Results 1  10
of
103
A.: ChernoffHoeffding bounds for applications with limited independence
 SIAM J. Discret. Math
, 1995
"... ..."
(Show Context)
Probabilistic generation of finite simple groups, II
, 2008
"... In earlier work it was shown that each nonabelian finite simple group G has a conjugacy class C such that, whenever 1 ̸ = x ∈ G, the probability is greater than 1/10 that G =〈x,y 〉 for a random y ∈ C. Much stronger asymptotic results were also proved. Here we show that, allowing equality, the bound ..."
Abstract

Cited by 45 (11 self)
 Add to MetaCart
In earlier work it was shown that each nonabelian finite simple group G has a conjugacy class C such that, whenever 1 ̸ = x ∈ G, the probability is greater than 1/10 that G =〈x,y 〉 for a random y ∈ C. Much stronger asymptotic results were also proved. Here we show that, allowing equality, the bound 1/10 can be replaced by 13/42; and, excluding an explicitly listed set of simple groups, the bound 2/3 holds. We use these results to show that any nonabelian finite simple group G has a conjugacy class C such that, if x1, x2 are nontrivial elements of G, then there exists y ∈ C such that G =〈x1,y〉=〈x2,y〉. Similarly, aside from one infinite family and a small, explicit finite set of simple groups, G has a conjugacy class C such that, if x1, x2, x3 are nontrivial elements of G, then there exists y ∈ C such that G =〈x1,y〉= 〈x2,y〉=〈x3,y〉. We also prove analogous but weaker results for almost simple groups.
Splitters and nearoptimal derandomization
"... We present a fairly general method for finding deterministic constructions obeying what we call krestrictions; this yields structures of size not much larger than the probabilistic bound. The structures constructed by our method include (n; k)universal sets (a collection of binary vectors of lengt ..."
Abstract

Cited by 37 (1 self)
 Add to MetaCart
We present a fairly general method for finding deterministic constructions obeying what we call krestrictions; this yields structures of size not much larger than the probabilistic bound. The structures constructed by our method include (n; k)universal sets (a collection of binary vectors of length n such that for any subset of size k of the indices, all 2k configurations appear) and families of perfect hash functions. The nearoptimal constructions of these objects imply the very efficient derandomization of algorithms in learning, of fixedsubgraph finding algorithms, and of near optimal threshold formulae. In addition, they derandomize the reduction showing the hardness of approximation of set cover. They also yield deterministic constructions for a localcoloring protocol, and for exhaustive testing of circuits.
Gibbs States Of The Hopfield Model In The Regime Of Perfect Memory
, 1994
"... : We study the thermodynamic properties of the Hopfield model of an autoassociative memory. If N denotes the number of neurons and M(N) the number of stored patterns, we prove the following results: If M N # 0 as N " 1, then there exists an infinite number of infinite volume Gibbs measures for ..."
Abstract

Cited by 22 (11 self)
 Add to MetaCart
: We study the thermodynamic properties of the Hopfield model of an autoassociative memory. If N denotes the number of neurons and M(N) the number of stored patterns, we prove the following results: If M N # 0 as N " 1, then there exists an infinite number of infinite volume Gibbs measures for all temperatures T ! 1 concentrated on spin configurations that have overlap with exactly one specific pattern. Moreover, the measures induced on the overlap parameters are Dirac measures concentrated on a single point. If M N ! ff, as N " 1 for ff small enough, we show that for temperatures T smaller than some T (ff) ! 1, the induced measures can have support only on a disjoint union of balls around the previous points, but we cannot construct the infinite volume measures through convergent sequences of measures. Subject Classification Numbers: 60K35, 82B44, 82C32 # Work partially supported by the Commission of the European Communities under contract No. SC1CT910695 1 email: bovier@iaa...
Boundeddegree graphs have arbitrarily large geometric thickness
, 2008
"... The geometric thickness of a graph G is the minimum integer k such that there is a straight line drawing of G with its edge set partitioned into k plane subgraphs. Eppstein [Separating thickness from geometric thickness. In Towards a Theory of Geometric Graphs, vol. 342 of Contemp. Math., AMS, 200 ..."
Abstract

Cited by 17 (7 self)
 Add to MetaCart
The geometric thickness of a graph G is the minimum integer k such that there is a straight line drawing of G with its edge set partitioned into k plane subgraphs. Eppstein [Separating thickness from geometric thickness. In Towards a Theory of Geometric Graphs, vol. 342 of Contemp. Math., AMS, 2004] asked whether every graph of bounded maximum degree has bounded geometric thickness. We answer this question in the negative, by proving that there exists ∆regular graphs with arbitrarily large geometric thickness. In particular, for all ∆ ≥ 9 and for all large n, there exists a ∆regular graph with geometric thickness at least c √ ∆n 1/2−4/∆−ǫ. Analogous results concerning graph drawings with few edge slopes are also presented, thus solving open problems by Dujmović et al. [Really straight graph drawings. In Proc. 12th
Revisiting the Efficiency of Malicious TwoParty Computation
 In Eurocrypt ’07, SpringerVerlag (LNCS 4515
, 2006
"... In a recent paper Mohassel and Franklin study the e#ciency of secure twoparty computation in the presence of malicious behavior. Their aim is to make classical solutions to this problem, such as zeroknowledge compilation, more practical. The authors provide several schemes which are the most e# ..."
Abstract

Cited by 12 (0 self)
 Add to MetaCart
(Show Context)
In a recent paper Mohassel and Franklin study the e#ciency of secure twoparty computation in the presence of malicious behavior. Their aim is to make classical solutions to this problem, such as zeroknowledge compilation, more practical. The authors provide several schemes which are the most e#cient to date. We propose a modification to their main scheme using expanders.
Randomly Sampling Molecules
 SIAM Journal on Computing
, 1996
"... We give the first polynomialtime algorithm for the following problem: Given a degree sequence in which each degree is bounded from above by a constant, select, uniformly at random, an unlabelled connected multigraph with the given degree sequence. We also give the first polynomialtime algorithm ..."
Abstract

Cited by 12 (3 self)
 Add to MetaCart
(Show Context)
We give the first polynomialtime algorithm for the following problem: Given a degree sequence in which each degree is bounded from above by a constant, select, uniformly at random, an unlabelled connected multigraph with the given degree sequence. We also give the first polynomialtime algorithm for the following related problem: Given a molecular formula, select, uniformly at random, a structural isomer having the given formula.
Localization for the Schrödinger operator with a Poisson random potential
, 2006
"... We prove exponential and dynamical localization for the Schrödinger operator with a nonnegative Poisson random potential at the bottom of the spectrum in any dimension. We also conclude that the eigenvalues in that spectral region of localization have finite multiplicity. We prove similar localizat ..."
Abstract

Cited by 10 (2 self)
 Add to MetaCart
(Show Context)
We prove exponential and dynamical localization for the Schrödinger operator with a nonnegative Poisson random potential at the bottom of the spectrum in any dimension. We also conclude that the eigenvalues in that spectral region of localization have finite multiplicity. We prove similar localization results in a prescribed energy interval at the bottom of the spectrum provided the density of the Poisson process is large enough.
Set systems without a simplex or a cluster
, 2007
"... A ddimensional simplex is a collection of d+1 sets with empty intersection, every d of which have nonempty intersection. A kuniform dcluster is a collection of d + 1 sets of size k with empty intersection and union of size at most 2k. We prove the following result which simultaneously addresses a ..."
Abstract

Cited by 10 (5 self)
 Add to MetaCart
A ddimensional simplex is a collection of d+1 sets with empty intersection, every d of which have nonempty intersection. A kuniform dcluster is a collection of d + 1 sets of size k with empty intersection and union of size at most 2k. We prove the following result which simultaneously addresses an old conjecture of Chvátal [7] and a recent conjecture of the second author [28]. For d ≥ 2 and ζ> 0 there is a number T such that the following holds for sufficiently large n. Let G be a kuniform set system on [n] = {1, · · · , n} with ζn < k < n/2 − T, and suppose either that G contains no ddimensional simplex or that G contains no dcluster. Then G  ≤ � � n−1 k−1 with equality only for the family of all ksets containing a specific element. In the nonuniform setting we obtain the following exact result that generalises an old question of Erdős and a result of Milner [11], who proved the case d = 2. Suppose d ≥ 2 and G is a set system on [n] that does not contain a ddimensional simplex, with n sufficiently large. Then G  ≤ 2n−1 + �d−1 � � n−1 i=0 i, with equality only for the family consisting of all sets that either contain some specific element or have size at most d − 1. Each of these results is proved via the corresponding stability result, which gives structural information on any G whose size is close to maximum. These in turn rely on a stability result that we obtain for intersecting families, which supersedes a result of Friedgut [17] that was proved using spectral techniques, and is based on a purely combinatorial result of Frankl.