O. Goldreich. A sample of samplers - a computational perspective on sampling (survey). In Electronic Colloquium on Computational Complexity (ECCC) (20), volume 4, 1997.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....extractors have played a fundamental and unifying role in the theory of pseudorandomness. In particular, it has been discovered that they are intimately related to a number of other important and widely studied objects, such as hash functions [9] expander graphs [14, 28, 18, 23, 4] samplers [7, 30], pseudorandom generators [26] and error correcting codes [26, 25, 24] In addition, extractors have been found to have a vast and ever growing collection of applications in diverse aspects of computational complexity, combinatorics, and cryptography. See the excellent surveys [13, 19] Like the ....

O. Goldreich. A sample of samplers: A computational perspective on sampling. Technical Report TR97-020, Elec. Colloq. on Comput. Complexity, May 1997.

....[RSW00] We will impose a slightly stronger requirement on the sampling method: for any [0, 1] valued function, its average on S should approximate its average on [n] with high probability. Such sampling procedures are known as averaging (or oblivious) samplers, and have been studied extensively [BR94, CEG95, Zuc97, Gol97]. Our definition di#ers slightly from the standard definition, to allow us to obtain some savings in parameters (discussed later) Definition 8 A function Samp : is a ( #, #) averaging sampler if for every function f : n] 0, 1] with average value i f(i) i 1 , i t ) ....

.... whose randomness complexity is within a constant factor of optimal [Zuc97] However, these constructions have a sample complexity that is only polynomial in the optimal bound, resulting in a t local extractor with t = poly(log(1 #) 1 #) It is an interesting open problem, posed in [Gol97], to construct averaging samplers whose sample and randomness complexities are both within a constant factor of optimal. Without the averaging constraint, there are samplers which achieve this [BGG93, GW97, Gol97] 6.4 Previous Constructions. Some previous constructions of cryptosystems in ....

[Article contains additional citation context not shown here]

Oded Goldreich. A sample of samplers: A computational perspective on sampling. Technical Report TR97-020, Electronic Colloquium on Computational Complexity, May 1997.

.... to approximate the expectation of a function with respect to a known distribution over its domain (see, e.g. Karger s thesis [13] Tight bounds on the required amount of randomness and number of domain elements examined have been obtained for this problem (see the recent survey by Goldreich [8]) In contrast, our analysis, which is motivated by learning problems like those of Haussler s model, concerns the number of examples needed to estimate all the expectations of a possibly infinite family of random variables for any distribution over the domain. Since the learner s environment can ....

O. Goldreich. A sample of samplers: a computational perspective on sampling. Electronic Colloquium on Computational Complexity, TR97-020, 1997.

....permutations in the development of extractors. 7 Applications of the Extractors 7.1 Averaging Samplers In this section, we describe how our high min entropy extractors yield improved sampling algorithms, as pointed out to us by Ronen Shaltiel. The reader is referred to the survey of Goldreich [Gol97] and the references therein for a detailed description of previous work on samplers. A sampler is a randomized algorithm which, given any function f : m) 0; 1] as an oracle, estimates (with high probability) the average value of f up to some desired accuracy . It is desirable to minimize ....

.... from an analogous restriction on the relationship between and the input length in [Zuc97] 25 Nonconstructively, there are averaging samplers using only t = O(log(1= 2 ) samples and n = m log(1= O(1) random bits [CEG95, Zuc97] and these bounds are essentially tight [CEG95] See [Gol97] for precise statements. Zuckerman [Zuc97] has shown that averaging samplers are essentially equivalent to extractors. We will only use the transformation from extractors to samplers: From an extractor E : n) d) m) define an averaging sampler S E : n) m) t with t = 2 d by S ....

Oded Goldreich. A sample of samplers: A computational perspective on sampling. Technical Report TR97-020, Electronic Colloquium on Computational Complexity, May 1997.

..... S k . As a result, 1 In the next section we will give a seemingly weaker (but in fact equivalent) formal definition. 2106 ANDREEV, CLEMENTI, ROLIM, AND TREVISAN our sampling algorithm is not oblivious according to the definition of Bellare and Rompel [BR94] however, it is nonadaptive. See [G97] for definitions of these notions and for a survey on sampling. Our main result can be stated in the following way. Theorem 3 (main theorem) For any # 0, there exist a polynomial p and a deterministic algorithm A such that the following holds. For any # 0, n 0, any (p(n #) p(n #) # ....

....to sample is not Boolean but takes real values in the range [0, 1] The proof of Theorem 3 contained in this paper can be easily generalized to the case of such functions. We choose, however, to state and prove only the case of Boolean functions since proofs are cleaner and since, as proved in [G97], sampling real valued functions is reducible to sampling Boolean functions. We can thus get the following result as a corollary of Theorem 3 and of [G97, Theorem 5.5] Corollary 24. For any # 0, there exist a polynomial p and a deterministic algorithm A such that the following holds. For any # ....

<F3.742e+05> O. Goldreich,<F4.081e+05> A sample of samplers---A computational perspective on sampling,<F3.815e+05> Electronic Colloquium on Computational Complexity, 1997, TR97-020.

....than if we were to sample points independently each time, costing k(n) bits each time. Using expander graphs, the current best ratio one can achieve between the logarithm of the number of bad points and the logarithm of the number of all sample points is a constant less than 3=4 ffl (see e.g. [7]) Zuckerman [13] improved this to any ffl arbitrarily close to 0. Theorem 1.1 (Zuckerman) For any constant ffl 0, there is a deterministic polynomial time algorithm Z such that, for any BPP algorithm, which uses k(n) random bits on input of size n and achieves error probability 1=4, Z ....

O. Goldreich. A sample of samplers---A computational perspective on samplers. Available ECCC, Electronic Colloquium on Computational Complexity, TR97-020, May 1997. http://www.eccc.uni-trier.de/eccc/.

....function f that is accessed as oracle. The source of this non obliviousness is the selection of a good set S j among the candidates S 1 ; S k . As a result, our sampling algorithm is not oblivious according to the definition of Bellare and Rompel [BR94] however it is non adaptive. See [G97] for definitions of these notions and for a survey on sampling. Our main result can be stated in the following way Theorem 3 (Main Theorem) For any fl 0, there exist a polynomial p and a deterministic algorithm A such that the following holds. For any ffl 0, n 0, any (p(n=ffl) p(n=ffl) ....

....want to sample is not Boolean but takes real values in the range [0; 1] The proof of Theorem 3 contained in this paper can be easily generalized to the case of such functions. We choose however to state and prove only the case of Boolean functions since proofs are cleaner and since, as proved in [G97], sampling real valued functions is reducible to sampling Boolean functions. We can thus get the following result as a corollary of Theorem 3 and of [G97, Theorem 5.5] Corollary 24 For any fl 0, there exist a polynomial p and a deterministic algorithm A such that the following holds. For any ....

O. Goldreich. A sample of samplers --- A computational perspective on sampling. Electronic Colloquium on Computational Complexity TR97-020, 1997.

....BPP. Furthermore, in several cases the construction of combinatorial objects having one sided random (i.e. hitting) properties has turned out to be more efficient than that of combinatorial objects having two sided random (i.e. discrepancy) properties (for a survey of these cases see Appendix C of [10]) This is for instance the case for extractors and OR dispersers ( 13] Another case in which one sided randomness seems to be easier to achieve is in the case of small linear subspaces of f0; 1g n [4] It is indeed possible to construct small hitting sets for this class of subsets (and, ....

Goldreich O. (1997), "A Sample of Samplers: A Computational Perspective on Sampling", ECCC, TR97-20.

....g 2 Z N ) can be chosen in a randomness efficient way (which is polynomial time) In particular, we present a method of choosing a random n bit prime using only a linear number of random bits. This translates to a hitting problem which can be solved efficiently using methods described in [G2]. 4.1 Our construction vs. the HSS construction Looking at Theorem 3.2, the first construction that comes to mind is a pseudorandom generator that takes a seed r of length dn=2e and outputs g r mod N (for a fixed pair hN; gi in P n ) However, the output of the above so called pseudorandom ....

....in order to yield a prime) An additional requirement is that the distribution of primes obtained in this way will be very close to uniform. Our goal now is to find an algorithm that hits W , whose randomness complexity is linear in n. The methods we use are described in the survey of Goldreich [G2] on samplers and will be adapted to (and analyzed in) our specific setting. A pairwise independent hitter Our first attempt uses a pairwise independent sequence of m uniformly distributed strings in f0; 1g 2n . Such a sequence can be generated in the following way: We associate f0; 1g 2n ....

O. Goldreich, A sample of samplers: A computational perspective on sampling, ECCC 4(020), 1997.

....for the k simulations of A. The number of random bits used is r (for the starting point) k log d (for determining the walk) which is r O(k) while the error probability drops exponentially in k. 1 For some applications it is better to perform a different function on the k results. Refer to [5] for examples. 1.3 Amplification of space bounded computations The randomness of a probabilistic algorithm may be thought as the ability to flip an unbiased coin at each stage of its execution. Frequently it is more convenient to think of the results of these coin flips as given to the algorithm ....

O. Goldreich. A sample of samplers -- a computational perspective on sampling (survey). Electronic Colloquium on Computational Complexity (ECCC), TR97-020, 1997.

.... error probability at most 2 Gammat(jxj) which uses O(t(jxj) Delta p(jxj) coins (for any polynomial t) More efficient amplification procedures, utilizing Expander Random Walks and other tricks, yield the same error bound while using only p(jxj) 4 o(1) Delta t(jxj) coins (see survey [6]) In particular, for any constant c 4, using a sufficiently large polynomial t, we get a procedure which uses c Delta t(jxj) coins and has error probability bounded by 2 Gammat(jxj) An alternative construction due to Zuckerman provides, for any constant c 1 and sufficiently large ....

O. Goldreich. A Sample of Samplers -- A Computational Perspective on Sampling. ECCC, TR97-020, May 1997.

No context found.

O. Goldreich. A sample of samplers - a computational perspective on sampling (survey). In Electronic Colloquium on Computational Complexity (ECCC) (20), volume 4, 1997.

No context found.

O. Goldreich. A sample of samplers -- a computational perspective on sampling (survey). Electronic Colloquium on Computational Complexity (ECCC), TR97-020, 1997.

No context found.

O. Goldreich. A sample of samplers - a computational perspective on sampling (survey). In Electronic Colloquium on Computational Complexity (ECCC) (20), volume 4, 1997.

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC