The Monte Carlo Algorithm With a Pseudorandom Generator

We analyze the Monte Carlo algorithm for the approximation of multivariate integrals when a pseudorandom generator is used. We establish lower and upper bounds on the error of such algorithms. We prove that as long as a pseudorandom generator is capable of producing only finitely many points, the Monte Carlo algorithm with such a pseudorandom generator fails for ?2 or continuous functions. It also fails for Lipschitz functions if the number of points does not depend on the number of variables. This is the case if a linear congruential generator is used with one initial seed. On the other hand, if a linear congruential generator of period m is used for each component, with independent uniformly distributed initial seeds, then the Monte Carlo algorithm with such a pseudorandom generator using n function values behaves as for the uniform distribution and its expected error is roughly n-1/2 as long as the number n of function values is less than m2 . X. Introduction Randomization is being widely used or proposed to solve both continuous and discrete problems. Examples include multivariate integration, algebraic eigenvalues, primality testing, byzantine agreement, and verification. When randomized algorithms are implemented on a computer, pseudorandom numbers must be used. In this paper we investigate whether the good properties of the Monte Carlo algorithm for multivariate integration hold if pseudorandom numbers are used. We suggest that such an analysis should be performed whenever randomization is used. One can identify two types of work regarding pseudorandom generators. In the first, they are studied in isolation from a particular problem. Excellent surveys are given in Knuth [11, Chapter 3] and Niederreiter [14, 15, 16]. More recently, polynomial-time unpredictability of pseudorandom generators has been studied by Yao [21], Blum and Micali [3], and others. In the second, the relation between pseudorandom generators and randomized algorithms is studied for a specific problem. Examples are provided by Bach [1], who studied finding square roots modulo a prime number and primality testing. Karloff and Raghavan [10] studied sorting, selection, and oblivious routing in networks. They showed that certain randomized algorithms work well with a linear congruential generator and a random seed. Received September 4, 1990. 1991 Mathematics Subject Classification. Primary 65C05, 65C10. ©1992 American Mathematical Society 0025-5718/92 $1.00+ $.25 per page