Monte Carlo algorithm

In computing, a Monte Carlo algorithm is a randomized algorithm whose output may be incorrect with a certain (typically small) probability.

The name refers to the grand casino in the Principality of Monaco at Monte Carlo, which is well-known around the world as an icon of gambling.

The related class of Las Vegas algorithms are also randomized, but in a different way: they take an amount of time that varies randomly, but always produce the correct answer. A Monte Carlo algorithm can be converted into a Las Vegas algorithm whenever there exists a procedure to verify that the output produced by the algorithm is indeed correct. If so, then the resulting Las Vegas algorithm is merely to repeatedly run the Monte Carlo algorithm until one of the runs produces an output that can be verified to be correct.

Whereas the answer returned by a deterministic algorithm is always expected to be correct, this is not the case for Monte Carlo algorithms. For decision problems, these algorithms are generally classified as either false-biased or true-biased. A false-biased Monte Carlo algorithm is always correct when it returns false; a true-biased algorithm is always correct when it returns true. While this describes algorithms with one-sided errors, others might have no bias; these are said to have two-sided errors. The answer they provide (either true or false) will be incorrect, or correct, with some bounded probability.

For instance, the Solovay–Strassen primality test is used to determine whether a given number is a prime number. It always answers true for prime number inputs; for composite inputs, it answers false with probability at least ½ and true with probability less than ½. Thus, false answers from the algorithm are certain to be correct, whereas the true answers remain uncertain; this is said to be a ½-correct false-biased algorithm.

For a Monte Carlo algorithm with one-sided errors, the failure probability can be reduced (and the success probability amplified) by running the algorithm k times. Consider again the Solovay–Strassen algorithm which is ½-correct false-biased. One may run this algorithm multiple times returning a false answer if it reaches a false response within k iterations, and otherwise returning true. Thus, if the number is prime then the answer is always correct, and if the number is composite then the answer is correct with probability at least 1−(1−½)^k = 1−2^−k.

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