Computation Model and Complexity Class

2 An algorithmic process that uses the result of a random draw to make an approximated decision has the ability to estimate reasonable solutions. Random problem classes use probabilities to describe their correctness in a polynomial run time. These definitions give rise to some further complexity classes. Like a (deterministic) Turing machine, but we allow the machine to flip coins during the computation, and transitions can depend on the outcomes of the flips. Equivalently, a randomized Turing machine can be implemented as a deterministic Turing machine with an additional read-only input tape for inputting a random stream of 0’s and 1’s.

3 Finite control Random bits Scatch tape(s) Input

4 Randomized Polynomial time (RP) Random polynomial algorithms accept input with probability 50% or more if the input is a member of the language. If the input is not a member, the algorithms accept the input with zero probability. More formal definition: Defintion2.6.1 The class RP consists of all languages L that have a randomized algorithm A running in worst- case polynomial time such that for any input x in ∑*,

5 Zero-error Probabilistic Polynomial time Defintion2.6.2 The class ZPP (for Zero-error Probabilistic time) is the class of languages that have Las Vegas algorithms running in expected polynomial time. Random algorithm with expected polynomial run-time, always gives right answer except it can also answer don't know. The definition of class ZPP is almost the same as the definition of P, except that ZPP allows the behavior of the TM to involve randomness, and the expected running time, rather that the worst-case running time is measured.

6 Probabilistic Polynomial time (PP) Probabilistic polynomial algorithms correctly accept input with probability greater than 50% and incorrectly accept input with probability less than 50%. Defintion2.6.3 The class PP (for Probabilistic Polynomial time )consists of all languages L that have a randomizd algorithm A running in worst-case polynomial time such that for any input x in ∑*,

7 Bounded-error Probabilistic Polynomial time (BPP) -Random algorithm that runs in polynomial time and has a bound on the error probability, say 2/3. Defintion2.6.4 The class BPP (for Bounded-error Probabilistic Polynomial time) consists of all languages L that have a randomized algorithm A running in worst- case polynomial time such that for any input x in ∑*,

8 The BPP is also known as the class of problems which can be solved by Monte Carlo algorithms.

9 Relationships to the classes P and NP Let us define the class co-RP to be the set of languages L such that L is in RP; i.e.,co-RP is the complements of the languages in RP. Theorem2.6.1 P is the subset of ZPP. Theorem2.6.2 RP is the subset of NP. －

10 Relationship of different classes NPRPZPPP co-NPco-RP

11 Homework P70 1,2,3