Nathan Linial, Michael Luby, Michael saks, David Zuckerman. Efficient construction of a small hitting set for combinatorial rectangles in high dimension. Proceedings of the 25th Annual symposium on Theory of computation.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....The Fibonacci lattice, depicted in Figure 1.a, is the 2 dimensional lattice with base f(1; OE) GammaOE; 1)g, where OE = Gamma1 p 5) 2 is the golden ratio. This lattice has some amazing properties and we will employ some of them here. It plays an important role in the discrepancy theory [8, 11, 4] and in the theory of Diophantine approximation [8] Our workloads must be finite, so they contain some small part of the Fibonacci lattice. In particular, we will consider workloads whose points are the Fibonacci lattice points of some appropriate rectangles and have sizes n = fk , where fk is ....

....redundancy Theta(log d Gamma1 n) is required to achieve access overhead less than B. Unfortunately, there is no simple generalization of the Fibonacci lattice in higher dimensions. Furthermore, no simple point set is known to have good discrepancy properties in higher dimensions (see however [11]) We also leave it as a major open problem to determine the tradeoff of random point sets. We have made some progress on this problem, but our results are too preliminary to be included in this abstract. Finally, there is some room of improvement on the upper bound of Theorem 2 to show that ....

N. Linial, M. Luby, M. Saks, and D. Zuckerman. Efficient Construction of a Small Hitting Set for Combinatorial Rectangles in High Dimension. Proceedings of the Twenty-Fifth Annual ACM Symposium on Theory of Computing (STOC), pp:258--267, San Diego, California, May 1993.

....usual geometric (axis aligned) rectangles. The PAC learning of rectangles has been studied because they have been found experimentally to yield excellent hypotheses for a variety of applied learning problems (see [38, 12] Also, pseudorandom sets for rectangles have been actively studied recently [29, 4, 14, 25, 11, 20, 6] because (i) they are a subproblem common to the derandomization of depth 2 (DNF) circuits and derandomizing Randomized Logspace (RL) and (ii) they approximate the distribution of n independent multivalued random variables. In this work, we present improved (and in some cases optimal) upper ....

N. Linial, M. Luby, M. Saks, and D. Zuckerman. Efficient Construction of a Small Hitting Set for Combinatorial Rectangles in High Dimension. In Proc. ACM Symposium on the Theory of Computing, pages 258--267, 1993.

.... time, randomized algorithms (RP) and recently were shown to derandomize the class of two side bounded error, polynomial time, randomized algorithms (BPP) ACR96] A deterministic construction of a hitting set for combinatorial rectangles of dimension d and volume at least ffl is presented in [LLSZ93]. For positive integers d and m, a combinatorial rectangle R within S = m] d is defined as R = R 1 Theta Delta Delta Delta Theta R d , where, for all i 2 [d] R i [m] The volume of R is Q i2[d] jR i j=m d . In [SSZ95] explicitly constructed dispersers were used for a polynomial ....

N. Linial, M. Luby, M. Sacks, and D. Zuckerman. Efficient construction of a small hitting set for combinatorial rectangles in high dimension. Proc. of the 25th ACM Symposium on Theory of Computing, pp. 258--267, 1993.

....the efficiency of our generators, but one can easily check that all the generators can be computed in simultaneous (md=ffl) O(1) time and O(log m log d log 1=ffl) space. It s worth mentioning that the hitting version of our problem has already been settled. Linial, Luby, Saks, and Zuckerman [8] gave an explicit generator using O(log m log log d log 1=ffl) bits that can hit any rectangle in R(m; d) of volume at least ffl. In fact the work of Armoni at al. 3] followed this result closely, and so does ours. 2 Preliminaries 2.1 Notations For a set V , we let 2 V denote the family of ....

N. Linial, M. Luby, M. Saks, and D. Zuckerman, Efficient construction of a small hitting set for combinatorial rectangles in high dimension, Combinatorica, 17, pages 215-234, 1997.

....Our paper presents a new method to efficiently construct Hitting Sets for the class of systems of boolean linear functions. Systems of boolean linear functions can be also considered as the algebraic generalization of boolean combinatorial rectangular functions studied by Linial et al. in [11]. In the restricted case of boolean rectangular functions, our method (even though completely different) achieves equivalent results to those obtained in [11] Our method gives also an interesting upper bound on the circuit complexity of the solutions of any system of linear equations defined over ....

.... functions can be also considered as the algebraic generalization of boolean combinatorial rectangular functions studied by Linial et al. in [11] In the restricted case of boolean rectangular functions, our method (even though completely different) achieves equivalent results to those obtained in [11]. Our method gives also an interesting upper bound on the circuit complexity of the solutions of any system of linear equations defined over a finite field. Furthermore, as preliminary result, we show a new upper bound on the circuit complexity of integer monotone functions that generalizes the ....

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Linial N., Luby M., Saks M., and Zuckerman D. (1993), "Efficient construction of a small hitting set for combinatorial rectangles in high dimension", in Proc. 25th ACM STOC, 258-267.

....usual geometric (axis aligned) rectangles. The PAC learning of rectangles has been studied because they have been found experimentally to yield excellent hypotheses for a variety of applied learning problems (see [36, 11] Also, pseudorandom sets for rectangles have been actively studied recently [28, 4, 13, 24, 10, 19, 6] because (i) they are a subproblem common to the derandomization of depth 2 (DNF) circuits and derandomizing Randomized Logspace (RL) and (ii) they approximate the distribution of n independent multivalued random variables. In this work, we present improved (and in some cases optimal) upper ....

N. Linial, M. Luby, M. Saks, and D. Zuckerman. Efficient Construction of a Small Hitting Set for Combinatorial Rectangles in High Dimension. In Proc. ACM Symposium on the Theory of Computing, pages 258-- 267, 1993.

....O(rnc 2:62 ) where r = maxfr 1 ; r n g. The cardinality of the resulting sample space is still jCj. The algorithm can also deal with more general constraints with no change. In particular, it can deal with combinatorial rectangles , as described by Even et. al [9] and by Linial et. al [12]. A combinatorial rectangle is an independence constraint over an event of the form fX i 1 2 R 1 ; X i k 2 R k g ; where R j is a subset of f0; r Gamma 1g (or of f0; r i j Gamma 1g in the more general case) The proof remains essentially unchanged, except for minor ....

N. Linial, M. Luby, M. Saks, and D. Zuckerman. Efficient construction of a small hitting set for combinatorial rectangles in high dimension. In Proceedings of the 25th Annual ACM Symposium on Theory of Computing, to appear, 1993.

....: Note that this size is poly(n) if d 2 O(log 2=3 n) and ffl 2 GammaO(log 2=3 n) Also, when d = n, our bound is better than that of [7] if ffl 2 Gammac 0 p log n , where c 0 0 is a certain absolute constant. We remark that Lu s construction builds on earlier work of [2, 5, 6, 10]. Given log L random bits to index a random element of the permutation family guaranteed by Corollary 1.1, and given any i 2 [n] we can deterministically construct (i) in time polylogarithmic in L. 2 Proof of Main Theorem Fix an arbitrary set K [n] of any size k d, and choose any x 2 K. We ....

N. Linial, M. Luby, M. Saks, and D. Zuckerman. Efficient construction of a small hitting set for combinatorial rectangles in high dimension. Combinatorica (17), 1997, pp. 215--234. 5

....least ffi. When the universe is understood, such a set will be referred to as an ffi hitting set. An easy probabilistic proof shows that there exist (a; d; ffi) hitting sets of size dln(2)ad=ffie. A constructive solution is offered by Linial, et al. who prove the following theorem: Theorem 4 ([LLSZ97]) There exists an (a; d; ffi) hitting set of cardinality poly(log(d)a=ffi) constructible in time poly(ad=ffi) 4 2.3 A Lemma about Poly logarithmic Decay In order to avoid logarithms of negative numbers, we define iterated logarithms as follows. For n 1 and k 2 N, log (k) n = 1 if log ....

Nathan Linial, Michael Luby, Michael Saks, and David Zuckerman. Efficient construction of a small hitting set for combinatorial rectangles in high dimension. Combinatorica, 17(2):215--234, 1997.

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N. Linial, M. Luby, M. Saks and D. Zuckerman. Efficient construction of a small hitting set for combinatorial rectangles in high dimension. In Proc. of 25th ACM Symposium on Theory of Computing, pp. 258-267, 1993.

....a 2 Gamma Omega Gamma k) approximation to P n;m . A proof can be found in either [9, 10, 5] 1.6 Subsequent Work In the five years which have elapsed since the conference presentation of this work (cf. 9] a few related works were done. We briefly describe the related results in [14, 5, 3]. The work of Chari et al. 5, Sec. 3] is most relevant to the current write up. It presents constructions which match or improve over the sizes of all constructions presented in [9] However, for some natural setting of the parameters, the size of the main construction of the current write up ....

....our construction has size polynomial in log(n) 1=ffl, and 2 k ) Thus, 4 [5, Sec. 3] yields no improvement when ffl 2 Gammak , which is the typical case when one requires a bound on the approximation in the Norm 1 measure (rather than in Max Norm as defined above) 2 Linial et al. [14] consider a one sided version of the discrepancy problem. That is, rather than constructing sets which approximate the volume of all rectangles they construct sets which hit all sufficiently big rectangles. Their construction is polynomial in all relevant parameters (including the bound on the ....

Linial, N., Luby, M., M. Saks, and D. Zuckerman, "Efficient Construction of a Small Hitting Set for Combinatorial Rectangles in High Dimension", Combinatorica (to appear). Preliminary version in 25th STOC, 1993. 12

....a 2 Gamma Omega Gamma k) approximation to P n;m . A proof can be found in either [9, 10, 5] 1.6 Subsequent Work In the five years which have elapsed since the conference presentation of this work (cf. 9] a few related works were done. We briefly describe the related results in [16, 5]. The work of Chari et al. 5, Sec. 3] is most relevant to the current write up. It presents constructions which match or improve over the sizes of all constructions presented in [9] However, for some natural setting of the parameters, the size of the main construction of the current write up ....

....(whereas our construction has size polynomial in log(n) 1=ffl, and 2 k ) Thus, 5, Sec. 3] yields no improvement when ffl 2 Gammak , which is the typical case when one requires a bound on the approximation in the Norm 1 measure (rather than in Max Norm as defined above) 3 Linial et al. [16] consider a one sided version of the discrepancy problem. That is, rather than constructing sets which approximate the volume of all rectangles they construct sets which hit all sufficiently big rectangles. Their construction is polynomial in all relevant parameters (including the bound on the ....

Linial, N., Luby, M., M. Saks, and D. Zuckerman, "Efficient Construction of a Small Hitting Set for Combinatorial Rectangles in High Dimension", Combinatorica (to appear). Preliminary version in 25th STOC, 1993.

....at least e. When the universe is understood, such a set will be referred to as an e hitting set. An easy probabilistic proof shows that there exist (k;d;e) hitting sets of size dln(2)kd=ee. A constructive solution is offered by Linial, et al. who prove the following theorem: Theorem 4 ([LLSZ97]) There exists a (k;d;e) hitting set of cardinality poly(ln(d)k=e) constructible in time poly(kd=e) 2.3. A Lemma about Poly Logarithmic Decay In order to avoid logarithms of negative numbers, we define iterated logarithms as follows. For n 1 and k 2 N, ln (k) n = 1 if ln (k Gamma1) n ....

Nathan Linial, Michael Luby, Michael Saks, and David Zuckerman. Efficient construction of a small hitting set for combinatorial rectangles in high dimension. Combinatorica, 17(2):215-- 234, 1997.

.... using poly(S) random bits can be simulated deterministically in SPACE(S) NZ] and to constructing expander graphs that beat the eigenvalue bound [WZ] The lemma about expander graphs used in the RP construction has been used to explicitly construct a hitting set for high dimensional rectangles [LLSZ]. Our results have recently been extended by Srinivasan and the author [SZ] to the case of subconstant ffi . In particular, they considered ffi sources outputting R bits such that any string has probability at most 2 GammaR ffl . For any fixed ffl 0, they gave an n O(log n) simulation of ....

N. Linial, M. Luby, M. Saks, and D. Zuckerman, "Efficient Construction of a Small Hitting Set for Combinatorial Rectangles in High Dimension," 25th ACM Symposium on Theory of Computing, 1993, pp. 258-267.

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Nathan Linial, Michael Luby, Michael saks, David Zuckerman. Efficient construction of a small hitting set for combinatorial rectangles in high dimension. Proceedings of the 25th Annual symposium on Theory of computation.

No context found.

Linial, N., Luby, M., Saks, M., and Zuckerman, D. (1993), Efficient construction of a small hitting set for combinatorial rectangles in high dimension, in "Proceedings of 25th ACM Symposium on Theory of Computation," pp.258-267.

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