Results 11  20
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62
Splitters and NearOptimal Derandomization
, 1995
"... We present a fairly general method for finding deterministic constructions obeying what we call k restrictions; 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 leng ..."
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Cited by 39 (2 self)
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We present a fairly general method for finding deterministic constructions obeying what we call k restrictions; 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 2 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 \Sigma\Pi\Sigma 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.
A New Parallel Algorithm For The Maximal Independent Set Problem
, 1989
"... A new parallel algorithm for the maximal independent set problem is constructed. It runs in O(log 4 n) time when implemented on a linear number of EREWprocessors. This is the first deterministic algorithm for the maximal independent set problem (MIS) whose running time is polylogarithmic and whose ..."
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Cited by 35 (2 self)
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A new parallel algorithm for the maximal independent set problem is constructed. It runs in O(log 4 n) time when implemented on a linear number of EREWprocessors. This is the first deterministic algorithm for the maximal independent set problem (MIS) whose running time is polylogarithmic and whose processortime product is optimal up to a polylogarithmic factor.
Approximations of General Independent Distributions
, 1992
"... We describe efficient constructions of small probability spaces that approximate the independent distribution for general random variables. Previous work on efficient constructions concentrate on approximations of the independent distribution for the special case of uniform booleanvalued random var ..."
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Cited by 32 (4 self)
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We describe efficient constructions of small probability spaces that approximate the independent distribution for general random variables. Previous work on efficient constructions concentrate on approximations of the independent distribution for the special case of uniform booleanvalued random variables. Our results yield efficient constructions of small sets with low discrepancy in high dimensional space and have applications to derandomizing randomized algorithms. 1 Introduction The problem of constructing small sample spaces that "approximate" the independent distribution on n random variables has received considerable attention recently (cf. [6, Chor Goldreich] [8, Karp Wigderson], [11, Luby], [1, Alon Babai Itai], [13, Naor Naor], [2, Alon Goldreich Hastad Peralta], [3, Azar Motwani Naor]). The primary motivation for this line of research is that random variables that are "approximately" independent suffices for the analysis of many interesting randomized algorithm and hence c...
Algorithmic Derandomization via Complexity Theory
 In Proceedings of the 34th annual ACM Symposium on Theory of Computing (STOC
, 2002
"... We point out how the methods of Nisan [Nis90, Nis92], originally developed for derandomizing spacebounded computations, may be applied to obtain polynomialtime and NC derandomizations of several probabilistic algorithms. Our list includes the randomized rounding steps of linear and semidefinit ..."
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Cited by 25 (1 self)
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We point out how the methods of Nisan [Nis90, Nis92], originally developed for derandomizing spacebounded computations, may be applied to obtain polynomialtime and NC derandomizations of several probabilistic algorithms. Our list includes the randomized rounding steps of linear and semidefinite programming relaxations of optimization problems, parallel derandomization of discrepancytype problems, and the JohnsonLindenstrauss lemma, to name a few.
Improved Algorithms via Approximations of Probability Distributions
 Journal of Computer and System Sciences
, 1997
"... We present two techniques for approximating probability distributions. The first is a simple method for constructing the smallbias probability spaces introduced by Naor & Naor. We show how to efficiently combine this construction with the method of conditional probabilities to yield improved NC alg ..."
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Cited by 24 (2 self)
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We present two techniques for approximating probability distributions. The first is a simple method for constructing the smallbias probability spaces introduced by Naor & Naor. We show how to efficiently combine this construction with the method of conditional probabilities to yield improved NC algorithms for many problems such as set discrepancy, finding large cuts in graphs, finding large acyclic subgraphs etc. The second is a construction of small probability spaces approximating general independent distributions, which is of smaller size than the constructions of Even, Goldreich, Luby, Nisan & Velickovi'c. Such approximations are useful, e.g., for the derandomization of certain randomized algorithms. Keywords. Derandomization, parallel algorithms, discrepancy, graph coloring, small sample spaces, explicit constructions. 1 Introduction Derandomization, the development of general tools to derive efficient deterministic algorithms from their randomized counterparts, has blossomed ...
Efficient Approximation of Product Distributions
 in Proceedings of the 24th Annual ACM Symposium on Theory of Computing
, 1998
"... We describe efficient constructions of small probability spaces that approximate the joint distribution of general random variables. Previous work on efficient constructions concentrate on approximations of the joint distribution for the special case of identical, uniformly distributed random var ..."
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Cited by 22 (1 self)
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We describe efficient constructions of small probability spaces that approximate the joint distribution of general random variables. Previous work on efficient constructions concentrate on approximations of the joint distribution for the special case of identical, uniformly distributed random variables. Preliminary version has appeared in the Proceedings of the 24th ACM Symp. on Theory of Computing (STOC), pages 1016, 1992. y Dept. of Electrical EngineeringSystems, TelAviv University, RamatAviv, TelAviv 69978, Israel. Email: guy@eng.tau.ac.il. z Department of Computer Science and Applied Mathematics, Weizmann Institute of Science, Rehovot, Israel. Email: oded@wisdom.weizmann.ac.il. Research partially supported by grant No. 8900312 from the United StatesIsrael Binational Science Foundation (BSF), Jerusalem, Israel. x International Computer Science Institute, Berkeley, CA 94704, USA. Email: luby@icsi.berkeley.edu. Research supported in part by National Science Founda...
Constructing a Maximal Independent Set in Parallel
 SIAM J. Disc. Math
, 1989
"... f a The problem of constructing in parallel a maximal independent set o given graph is considered. A new deterministic NC algorithm imple  t mented in the EREW PRAM model is presented. On graphs with n ver ices and m edges, it uses O ((n +m )/logn ) processors and runs in O (log n ) 3  c time. T ..."
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Cited by 21 (1 self)
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f a The problem of constructing in parallel a maximal independent set o given graph is considered. A new deterministic NC algorithm imple  t mented in the EREW PRAM model is presented. On graphs with n ver ices and m edges, it uses O ((n +m )/logn ) processors and runs in O (log n ) 3  c time. This reduces by a factor of logn both the running time and the pro essor count of the previously fastest deterministic algorithm which solves the problem using a linear number of processors. Key words: parallel computation, NC, graph, maximal independent set, 1 deterministic. . Introduction The problem of constructing in parallel a maximal independent set of a given graph, t MIS , has been investigated in several recent papers. Karp and Wigderson proved in [KW] hat the problem is in NC . Their algorithm finds a maximal independent set of an n  vertex graph in O (log n ) time and uses O (n /log n ) processors. In successive papers, the 4 3 3  s authors proposed algorithms which either...
Efficient ReadRestricted Monotone CNF/DNF Dualization by Learning with Membership Queries
, 1998
"... We consider exact learning monotone CNF formulas in which each variable appears at most some constant k times ("readk" monotone CNF). Let ..."
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Cited by 20 (1 self)
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We consider exact learning monotone CNF formulas in which each variable appears at most some constant k times ("readk" monotone CNF). Let
Testing kwise and almost kwise independence
 In 39th Annual ACM Symposium on Theory of Computing
, 2007
"... In this work, we consider the problems of testing whether a distribution over {0, 1} n is kwise (resp. (ɛ, k)wise) independent using samples drawn from that distribution. For the problem of distinguishing kwise independent distributions from those that are δfar from kwise independence in statis ..."
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Cited by 20 (8 self)
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In this work, we consider the problems of testing whether a distribution over {0, 1} n is kwise (resp. (ɛ, k)wise) independent using samples drawn from that distribution. For the problem of distinguishing kwise independent distributions from those that are δfar from kwise independence in statistical distance, we upper bound the number of required samples by Õ(nk /δ 2) and lower bound it by Ω(n k−1 2 /δ) (these bounds hold for constant k, and essentially the same bounds hold for general k). To achieve these bounds, we use Fourier analysis to relate a distribution’s distance from kwise independence to its biases, a measure of the parity imbalance it induces on a set of variables. The relationships we derive are tighter than previously known, and may be of independent interest. To distinguish (ɛ, k)wise independent distributions from those that are δfar from (ɛ, k)wise independence in statistical distance, we upper bound the number of required samples by O ` k log n δ2ɛ2 ´ and lower bound it by
Discrepancy Sets and Pseudorandom Generators for Combinatorial Rectangles
, 1996
"... A common subproblem of DNF approximate counting and derandomizing RL is the discrepancy problem for combinatorial rectangles. We explicitly construct a poly(n)size sample space that approximates the volume of any combinatorial rectangle in [n] n to within o(1) error (improving on the construction ..."
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Cited by 19 (4 self)
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A common subproblem of DNF approximate counting and derandomizing RL is the discrepancy problem for combinatorial rectangles. We explicitly construct a poly(n)size sample space that approximates the volume of any combinatorial rectangle in [n] n to within o(1) error (improving on the constructions of [EGLNV92]). The construction extends the techniques of [LLSZ95] for the analogous hitting set problem, most notably via discrepancy preserving reductions. 1 Introduction In a general discrepancy problem, we are given a family of sets and want to construct a small sample space that approximates the volume of an arbitrary set in the family. This problem is closely related to other important issues in combinatorial constructions such as the problem of constructing small sample spaces that approximate the independent distributions on many multivalued random variables [KW84, Lub85, ABI86, CG89, NN90, AGHP90, EGLNV92, Sch92, KM93, KK94], and the problem of constructing pseudorandom generat...