A typical approach to estimate an unknown quantity is to design an experiment that produces a random variable Z distributed in [0; 1] with E[Z] = , run this experiment independently a number of times and use the average of the outcomes as the estimate. In this paper, we consider the case when no a priori information about Z is known except that is distributed in [0; 1]. We describe an approximation algorithm AA which, given ffl and ffi, when running independent experiments with respect to any Z, produces an estimate that is within a factor 1 + ffl of with probability at least 1 \Gamma ffi. We prove that the expected number of experiments run by AA (which depends on Z) is optimal to within a constant factor for every Z. An announcement of these results appears in P. Dagum, D. Karp, M. Luby, S. Ross, "An optimal algorithm for Monte-Carlo Estimation (extended abstract)", Proceedings of the Thirtysixth IEEE Symposium on Foundations of Computer Science, 1995, pp. 142-149 [3]. Section ...

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Given d, m and epsilon, we deterministically produce a sequence of points S that hits every combinatorial rectangle in [m]^d of volume at least epsilon. Both the running time of the algorithm and |S| are polynomial in m log(d)/epsilon. This algorithm has applications to deterministic constructions of small sample spaces for general multivalued random variables. 1 Introduction The problem we consider in this paper is motivated by the following basic witness finding problem: design an efficient algorithm that on Hebrew University, Computer Science Department, Jerusalem Israel. Research partially done while visiting the International Computer Science Institute. y International Computer Science Institute and UC Berkeley. Research partially supported by NSF Grant CCR-9016468 and grant No. 89-00312 from the United States-Israel Binational Science Foundation (BSF), Jerusalem, Israel. z Department of Mathematics, Rutgers University and Department of CSE, UCSD. Supported in part by NSF under grant C...

) Abstract A subset H ` f0; 1g n is a Hitting Set for a class R of boolean functions with n inputs if, for any function f 2 R such that Pr (f = 1) ffi (where ffi 2 (0; 1) is some fixed value), there exists an element ~ h 2 H such that f( ~ h) = 1. The efficient construction of Hitting Sets for non trivial classes of boolean functions is a fundamental problem in the theory of derandomization. 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 a fini...

We describe a deterministic algorithm which, on input integers d, m and real number eC (0, 1), produces a subset S of [m] d- = {1,2,3....,m} d that hits every combinatorial rectangle in [m]d of volume at least e, i.e., every subset of [m]d the form R 1 • R2 •... x R d of size at least em d. The cardinality of S is polynomial in rn(logd)/e, and the time to constrnct it is polynomial in md/~. The construction of such sets has applications in derandomization methods based on small sample spaces for general multivalued random variables. 1.