Zero-error Probabilistic Polynomial time

In complexity theory, ZPP (zero-error probabilistic polynomial time) is the complexity class of problems for which a probabilistic Turing machine exists with these properties:

In other words, if the algorithm is allowed to flip a truly-random coin while it is running, it will always return the correct answer and, for a problem of size n, there is some polynomial p(n) such that the average running time will be less than p(n), even though it might occasionally be much longer. Such an algorithm is called a Las Vegas algorithm.

Alternatively, ZPP can be defined as the class of problems for which a probabilistic Turing machine exists with these properties:

The two definitions are equivalent.

The definition of ZPP is based on probabilistic Turing machines, but, for clarity, note that other complexity classes based on them include BPP and RP. The class BQP is based on another machine with randomness: the quantum computer.

The class ZPP is exactly equal to the intersection of the classes RP and co-RP. This is often taken to be the definition of ZPP. To show this, first note that every problem which is in both RP and co-RP has a Las Vegas algorithm as follows:

Note that only one machine can ever give a wrong answer, and the chance of that machine giving the wrong answer during each repetition is at most 50%. This means that the chance of reaching the kth round shrinks exponentially in k, showing that the expected running time is polynomial. This shows that RP intersect co-RP is contained in ZPP.

To show that ZPP is contained in RP intersect co-RP, suppose we have a Las Vegas algorithm C to solve a problem. We can then construct the following RP algorithm:

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