Randomized Algorithm

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A Randomized Algorithm is an algorithm that uses a random function in its conditional logic.



References

2013

  • (Wikipedia, 2013) ⇒ http://en.wikipedia.org/wiki/Randomized_algorithm
    • A randomized algorithm is an algorithm which employs a degree of randomness as part of its logic. The algorithm typically uses uniformly random bits as an auxiliary input to guide its behavior, in the hope of achieving good performance in the "average case" over all possible choices of random bits. Formally, the algorithm's performance will be a random variable determined by the random bits; thus either the running time, or the output (or both) are random variables.

      One has to distinguish between algorithms that use the random input to reduce the expected running time or memory usage, but always terminate with a correct result (Las Vegas algorithms) in a bounded amount of time, and probabilistic algorithms, which, depending on the random input, have a chance of producing an incorrect result (Monte Carlo algorithms) or fail to produce a result either by signalling a failure or failing to terminate.

      In the second case, random performance and random output, the term "algorithm" for a procedure is somewhat questionable. In the case of random output, it is no longer formally effective.[1] However, in some cases, probabilistic algorithms are the only practical means of solving a problem.[2]

      In common practice, randomized algorithms are approximated using a pseudorandom number generator in place of a true source of random bits; such an implementation may deviate from the expected theoretical behavior.

  1. "Probabilistic algorithms should not be mistaken with methods (which I refuse to call algorithms), which produce a result which has a high probability of being correct. It is essential that an algorithm produces correct results (discounting human or computer errors), even if this happens after a very long time." Henri Cohen (2000). A Course in Computational Algebraic Number Theory. Springer-Verlag, p. 2.
  2. "In testing primality of very large numbers chosen at random, the chance of stumbling upon a value that fools the Fermat test is less than the chance that cosmic radiation will cause the computer to make an error in carrying out a 'correct' algorithm. Considering an algorithm to be inadequate for the first reason but not for the second illustrates the difference between mathematics and engineering." Hal Abelson and Gerald J. Sussman (1996). Structure and Interpretation of Computer Programs." MIT Press, section 1.2.


2005

1995