# RP

This article relates to the theory of computation. For alternate uses, see RP (disambiguation).

In complexity theory, RP ("randomized polynomial time") is the complexity class of problems for which a probabilistic Turing machine exists with these properties:

RP algorithm (1 run)
Correct
YES NO
YES ≥1/2 ≤1/2
NO 0 1
RP algorithm (n runs)
Correct
YES NO
YES ≥1-½n ≤½n
NO 0 1
Co-RP algorithm (1 run)
Correct
YES NO
YES 1 0
NO ≤1/2 ≥1/2
• It always runs in polynomial time in the input size
• If the correct answer is NO, it always returns NO
• If the correct answer is YES, then it returns YES with probability at least 1/2

In other words, the algorithm is allowed to flip a truly-random coin while it's running. The only case in which the algorithm can return YES is if the actual answer is YES; therefore if it returns YES, then the correct answer is definitely YES, and the algorithm terminates. However, the algorithm can return NO, no matter what the actual answer is. If it returns NO, it might be wrong.

If the correct answer is YES and the algorithm is run n times then it will return YES at least once with probability at least 1 - ½n. So if the algorithm is run 100 times, then the chance of it giving the wrong answer every time is lower than the chance that cosmic rays corrupted the memory of the computer running the algorithm. In this sense, if a source of random numbers is available, most algorithms in RP are highly practical.

The fraction 1/2 in the definition is arbitrary. The set RP will contain exactly the same problems, even if the 1/2 is replaced by any constant nonzero probability less then 1; here constant means independent of the input to the algorithm.

The definition of RP says YES is always right and NO is usually right. The complexity class Co-RP is similarly defined, except that NO is always right and YES is usually right. In other words, it accepts all YES instances but can either accept or reject NO instances. The class BPP describes algorithms that can give incorrect answers on both YES and NO instances. The intersection of the sets RP and Co-RP is called ZPP. Some authors use the names R and Co-R rather than RP and Co-RP.

P is a subset of RP, which is a subset of NP. Similarly, P is a subset of Co-RP which is a subset of Co-NP. It is not known whether any of these subsets are strict. However, it is believed that either P ≠ RP or RP ≠ NP, since otherwise P = NP, which is widely believed to be false. It is also not known whether RP = Co-RP, or whether RP is a subset of the intersection of NP and Co-NP.

 Important complexity classes (more) P | NP | Co-NP | NP-C | Co-NP-C | NP-hard | UP | #P | #P-C | L | NC | P-C PSPACE | PSPACE-C | EXPTIME | EXPSPACE | BQP | BPP | RP | ZPP | PCP | IP | PH

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