. A. Joffe. On a set of almost deterministic k-independent random variables. Ann. Probability 2(1974), 161-162.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....answering with a special ABORT message) 3. If j = 1, then V g;h answers with the verifier s fixed initiation message for session i (i.e. v 1 ) We stress that functions in such families can be described by strings of polynomial length in a way that enables polynomial time evaluation (cf. [24, 9, 10, 1]) In particular, Vg;h checks whether the query is of the prescribed format (as described in Section 2.5, and as determined by the schedule) and that the contents of its messages is consistent with V g;h s prior answers. That is, for every proper prefix q = b1 ; a1 ; bu ; au) of ....

A. Joffe. On a set of Almost Deterministic k-Independent Random Variables. The annals of Probability, 1974, Vol. 2, No. 1, pages 161-162.

....and compare them with a more general randomness. See Blum, Cucker, Shub, Smale [7] Novak ( 17] 18] 19] 21] and Traub, Wo zniakowski [24] The use of random bits for the summation problem and for related problems was studied in Chor, Goldreich [8] Goldreich, Wigderson [9] and Joffe [13]. 2 The Summation Problem We are interested in the approximate computation of a mapping S : F R; 1) where F is a class of real valued functions on a set D and S is an arbitrary mapping the solution operator , mapping an input (instance) f 2 F of our numerical problem to the exact ....

....: N Gamma 1g. Furthermore, Z k and Z l are independent if and only if gcd(N; k Gamma l) 1: c) The idea behind algorithm A n , to construct pairwise independent indices, was independently found by different authors. See Bakhvalov [4] Chor, Goldreich [8] Goldreich, Wigderson [9] Joffe [13], Sugita, Takanobu [22] and papers mentioned by these authors for more information. Algorithm A n combines the optimality of algorithm A n (optimality in the sense error versus number of function values) with the small amount of randomness of A n . This latter algorithm seems to be new. ....

Joffe, A.: On a set of almost deterministic k-independent random variables. Ann. Probab., 2, 161--162 (1974)

....Informatik, Saarbrucken. Basic Research in Computer Science, Centre of the Danish National Research Foundation. 1 Introduction During the last years there is a growing interest in techniques for removing randomness from parallel (and sequential) algorithms. These techniques were originated by [7, 8] and generalized in [1, 2, 4, 6, 9, 10, 11] The approach usually followed can be summarized as follows: The random variables which are considered are defined over a smaller probability space, specially designed, containing only a polynomial number of sample points. In that space, the random ....

A. Joffe, On a set of almost deterministic k-independent random variables, Ann. Probability, 2 (1974) 161-162. 9

....amount of work on derandomizing randomized algorithms. A common technique to reducing randomness is to use k wise independent random variables rather than totally independent ones. The generation of k wise independent and approximately k wise independent random variables has been well studied [Jof74, CG89, NN93, EGL 98, CRS00] Early applications of k wise independence in derandomization can be found in [KW85, Lub86, ABI86, Lub93, BR91] Very recently Klivans and Spielman [KS01] gave a randomness ecient method for testing if a polynomial is identically zero. In all of these algorithms a ....

....the sample. Furthermore, we would like it to work on the EREW PRAM, which is more restrictive than the CRCW PRAM, and is more relevant to real parallel machines [Val90, GMR99] We solve both these problems using Jo e s method for generating k wise independent variables, given below. Lemma 2.4. Jof74] Let q be prime, a 0 ; a k 1 be chosen uniformly from Z q , and X(i) P j a j i j (mod q) Then X(0) X(q 1) are uniformly distributed over Z q and k wise independent. That is, for generating pairwise independent variables we require two random coecients, a 0 and a 1 . We ....

A. Joe. On a set of almost deterministic k- independent random variables. Ann. Probability, 2(1):161-162, 1974.

....amount of work on derandomizing randomized algorithms. A common technique to reducing randomness is to use k wise independent random variables rather than totally independent ones. The generation of k wise independent and approximately k wise independent random variables has been well studied [Jof74, CG89, NN93, EGL 98, CRS00] Several results are known on derandomizing randomized algorithms that use k wise independence to obtain deterministic algorithms (see [KW85, Lub86, ABI86, Lub93, BR91] Very recently Klivans and Spielman [KS01] gave a randomness ecient method for testing if a ....

....elements in linear time in the size of the sample. Furthermore, we would like it to work on the EREW PRAM, which is a much more realistic model than the CRCW PRAM. We solve both of these problems using Jo e s method for generating k wise independent variables, given below. Lemma 2. 4 (Jo e [Jof74] Let q be prime, a 0 ; a 1 ; a k 1 be chosen uniformly at random from Z q , and X(i) P k 1 j=0 a j i j (mod q) Then X(0) X(q 1) are uniformly distributed over Z q and k wise independent. That is, for generating pairwise independent variables we require two random ....

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A. Joe. On a set of almost deterministic k-independent random variables. Ann. Probability, 2(1):161-162, 1974.

....on Randomization Methods in Algorithm Design. 1 1 Introduction Constructing pseudorandom permutation families is often more difficult than constructing pseudorandom function families. For example, there are polynomial size constructions of k wise independent function families for constant k [8, 9, 1, 12]. On the other hand, although there are polynomial size 3 wise independent permutation families (see, e.g. 14] there are only exponential size constructions known for higher k. In fact, the only subgroups of the symmetric group that are 6 wise independent are the alternating group and the ....

A. Joffe. On a set of almost deterministic k-independent random variables. Annals of Probability 2(1), 1974, pp. 161--162.

.... accept is also negligible. Lemma 2 If x 2 L, then M outputs accept with high probability. Proof Assume, again, that M will use no more that q distinct random strings s 1 ; s q in its queries (where q is polynomial in n) Let H be a family of q wise independent hash functions [Jof74, WC81, CG89] Consider a family of (malicious) veri ers indexed by H. For h 2 H, V h is the veri er that, on input random string s, generates (r; h(s) and then behaves like the honest veri er V on random strings r and . By completeness of (P; V) and because V h looks to the honest prover P just ....

A. Joe. On a set of almost deterministic k-independent random variables. Annals of Probability, 2:161-162, 1974.

.... of the arguments, but the converse is not true in general because a sequence generator need not have the property that arbitrary digits can be accessed eciently (only consecutive digits must be eciently computable) The usefulness of local randomness has previously been observed (e.g. 1] 3] [6], 7] and was referred to as k wise independence. However, our treatment is more general in that (1) families of functions that are only almost locally random of degree k and (2) polynomial time computable functions with superpolynomial degree of local randomization are considered, allowing ....

A. Joe, On a set of almost deterministic k-independent random variables, The Annals of Probability, Vol. 2, No. 1, pp. 161-162, 1974.

....pseudorandom permutations; explicit constructions. 1 Introduction Constructing pseudorandom permutation families is often more di#cult than constructing pseudorandom function families. For example, there are polynomial size constructions of k wise independent function families for constant k [5, 6, 1, 8]. On the other hand, although there are polynomial size 3 wise independent permutation families (see, e.g. 10] there are only exponential size constructions known for higher k. In fact, the only subgroups of the symmetric group that are 6 wise independent are the alternating group and the ....

A. Jo#e. On a set of almost deterministic k-independent random variables. Annals of Probability 2(1), 1974, pp. 161--162.

....be performed using polynomial work, then the construction can be performed using polynomial work. The parallelization is straighforward: all the possible samples are tested in parallel. 8 Construction. For de niteness, we assume the construction by Jo e of a k wise independent probability space [37]: Let be a prime number with n 2n; the sample space is ; k) Z k (the k fold product of the nite eld Z = f0; 1g) For 1 i and (a 0 ; a k 1 ) 2 ; k) let X i = P k 1 j=0 a j i j mod and let I i = 1 if 0 X i 2pn. The 0 1 random variables I 1 ; ....

A. Joe. On a set of almost deterministic k-independent random variables. Annals of Probability 2 (1974), 161-162. 39

....Saarbrucken. y Basic Research in Computer Science, Centre of the Danish National Research Foundation. 1 1 Introduction During the last years there is a growing interest in techniques for removing randomness from parallel (and sequential) algorithms. These techniques were originated by [7, 8] and generalized in [1, 2, 4, 6, 9, 10, 11] The approach usually followed can be summarized as follows: The random variables which are considered are defined over a smaller probability space, specially designed, containing only a polynomial number of sample points. In that space, the random ....

A. Joffe, On a set of almost deterministic k-independent random variables, Ann. Probability, 2 (1974) 161-162.

....O(log(m n) O(log n) and work O(n k 1 m) in the EREW PRAM model. Proof: First, the Chernoff Hoeffding bound for k wise independence guarantees the existence of a lattice vector q such that Delta i = O( p i m 1=k ) assuming that k is a constant) Second, use the construction in [16] of a k wise independent probability space of size O(n k ) 2 All points in D can be checked in parallel using time log(m n) and total work O(mn k 1 ) To simplify later expressions, we assume that m is polynomial in n, so that log(n m) O(log n) In any case, the resulting work bound is ....

A. Joffe. On a set of almost deterministic k-independent random variables. Annals of Probability 2 (1974) 161--162.

.... space of size O( nj) 4 ) O( n log(n Gamma1 ffi Gamma1 ) 4 ) with the assumed marginals (distributions of each variable Ku ) By slight modification of the marginals, earlier and somewhat simpler methods based on linear error correcting codes can likely be used as well, see [39, 50, 1, 9]. This would require modification of the analysis of the sampling process. 3 The case OE = 2 2 3.1 Preliminaries and related literature We focus now on the central special case in which the cost function is the square of Euclidean distance. We denote the Euclidean distance between u and v ....

A. Joffe. On a set of almost deterministic k-independent random variables. Ann. Probability, 2:161--162, 1974.

....be performed using polynomial work, then the construction can be performed using polynomial work. The parallelization is straighforward: all the possible samples are tested in parallel. 8 Construction. For de niteness, we assume the construction by Jo e of a k wise independent probability space [40]: Let be a prime number with n 2n; the sample space is ; k) Z k (the k fold product of the nite eld Z = f0; 1g) For 1 i and (a 0 ; a k 1 ) 2 ; k) let X i = P k 1 j=0 a j i j mod and let I i = 1 if 0 X i 2pn. The 0 1 random variables I 1 ; ....

A. Joe. On a set of almost deterministic k-independent random variables. Annals of Probability 2 (1974), 161-162.

....c 1 ffl CD ffl : 4.2 Probability Spaces with Limited Independence To compute good samples in parallel fast and deterministically, we follow the approach of sampling using probability spaces whose support has polynomial size. More specifically, we use t wise independent probability spaces [28, 30, 4]. 9 A probability distribution for the indicator (0 1) random variables I 1 ; I n is t wise independent if for any indices j 1 ; j t , and 0 1 values c 1 ; c t : PrfI j1 = c 1 ; I j t = c t g = Q t i=1 PrfI j i = c i g. The following fact is essential for our ....

....distribution for the indicator (0 1) random variables I 1 ; I n is t wise independent if for any indices j 1 ; j t , and 0 1 values c 1 ; c t : PrfI j1 = c 1 ; I j t = c t g = Q t i=1 PrfI j i = c i g. The following fact is essential for our algorithms [28, 29]. 8 This construction has been extensively used in computational geometry [14, 15, 32, 7] The idea of two levels of sampling also appears in the context of hashing [19] and possibly elsewhere) 9 In our algorithm, there is no advantage in using other more efficient constructions of ....

A. Joffe. On a set of almost deterministic k-independent random variables. Annals of Probability, 2 (1974) 161--162.

....area is Wigderson and Zuckerman [WZ93] where also more references can be found. Explicit quasi random graph (expanders) were applied in geometric algorithms by Ajtai and Megiddo [AM92] and by Katz and Sharir [KS93a] KS93b] Small k wise independent probability spaces were constructed by Joffe [Jof74] an asymptotically optimal size was achieved by Alon et al. ABI86] see also [KM94a] Constructions of (exactly) k wise independent probability spaces are also covered in [AS93] A nice survey of applications of 2 wise independence is Luby and Wigderson [LW95] Bounded independence was used ....

A. Joffe. On a set of almost deterministic k-independent random variables. Annals of Probability, 2:161--162, 1974.

....a bit by bit rounding approach. Unfortunately, under the requirement that the algorithm be in NC, the best discrepancies obtained are Delta i = O( q 1 ffl i log m) Using the Chernoff Hoeffding bounds for arbitrary p j s and the construction of k wise independent probability spaces in [22], it is possible to avoid the bit by bit rounding and obtain a faster and simpler algorithm (checking all the points in the probability space in a straightforward manner) though with worse bounds for the discrepancies obtained and the number of processors used. This algorithm, with k = 2, turns ....

....k 1 ) processors in the EREW PRAM model. The lemma is verified as follows. First, the Chernoff Hoeffding bound for k wise independence guarantees the existence of a lattice vector q such that Delta i = O( p i m 1=k ) assuming that k is a constant) Second, use the construction in [22] of a k wise independent probability space D of size O(n k ) 5 All points in D can be checked in parallel using time O(log(m n) and total work O(mn k 1 ) 2.2 Modelling Rounding with Levelled RFAs Limiting the Precision. In order to derandomize the rounding procedure while getting ....

A. Joffe. On a set of almost deterministic k-independent random variables. Annals of Probability 2 (1974), 161--162.

.... : we have P r( V i2I (X i = v i ) Q i2I P r(X i = v i ) Efficient constructions of sample spaces S f0; 1g n of size usually much smaller than 2 n , such that the distribution induced on f0; 1g n by sampling uniformly at random from S is k wise independent, are given, e.g. in [16, 19, 1]. The idea here is to analyze a given randomized algorithm and to show that its behavior is good enough if the X i s are k wise independent for a suitably large k = k(n) rather than completely independent. One can then search over all points in k wise independent sample space and ....

A. Joffe, On a set of almost deterministic k--independent random variables, The Annals of Probability, 2(1):161--162, 1974.

....if for any set I f1; ng of at most k indices and for any choice v 1 ; v n we have Pr[ i2I (X i = v i ) Y i2I Pr[X i = v i ] In this paper we will be primarily concerned with random bits, i.e. uniform binary random variables. Efficient constructions are given in [7, 5, 1] for sample spaces S f0; 1g n such that the distribution induced on f0; 1g n , by sampling uniformly from S, is k wise independent. These constructions have been used to design several deterministic algorithms from their randomized counterparts such as the NC algorithms for maximal ....

....setting X j equal to Y k where the color class of the vertex j is C k . Whenever vertices i and j are adjacent they belong to different color classes and hence the corresponding variables X i and X j are pairwise independent. We can explicitly construct such an S 0 with jS 0 j = jSj = O( G) [1, 5, 7]. 2 If G is a bipartite graph the above sample space is of constant size where as the construction in [8] has size O( Delta) Thus, for k = 2, Schulman s constructions essentially approximate the chromatic number by the maximum degree. Since G can be colored in NC with Delta 1 colors[7] we ....

A. Joffe. On a set of almost deterministic k--independent random variables. The Annals of Probability, 2(1):161--162, 1974.

...., asserting that the events of the form [X i 1 = b i 1 ; X i 2 = b i 2 ; X i d = b i d ] should have the same probability as if the variables were independent. History Most previous work in derandomization has focused on constructing small d wise independent distributions for small d [Jof74, Lub86, ABI86, KM94, NN93, AGHP90, AMN92, EGL 92, BR91, MNN89] Furthermore, the emphasis has been on constructions in NC, so as to allow a derandomization of parallel algorithms. Most of these works construct a distribution that only approximately satisfies some of the required constraints. ....

....KM94, NN93, AGHP90, AMN92, EGL 92, BR91, MNN89] Furthermore, the emphasis has been on constructions in NC, so as to allow a derandomization of parallel algorithms. Most of these works construct a distribution that only approximately satisfies some of the required constraints. The early works [Jof74, Lub86, ABI86] generate distributions that err on the probabilities Pr(X i = b i ) but precisely satisfy the actual independence constraints such as those asserting that Pr(X i = b i ; X j = b j ) Pr(X i = b i ) Delta Pr(X j = b j ) This approach is inherently limited, since it was shown by ....

A. Joffe. On a set of almost deterministic k-independent random variables. Annals of Probability, 2:161--162, 1974.

....fully independent bits (actually 1 log 2 n bits suffice) There are only n 2 choices for the seed, so we can try all the sequences and pick the best one in polynomial time. This was an instance of complete derandomization. Generalizations to higher degrees of independence also exist (Joffe [Jof], cf. ABI] CGHFRS] A similar argument with triplewise independence finds a truth value assigment that satisfies 7m=8 of a given set of m 3 clauses. A notion of k wise near independence, a particularly useful derandomization tool, was introduced in [NaN] 3.3.2 Isoperimetry, expanders, ....

Joffe, A.: On a set of almost deterministic k-independent random variables. Annals of Prob. 2 (1974), 161--162.

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. A. Joffe. On a set of almost deterministic k-independent random variables. Ann. Probability 2(1974), 161-162.

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A. Joffe. On a set of almost deterministic k-independent random variables. Ann. Probability 2 (1974), 161-162.

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A. Joffe. On a set of Almost Deterministic k-Independent Random Variables. The annals of Probability, 1974, Vol. 2, No. 1, pages 161-162.

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A. Joffe.: "On a Set of Almost Deterministic k-Independent Random Variables", the Annals of Probability, 1974,Vol. 2, No. 1, pp. 161-162.

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