BPP (complexity) - Introduction

Introduction

BPP is one of the largest practical classes of problems, meaning most problems of interest in BPP have efficient probabilistic algorithms that can be run quickly on real modern machines. For this reason it is of great practical interest which problems and classes of problems are inside BPP. BPP also contains P, the class of problems solvable in polynomial time with a deterministic machine, since a deterministic machine is a special case of a probabilistic machine.

In practice, an error probability of 1/3 might not be acceptable, however, the choice of 1/3 in the definition is arbitrary. It can be any constant between 0 and 1/2 (exclusive) and the set BPP will be unchanged. It does not even have to be constant: the same class of problems is defined by allowing error as high as 1/2 − n−c on the one hand, or requiring error as small as 2−nc on the other hand, where c is any positive constant, and n is the length of input. The idea is that there is a probability of error, but if the algorithm is run many times, the chance that the majority of the runs are wrong drops off exponentially as a consequence of the Chernoff bound. This makes it possible to create a highly accurate algorithm by merely running the algorithm several times and taking a "majority vote" of the answers. For example, if one defined the class with the restriction that the algorithm can be wrong with probability at most 1/2100, this would result in the same class of problems.