RP (complexity)
Encyclopedia
In complexity theory
, RP ("randomized polynomial time") is the complexity class
of problems for which a probabilistic Turing machine
exists with these properties:
In other words, the algorithm
is allowed to flip a truly random coin while it is running. The only case in which the algorithm can return YES is if the actual answer is YES; therefore if the algorithm terminates and produces YES, then the correct answer is definitely YES; however, the algorithm can terminate with NO regardless of the actual answer. That is, if the algorithm returns NO, it might be wrong.
Some authors call this class R, although this name is more commonly used for the class of recursive language
s.
If the correct answer is YES and the algorithm is run n times with the result of each run statistically independent
of the others, then it will return YES at least once with probability at least . 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 than 1; here constant means independent of the input to the algorithm.
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 these inclusions are strict. However, if the commonly believed conjecture P = BPP is true, then RP, co-RP, and P collapse (are all equal). Assuming in addition that P ≠ NP, this then implies that RP is strictly contained in NP. It is not known whether RP = co-RP, or whether RP is a subset of the intersection of NP and co-NP, though this would be implied by P = BPP.
A natural example of a problem in co-RP currently not known to be in P is Polynomial Identity Testing
, the problem of deciding whether a given multivariate arithmetic expression over the integers is the zero-polynomial. For instance, is the zero-polynomial while
is not.
An alternative characterization of RP that is sometimes easier to use is the set of problems recognizable by nondeterministic Turing machines where the machine accepts if and only if at least some constant fraction of the computation paths, independent of the input size, accept. NP on the other hand, needs only one accepting path, which could constitute an exponentially small fraction of the paths. This characterization makes the fact that RP is a subset of NP obvious.
Computational complexity theory
Computational complexity theory is a branch of the theory of computation in theoretical computer science and mathematics that focuses on classifying computational problems according to their inherent difficulty, and relating those classes to each other...
, RP ("randomized polynomial time") is the complexity class
Complexity class
In computational complexity theory, a complexity class is a set of problems of related resource-based complexity. A typical complexity class has a definition of the form:...
of problems for which a probabilistic Turing machine
Probabilistic Turing machine
In computability theory, a probabilistic Turing machine is a non-deterministic Turing machine which randomly chooses between the available transitions at each point according to some probability distribution....
exists with these properties:
| RP algorithm (1 run) | ||
|---|---|---|
| Answer produced | ||
| Correct answer |
YES | NO |
| YES | ≥ 1/2 | ≤ 1/2 |
| NO | 0 | 1 |
| RP algorithm (n runs) | ||
| Answer produced | ||
| Correct answer |
YES | NO |
| YES | ≥ 1 − 2−n | ≤ 2−n |
| NO | 0 | 1 |
| co-RP algorithm (1 run) | ||
| Answer produced | ||
| Correct answer |
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 (otherwise, it returns NO).
In other words, the algorithm
Algorithm
In mathematics and computer science, an algorithm is an effective method expressed as a finite list of well-defined instructions for calculating a function. Algorithms are used for calculation, data processing, and automated reasoning...
is allowed to flip a truly random coin while it is running. The only case in which the algorithm can return YES is if the actual answer is YES; therefore if the algorithm terminates and produces YES, then the correct answer is definitely YES; however, the algorithm can terminate with NO regardless of the actual answer. That is, if the algorithm returns NO, it might be wrong.
Some authors call this class R, although this name is more commonly used for the class of recursive language
Recursive language
In mathematics, logic and computer science, a formal language is called recursive if it is a recursive subset of the set of all possible finite sequences over the alphabet of the language...
s.
If the correct answer is YES and the algorithm is run n times with the result of each run statistically independent
Statistical independence
In probability theory, to say that two events are independent intuitively means that the occurrence of one event makes it neither more nor less probable that the other occurs...
of the others, then it will return YES at least once with probability at least . 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 than 1; here constant means independent of the input to the algorithm.
Related complexity classes
The definition of RP says that a YES answer is always right and that a NO answer 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, and thus contains both RP and co-RP. The intersection of the sets RP and co-RP is called ZPP. Just as RP may be called R, some authors use the name co-R rather than co-RP.Connection to P and NP
PP (complexity)
In computational complexity theory, P, also known as PTIME or DTIME, is one of the most fundamental complexity classes. It contains all decision problems which can be solved by a deterministic Turing machine using a polynomial amount of computation time, or polynomial time.Cobham's thesis holds...
is a subset of RP, which is a subset of NP
NP (complexity)
In computational complexity theory, NP is one of the most fundamental complexity classes.The abbreviation NP refers to "nondeterministic polynomial time."...
. Similarly, P is a subset of co-RP which is a subset of co-NP. It is not known whether these inclusions are strict. However, if the commonly believed conjecture P = BPP is true, then RP, co-RP, and P collapse (are all equal). Assuming in addition that P ≠ NP, this then implies that RP is strictly contained in NP. It is not known whether RP = co-RP, or whether RP is a subset of the intersection of NP and co-NP, though this would be implied by P = BPP.
A natural example of a problem in co-RP currently not known to be in P is Polynomial Identity Testing
Schwartz-Zippel lemma and testing polynomial identities
In mathematics, the Schwartz–Zippel lemma is a tool commonly used in probabilistic polynomial identity testing, i.e. in the problem of determining whether a given multivariate polynomial is the...
, the problem of deciding whether a given multivariate arithmetic expression over the integers is the zero-polynomial. For instance, is the zero-polynomial while
is not.
An alternative characterization of RP that is sometimes easier to use is the set of problems recognizable by nondeterministic Turing machines where the machine accepts if and only if at least some constant fraction of the computation paths, independent of the input size, accept. NP on the other hand, needs only one accepting path, which could constitute an exponentially small fraction of the paths. This characterization makes the fact that RP is a subset of NP obvious.