N. Alon, L. Babai and A. Itai. A fast and Simple Randomized Algorithm for the Maximal Independent Set Problem. J. of Algorithms, Vol. 7, pages 567--583, 1986.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....random from G. We call G k wise independent if for all distinct x 1 ; x 2 ; x k 2 D, the random variables g(x 1 ) g(x 2 ) g(x k ) are independent and uniformly distributed in R. Families of k wise independent functions have been studied and applied extensively in recent years (cf. [3, 6, 7, 8, 14, 18]) A useful property is that if G is k wise independent, 6 then for all 1 i k, for all x 1 ; x 2 ; x i 2 D, and v 1 ; v 2 ; v i Gamma1 2 R: the distribution of g(x i ) given that (g(x 1 ) v 1 ; g(x 2 ) v 2 ; g(x i Gamma1 ) v i Gamma1 ) is uniform in R. For our ....

....] in some arbitrary manner and let g i be the polynomial of degree k Gamma 1 whose jth coefficient is a j Delta i b j . i.e. Proposition 4.6 A function g i chosen in this manner is k wise independent. Proof: As each g i is a random polynomial of degree k Gamma 1, this follows from [3]. 2 From the pair wise independence of polynomials of degree 1, and the random choice of the a and b values above, the set of the 2k coefficients of g i and g j are independent and we have the following (which was used in Lemma 4.3) Proposition 4.7 For all 1 i j , the polynomials g i and g ....

N. Alon, L. Babai and A. Itai, A fast and simple randomized algorithm for the maximal independent set problem, Journal of Algorithms 7, pp. 567-583, 1987.

....we mention several suggestions by Noga Alon [1] which could provide interesting directions for further work. The problem of guessing secrets is closely related to the study of small sample spaces supporting k wise independent (or nearly independent) random variables, which has a rich literature [2, 21, 3, 4]. The problem of interest there is to find a sample space as small as possible, and n binary random variables defined on it, with the property, called k wise independence, that for any choice of k random variables X 1 ; X k , the probabilities satisfy: P rob(X 1 : X k = a 1 : a k ....

N. Alon, L. Babai and A. Itai, A fast and simple randomized algorithm for the maximal independent set problem, J. Algorithms, 7 (1986), 567-583.

....we mention several suggestions by Noga Alon [1] which could provide interesting directions for further work. The problem of guessing secrets is closely related to the study of small sample spaces supporting k wise independent (or nearly independent) random variables, which has a rich literature [2, 21, 3, 4]. The problem of interest there is to find a sample space as small as possible, and n binary random variables defined on it, with the property, called k wise independence, that for any choice of k random variables X 1 , X k , the probabilities satisfy: Prob(X 1 . X k = a 1 . a k ) 1 2 ....

N. Alon, L. Babai and A. Itai, A fast and simple randomized algorithm for the maximal independent set problem, J. Algorithms, 7 (1986), 567-583.

....Then, any k bits of y is the result of multiplying a k Theta matrix of rank k by a uniformly chosen vector of length . By the argument above, such a set of k bits is uniformly random. Hence, y is k wise independent. 3 An Alternate Construction This construction, due to Alon, Babai, and Itai [ABI86], gives a k wise independent distribution over a sample space of size about n bk=2c , saving a square root factor over the previous construction. Assume n = 2 r Gamma 1, and work over GF (2 r ) Let a 1 ; a 2 r Gamma1 be the nonzeroes of GF (2 r ) Then, the van der Monde matrix ....

N. Alon, L. Babai, and A. Itai, "A Fast and Simple Randomized Algorithm for the Maximal Independent Set Problem," J. of Algorithms, 1986, pp. 567-583.

....over F n in polynomial time, using a random seed of length t Delta log 2 jF j. Specifically, the seed is used to specify a polynomial of degree t Gamma 1 over F , and the i th element in the output sequence is the result of evaluating this polynomial at the i th field element (cf. [2, 7]) Small bias generators. Here, we consider distributions of n long sequences over f0; 1g. For ffl 2 [0; 1] such a distribution is called ffl bias if for every non empty subset I, the exclusive or of the bits at locations I equals 1 with probability at least (1 Gamma ffl) Delta 1 2 and at ....

N. Alon, L. Babai and A. Itai. A fast and Simple Randomized Algorithm for the Maximal Independent Set Problem. J. of Algorithms, Vol. 7, pages 567--583, 1986.

....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 for derandomization e.g. in Karp and Wigderson [KW85] A combination ....

N. Alon, L. Babai, and A. Itai. A fast and simple randomized algorithm for the maximal independent set problem. Journal of Algorithms, 7:567--583, 1986.

....a random list satisfying Items (1) and (2) above is to use a sequence of pairwise independent random variables. Actually, our analysis becomes even easier if we use a sequence of 3 wise independent random variables. It will be most convenient to use the construction given in Alon et al. [3] which works in the field GF(2 m ) since this field corresponds naturally to the set of m bit strings. The construction uses t 2 m arbitrary elements of the field, denoted ff 1 ; ff 2 ; ff t . A sequence of t elements is represented by a triplet of field elements, denoted (u; v; w) ....

Alon, N., L., Babai and A. Itai, "A Fast and Simple Randomized Algorithm for the Maximal Indepedent Set Problem", J. of Algorithms, Vol. 7, 1986, pp. 567--583.

....a random list satisfying Items (1) and (2) above is to use a sequence of pairwise independent random variables. Actually, our analysis becomes even easier if we use a sequence of 3 wise independent random variables. It will be most convenient to use the construction given in Alon et al. [4] which works in the field GF(2 m ) since this field corresponds naturally to the set of m bit strings. The construction uses t 2 m arbitrary elements of the field, denoted ff 1 ; ff 2 ; ff t . A sequence of t elements is represented by a triplet of field elements, denoted (u; v; w) ....

Alon, N., L., Babai and A. Itai, "A Fast and Simple Randomized Algorithm for the Maximal Independent Set Problem", J. of Algorithms, Vol. 7, 1986, pp. 567--583.

....principal motivation has been to derandomize probabilistic algorithms. For example, approximations can be used to simulate random sampling and generate pseudo random permutations deterministically, in a manner vastly superior to what other derandomization techniques alone can offer in geometry [2, 3, 12, 14, 16, 19]. In addition, nets and approximations provide the tools for deterministic quasi Monte Carlo integration (which, for example, played a key role in solving the convex hull problem [8] Matousek et al. 13] recently demonstrated the close connection between approximations and discrepancy theory. ....

Alon, N., Babai, L., Itai, A. A fast and simple randomized algorithm for the maximal independent set problem, J. Alg., 7 (1986), 567--583.

....However, this result is highly non constructive and it appears that it cannot be used to actually de randomize algorithms. Two methods for searching the sample space have emerged in recent years and are briefly described below. The first technique is an exhaustive search of the sample space [KW, Lu1, ABI]. Since there is a good point in Omega for each possible input, trying every w 2 Omega is guaranteed to yield a good point. The problem with this approach is that generally the size of the sample space is exponential in the size of the input. This difficulty was overcome in certain cases (the ....

....set covers in hypergraphs. We show that this random sampling can be de randomized in NC using our methods and again the set balancing problem plays an important role. Finally, we give two more problems to which our techniques can be applied to, namely finding large independent sets in hypergraphs [ABI] and constructing Ramsey graphs. The rest of this paper is organized as follows. In Section 2 we describe the set balancing and lattice approximation problems and randomized algorithms for them. In Section 3 we present a brief overview of the method of conditional probabilities by describing its ....

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N. Alon, L. Babai and A. Itai, "A fast and simple randomized algorithm for the maximal independent set problem," J. of Algorithms, vol. 7 (1986), pp. 567-583.

....2k Gammam Delta n 2 and the proposition follows. The less obvious case is when m k log 2 n. Here we use a family of n wise independent functions mapping f0; 1g n onto f0; 1g , where def = m Gamma log 2 n. Function in such a family can be evaluated by poly(n) size circuits; cf. [1]. We consider the collisions caused by a uniformly chosen function from this family applied to S. Specifically, Lemma 2.1 Let H be a family of functions fh : f0; 1g n 7 f0; 1g g so that Prob h2H ( n i=1 h(ff i ) fi i ) 2 Gamman , for every n distinct ff 1 ; ff n 2f0; 1g n ....

N. Alon, L. Babai, and A. Itai, "A fast and Simple Randomized Algorithm for the Maximal Independent Set Problem", J. of Algorithms, Vol. 7, 1986, pp. 567--583.

....a random list satisfying Items (1) and (2) above is to use a sequence of pairwise independent random variables. Actually, our analysis becomes even easier if we use a sequence of 3 wise independent random variables. It will be most convenient to use the construction given in Alon et al. [ABI] which works in the field GF(2 m ) since this field corresponds naturally to the set of m bit strings. The construction uses t 2 m arbitrary elements of the field, denoted ff 1 ; ff 2 ; ff t . A sequence of t elements is represented by a triplet of field elements, denoted (u; v; w) ....

Alon, N., L., Babai and A. Itai, "A Fast and Simple Randomized Algorithm for the Maximal Indepedent Set Problem", J. of Algorithms, Vol. 7, 1986, pp. 567--583.

....that that of [13] is that here edges are selected k wise independently rather than totally independently. This is, of course, the key to the derandomization. When is a constant (that is, independent of H) k is in each call to OneNibble also a constant. The results of e.g. Alon, Babai, and Itai [1] then imply that only logarithmically many totally independent bits are neccessary to generate the polynomially many k wise independent bits required for the edge selections. Thus, with polynomially many processors, all of the possible choices of the truly independent bits can be tried out in ....

N. Alon, L. Babai, and A. Itai, A fast and simple randomized algorithm for the maximal indepedent set problem, Journal of Algorithms 7 (1986), 567--583.

....that some objects are distributed in a k wise independent manner then one can replace the algorithm s random tape by a string selected from a k wise independent distribution. Recalling that k wise independent distributions over f0; 1g can be generated using only O(k log n) bits (see, e.g. [1]) this yields a significant saving in the randomness complexity as well as to derandomization in time n O(k) This number of random bits is essentially optimal; see [3] 1] Further saving is possible whenever the analysis of the randomized algorithm can be carried out also in case its ....

.... Recalling that k wise independent distributions over f0; 1g can be generated using only O(k log n) bits (see, e.g. 1] this yields a significant saving in the randomness complexity as well as to derandomization in time n O(k) This number of random bits is essentially optimal; see [3] [1]. Further saving is possible whenever the analysis of the randomized algorithm can be carried out also in case its random tape is only almost k wise independent (i.e. every k bits are distributed almost uniformly) The reason being that the latter distributions can be generated using fewer ....

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N. Alon, L. Babai and A. Itai. A fast and Simple Randomized Algorithm for the Maximal Independent Set Problem. J. of Algorithms, Vol. 7, pages 567--583, 1986.

....properties is of major theoretical and practical importance. A typical property is that the probability distribution, induced on every k bit locations in a string randomly selected in the sample space, should be uniform. Such a sample space is called k wise independent. Alon, Babai and Itai [5] presented an efficient construction of k wise independent sample spaces of size approximately n k=2 , where n is (as above) the length of the strings in the sample space. This result is very close to best possible, in view of the lower bound of Chor. et al 11] Hence, k wise independent ....

....sample space. A sample space is called linear if its elements are obtained by a linear transformation of their succinct representation (equivalently, the sample space is a linear subspace) For example, the construction of a k wise independent sample space presented by Alon, Babai and Itai [5] is linear. To be more precise they construct a sample space on n bits, where n = 2 t Gamma1, which is generated by td 1 bits and is (2d 1) wise independent. Naor and Naor observed that a sample space which is almost unbiased with respect to linear Boolean tests can be used to sample succinct ....

[Article contains additional citation context not shown here]

N. Alon, L. Babai, and A. Itai, "A fast and Simple Randomized Algorithm for the Maximal Independent Set Problem", J. of Algorithms, Vol. 7, 1986, pp. 567--583.

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N. Alon, L. Babai and A. Itai. A fast and Simple Randomized Algorithm for the Maximal Independent Set Problem. J. of Algorithms, Vol. 7, pages 567--583, 1986.

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N. Alon, L. Babai and A. Itai. A fast and Simple Randomized Algorithm for the Maximal Independent Set Problem. J. of Algorithms, Vol. 7, pages 567--583, 1986.

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N. Alon, L. Babai and A. Itai. A fast and Simple Randomized Algorithm for the Maximal Independent Set Problem. J. of Algorithms, Vol. 7, pages 567-583, 1986.

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N.Alon, L Babai, A. Itai; A fast and simple randomized algorithm for the maximal independent set problem. J. Algo., 7 (1987), 567 - 583.

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N.Alon, L Babai, A. Itai; A fast and simple randomized algorithm for the maximal independent set problem. J. Algo., 7 (1987), 567 - 583.

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