random or the input is random. Part I concerns settings where the problem’s output is random. A sampling problem associates to each input x a probability distribution D(x), and the goal is to output a sample from D(x) (or at least get statistically close) when given x. Although sampling algorithms are fundamental tools in statistical physics, combinatorial optimization, and cryptography, and algorithms for a wide variety of sampling problems have been discovered, there has been comparatively little research viewing sampling throughthelens ofcomputational complexity. We contribute to the understanding of the power and limitations of efficient sampling by proving a time hierarchy theorem which shows, roughly, that “a little more time gives a lot more power to sampling algorithms.” Part II concerns settings where the algorithm’s output is random. Even when the specificationofacomputational problem involves no randomness, onecanstill consider randomized
MATHEMATICAL ENGINEERING TECHNICAL REPORTS
, 2007
"... Generalized zigzag products of regular digraphs and bounds on their spectral expansions ..."
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Generalized zigzag products of regular digraphs and bounds on their spectral expansions