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The algorithms discussed so far in this book are all deterministic. That is, each algorithm always proceeds in the same way whenever given the same input. However, there are many situations where we would like an algorithm to make a random choice at one or more steps. Such a situation arises when a deterministic algorithm would have to go through a huge number, or even an unknown number, of possible cases. Algorithms that make random choices at one or more steps are called probabilistic algorithms. We will discuss a particular class of probabilistic algorithms in this section, namely, Monte Carlo algorithms, for decision problems. Monte Carlo algorithms always produce answers to problems, but a small probability remains that these answers may be incorrect. However, the probability that the answer is incorrect decreases rapidly when the algorithm carries out sufficient computation. Decision problems have either “true” or “false” as their answer. The designation “Monte Carlo” is a reference to the famous casino in Monaco; the use of randomness and the repetitive processes in these algorithms make them similar to some gambling games. This name was introduced by the inventors of Monte Carlo methods, including Stan Ulam, Enrico Fermi, and John von Neumann.

A Monte Carlo algorithm for a decision problem uses a sequence of tests. The probability that the algorithm answers the decision problem correctly increases as more tests are carried out. At each step of the algorithm, possible responses are “true,” which means that the answer is “true” and no additional iterations are needed, or “unknown,” which means that the answer could be either “true” or “false.” After running all the iterations in such an algorithm, the final answer produced is “true” if at least one iteration yields the answer “true,” and the answer is “false” if every iteration yields the answer “unknown.” If the correct answer is “false,” then the algorithm answers “false,” because every iteration will yield “unknown.” However, if the correct answer is “true,” then the algorithm could answer either “true” or “false,” because it may be possible that each iteration produced the response “unknown” even though the correct response was “true.” We will show that this possibility becomes extremely unlikely as the number of tests increases.

The algorithms discussed so far in this book are all deterministic. That is, each algorithm always proceeds in the same way whenever given the same input. However, there are many situations where we would like an algorithm to make a random choice at one or more steps. Such a situation arises when a deterministic algorithm would have to go through a huge number, or even an unknown number, of possible cases. Algorithms that make random choices at one or more steps are called probabilistic algorithms. We will discuss a particular class of probabilistic algorithms in this section, namely, Monte Carlo algorithms, for decision problems. Monte Carlo algorithms always produce answers to problems, but a small probability remains that these answers may be incorrect. However, the probability that the answer is incorrect decreases rapidly when the algorithm carries out sufficient computation. Decision problems have either “true” or “false” as their answer. The designation “Monte Carlo” is a reference to the famous casino in Monaco; the use of randomness and the repetitive processes in these algorithms make them similar to some gambling games. This name was introduced by the inventors of Monte Carlo methods, including Stan Ulam, Enrico Fermi, and John von Neumann.

A Monte Carlo algorithm for a decision problem uses a sequence of tests. The probability that the algorithm answers the decision problem correctly increases as more tests are carried out. At each step of the algorithm, possible responses are “true,” which means that the answer is “true” and no additional iterations are needed, or “unknown,” which means that the answer could be either “true” or “false.” After running all the iterations in such an algorithm, the final answer produced is “true” if at least one iteration yields the answer “true,” and the answer is “false” if every iteration yields the answer “unknown.” If the correct answer is “false,” then the algorithm answers “false,” because every iteration will yield “unknown.” However, if the correct answer is “true,” then the algorithm could answer either “true” or “false,” because it may be possible that each iteration produced the response “unknown” even though the correct response was “true.” We will show that this possibility becomes extremely unlikely as the number of tests increases.

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