S. Chari, P. Rohatgi and A. Srinivasan. Improved algorithms via approximations of probability distributions (Extended Abstract). In Proceedings of the Twenty-Sixth Annual ACM Symposium on the Theory of Computing, ACM Press, New York, NY, 1994: 584-- 592.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....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 reduction in randomness is ....

Chari, Rohatgi, and Srinivasan. Improved algorithms via approximations of probability distributions. J. Comput. Syst. Sci., 61, 2000.

....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 multivariate polynomial is ....

Chari, Rohatgi, and Srinivasan. Improved algorithms via approximations of probability distributions. J. of Comput. and Syst. Sci., 61, 2000.

....class PP. A similar statement would be also useful in our context, since current upper bounds for formulas in 3 CNF with a unique satisfying assignment [PPSZ98] see section 3) are better than the bounds for arbitrary formulas [Sch99] see section 5) There are many di erent proofs of the lemma [BF97, Cha94, CRS93, MVV87]. Also, there are several proofs of a close result: the existence of a reduction to formulas with odd (or zero) number of satisfying assignments [Gup93, NRS95] In this section we give two new proofs. Both our proofs are based on the idea used in [BF97] In [BF97] this idea is combined with the ....

S. Chari, P. Rohatgi, A. Srinivasan, Improved algorithms via approximations of probability distributions, Journal of Computer and System Sciences 61 (2000), 81-107.

....of size poly(log n; 2 k ; 1=ffl) while the other two constructions yield (ffl; k) independent spaces S f1; 2; mg n of size O i (n=ffl) log (1=ffl) j and O i (n=ffl) log n j , respectively. The results of [EGLNV] were extended and applied by Chari, Rohatgi and Srinivasan [CRS]. 2 Again using the notion of discrepancy and projections, they constructed an (ffl; k) independent sample space S of size poly(log n; 1=ffl; minf2 k ; k log(1=ffl) g) The considerations in this paper are motivated by the problem from Azar, Motwani and Naor [AMN] In contrast to the work ....

.... the notion of discrepancy and projections, they constructed an (ffl; k) independent sample space S of size poly(log n; 1=ffl; minf2 k ; k log(1=ffl) g) The considerations in this paper are motivated by the problem from Azar, Motwani and Naor [AMN] In contrast to the work of [EGLNV] and [CRS] where the discrepancy of axis aligned rectangles is used, we offer a different approach for investigating approximations of probability spaces by using basic Linear Algebra. The intention behind our considerations is to give more insight towards the understanding of the underlying concepts for ....

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S. Chari, P. Rohatgi and A. Srinivasan, Improved Algorithms via Approximations of Probability Distributions, Proc. 26th Annual Symposium on the Theory of Computing STOC'96, 1996, 584-592.

....) O(log n) such that at least one sample from this space will lead to a cut in which at least m(1 #) 2 edges cross the cut. Exhaustive search (e.g. using (n m) log n processors and O(log n) time in total) of such a space leads to an e#cient deterministic NC algorithm for our problem. See [21, 11] for extensions to the more general problem of finding heavy codewords in linear codes, and for more e#cient algorithms for smaller values of #. The above application shows how small bias spaces, combined with the fact that randomized algorithms are typically robust to small changes in the ....

....changes in the underlying probabilities, leads to fruitful derandomization schemes. Furthermore, the work of [4] presents a nice approximate method of conditional probabilities, which works well with small bias sample spaces to lead to e#cient derandomization approaches. The reader is referred to [4, 6, 11] for a study of this useful approach. Finally, explicit constructions of small bias spaces have had an impact on constructing various combinatorial objects; one of these that has seen many applications (e.g. in computational learning theory and hardness of approximation results) is the ....

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S. Chari, P. Rohatgi, and A. Srinivasan. Improved algorithms via approximations of probability distributions. To appear in Journal of Computer and System Sciences.

....be made into a deterministic sequential algorithm that computes such a sample R [27] it uses the so called method of conditional probabilities. In parallel, several approaches have been used (k wise independence combined with the method of conditional probabilities and relaxed to biased spaces [6, 23, 25, 7]) However, so far these efforts to compute a sample in parallel have resulted only in O( p jSj 1 ffl log m) discrepancies. The situation is similar for other discrepancy like problems, like the lattice approximation problem [27, 23] and some sampling problems in computational geometry [12, ....

....C) Theorem 4. 1 The DP can be solved (deterministically) in the EREW PRAM model using time O(log 2 n) and work O(n 8 m 6 min(m 4 ; n 13 ) Though the work bound is still quite high, it is not much worst than the previous best bound (not achieving the optimal discrepancy) in [7]. There, a set of discrepancy C p jSj 1 ffl log n is computed in time O(log n) and work O(n 1 4=ffl ) with m = n) So, for ffl = 1=4, the work is already O(n 17 ) 4.3 Lattice approximation via discrepancy The algorithm for the LAP in [23] is obtained by a reduction to the DP. The ....

S. Chari, P. Rohatgi and A. Srinivasan. Improved Algorithms via Approximations of Probability Distributions. In Proc. ACM Sympos. Theory Comput. (1994) 584--592.

.... most t log 1=ffl nontrivial dimensions and each dimension being an interval, we also give a pseudorandom generator using O(log log d log 1=ffl log 1=2 t log 1=ffl ) bits, which again improves the previous upper bound O(log log d log 1=ffl log t log 1=ffl ) by Chari, Rohatgi, and Srinivasan [4]. 1 Introduction Pseudorandom generators for combinatorial rectangles have been actively studied recently, because they are closely related to some fundamental problems in theoretical computer science, such as derandomizing RL, DNF approximate counting, and approximating the distributions of ....

....an interval. Even et al. 5] observed that the problem of approximating the distribution of independent multivalued random variables can be reduced to this case. They gave a generator using O(k log d log 1=ffl) bits. This is good when k = O(log 1=ffl) For the case k log 1=ffl, Chari et al. [4] gave a generator using O(log log d log 1=ffl log k log 1=ffl ) bits. Here, we improve this again to O(log log d log 1=ffl log 1=2 k log 1=ffl ) We will not emphasize the efficiency of our generators, but one can easily check that all the generators can be computed in simultaneous ....

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S. Chari, P. Rohatgi, and A. Srinivasan, Improved algorithms via approximations of probability distributions, In Proceedings of the 26th Annual ACM Symposium on Theory of Computing, pages 584-592, 1994.

....[36] using the method of conditional probabilities this was derandomized to obtain a deterministic sequential algorithm computing such a sample R. In parallel, several approaches have been used (k wise independence combined with the method of conditional probabilities and relaxed to biased spaces [11, 32, 34, 12]) However, so far these efforts to compute a sample in parallel have resulted only in discrepancies O( p jSj 1 ffl log jSj) Results. In this paper, we describe NC algorithms (specifically, the algorithms run in O(log 2 n) time using O(n C ) processors for some constant C in the EREW ....

....in the EREW PRAM model in O(log n log(n m) O(log 2 n) time using O(n 7 m 10 log 6 n) processors. This is still much greater than the work O(nm) that is needed to compute a good discrepancy set sequentially. For m = n, the previous best parallel algorithm of Chari et al., [12] computes a set of discrepancy O( p jSj 1 ffl log n) in O(log n) time using O(n 1 4=ffl ) processors (for ffl = 1=4, that is O(n 17 ) processors, the same bound as ours) 4.3 Lattice Approximation Via Discrepancy The algorithm for the lattice approximation problem by Motwani, Naor and ....

S. Chari, P. Rohatgi and A. Srinivasan. Improved Algorithms via Approximations of Probability Distributions. In Proc. ACM Sympos. Theory Comput. (1994), 584--592. Revised version as DIMACS Technical Report 97-01.

.... Z n p ) within an additive error of O( n log k p p ) In fact, by replacing F by a smooth function that approximates it this error term can be improved to O( n log k (p=n) p ) Since for this example this is a weaker estimate than those obtained by the constructions in [EGL 92] and [CRS94] we omit the details. 4 Conclusion and Open questions We showed that the set A n p is an n Gamma1 p Gamma1 discrepancy set, and jA n p j = O(p 2 ) The modified construction described in Subsection 3.2 provides an ffl discrepancy set of size polynomial in log p= and linear in n. For the ....

Suresh Chari, Pankaj Rohatgi, and Aravind Srinivasan. Improved algorithms via approximations of probability distributions. In Proceedings of the 26 th Annual ACM Symposium on Theory of Computing, pages 584-592, May 1994.

.... than a factor of O(log n) However, by using results from the newly developing theory of approximating probability distri16 butions (Naor Naor [29] Azar, Motwani Naor [5] Alon, Goldreich, Hastad Peralta [2] Even, Goldreich, Luby, Nisan Velickovi c [14] and Chari, Rohatgi Srinivasan [10]) we get optimal results in the case where the X i s are binary with Pr(X i = 1) 1=2. A sample space X for n bit vectors was defined to be k wise ffl biased by Naor Naor [29] see also Vazirani [49] if 8S f1; 2; ng; 1 jSj k; jPr( M i2S x i = 1) Gamma Pr( M i2S x i = ....

S. Chari, P. Rohatgi, and A. Srinivasan. Improved algorithms via approximations of probability distributions. In Proc. ACM Symposium on Theory of Computing, 1994. To appear.

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S. Chari, P. Rohatgi and A. Srinivasan. Improved algorithms via approximations of probability distributions (Extended Abstract). In Proceedings of the Twenty-Sixth Annual ACM Symposium on the Theory of Computing, ACM Press, New York, NY, 1994: 584-- 592.

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S. Chari, P. Rohatgi and A. Srinivasan. Improved algorithms via approximations of probability distributions. In Proc. of 26th ACM Symposium on Theory of Computing, pp. 584-592, 1994.

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