Derandomization converts a randomized construction or algorithm into a deterministic one while preserving its desired guarantee. The method of conditional probabilities does this by maintaining a computable conditional expectation until all choices are fixed.
For a random objective with an easily computable conditional expectation, reveal one random choice at a time and choose an outcome whose conditional expected objective is at least the current value. Such an outcome exists because the current value is an average of the child values. When all choices are fixed, the actual objective is at least the initial expectation. Biased as well as fair choices work. Efficient expectation updates make the resulting deterministic procedure a polynomial-time algorithm.
A probabilistic Turing machine is a Turing machine whose transitions may depend on independent random bits.
A randomized decision algorithm has one-sided error when one answer is always correct and only the other answer can be mistaken.
contains decision problems with a polynomial-time randomized algorithm that rejects every negative instance and accepts every positive instance with probability at least one half.
consists of complements of languages in RP. Its algorithms always accept positive instances and reject negative instances with probability at least one half.
. Equivalently, it consists of problems having an always-correct randomized algorithm with polynomial expected running time.
Independent repetitions reduce a randomized algorithm's error probability while preserving polynomial running time.
Polynomial identity testing asks whether a polynomial represented implicitly, such as by an arithmetic circuit or a remainder computation, is the zero polynomial.
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