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In [[computing]], a '''Las Vegas algorithm''' is a [[randomized algorithm]] that always gives [[correctness (computer science)|correct]] results; that is, it always produces the correct result or it informs about the failure.  In other words, a Las Vegas algorithm does not gamble with the correctness of the result; it gambles only with the resources used for the computation. A simple example is randomized [[quicksort]], where the pivot is chosen randomly, but the result is always sorted. The usual definition of a Las Vegas algorithm includes the restriction that the ''expected'' run time always be finite, when the expectation is carried out over the space of random information, or entropy, used in the algorithm.


Las Vegas algorithms were introduced by [[László Babai]] in 1979, in the context of the [[graph isomorphism problem]], as a stronger version of [[Monte Carlo algorithm]]s.<ref>[[László Babai]], [http://people.cs.uchicago.edu/~laci/lasvegas79.pdf  Monte-Carlo algorithms in graph isomorphism testing], Université de Montréal, D.M.S. No. 79-10.</ref><ref>[[Leonid Levin]], [http://arxiv.org/abs/cs.CR/0012023 The Tale of One-way Functions], ''Problems of Information Transmission'', vol. 39 (2003), 92-103.</ref><ref>Dan Grundy, [http://www.cs.kent.ac.uk/people/staff/eab2/crypto/thesis.web.pdf Concepts and Calculation in Cryptography], University of Kent, Ph.D. thesis, 2008</ref> Las Vegas algorithms can be used in situations where the number of possible solutions is relatively limited, and where verifying the correctness of a candidate solution is relatively easy while actually calculating the solution is complex.


The name refers to the city of [[Las Vegas, Nevada]], which is well known within the United States as an icon of gambling.
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== Complexity class ==
 
The [[complexity class]] of [[decision problem]]s that have Las Vegas algorithms with [[expected value|expected]] polynomial runtime is '''[[Zero-error_Probabilistic_Polynomial_time | ZPP]]'''.
 
It turns out that
 
:<math>\textrm{ZPP} = \textrm{RP} \cap \,\text{co}\,\textrm{-RP}, \,\!</math>
 
which is intimately connected with the way Las Vegas algorithms are sometimes constructed. Namely the class '''[[RP (complexity)|RP]]''' consists of all decision problems for which a randomized polynomial-time algorithm exists that always answers correctly when the correct answer is "no", but is allowed to be wrong with a certain probability bounded away from one when the answer is "yes". When such an algorithm exists for both a problem and its complement (with the answers "yes" and "no" swapped), the two algorithms can be run simultaneously and repeatedly: run each for a constant number of steps, taking turns, until one of them returns a definitive answer. This is the standard way to construct a Las Vegas algorithm that runs in expected polynomial time. Note that in general there is no worst case upper bound on the run time of a Las Vegas algorithm.
 
== Relation to Monte Carlo algorithms ==
 
Las Vegas algorithms can be contrasted with [[Monte Carlo algorithm]]s, in which the resources used are bounded but the answer is not guaranteed to be correct 100% of the time. By an application of [[Markov's inequality]], a Las Vegas algorithm can be converted into a Monte Carlo algorithm via early termination.
 
== See also ==
* [[Randomness]]
* [[Monte Carlo algorithm]]
 
==Notes ==
{{reflist}}
 
==References==
* ''Algorithms and Theory of Computation Handbook'', CRC Press LLC, 1999, "Las Vegas algorithm", in ''Dictionary of Algorithms and Data Structures'' [online], Paul E. Black, ed., U.S. [[National Institute of Standards and Technology]]. 17 July 2006. (accessed May 09, 2009) Available from: [http://www.nist.gov/dads/HTML/lasVegas.html]
 
[[Category:Randomized algorithms]]

Revision as of 17:36, 14 February 2014


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