With high probability

Description of limiting behavior in probabilistic algorithms

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In mathematics, an event that occurs with high probability (often shortened to w.h.p. or WHP) is one whose probability depends on a certain number n and goes to 1 as n goes to infinity, i.e. the probability of the event occurring can be made as close to 1 as desired by making n big enough.

## Applications

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The term WHP is especially used in computer science, in the analysis of probabilistic algorithms. For example, consider a certain probabilistic algorithm on a graph with n nodes. If the probability that the algorithm returns the correct answer is  1 − 1 / n {\displaystyle 1-1/n} , then when the number of nodes is very large, the algorithm is correct with a probability that is very near 1. This fact is expressed shortly by saying that the algorithm is correct WHP.

Some examples where this term is used are:

* Miller–Rabin primality test: a probabilistic algorithm for testing whether a given number n is prime or composite. If n is composite, the test will detect n as composite WHP. There is a small chance that we are unlucky and the test will think that n is prime. But, the probability of error can be reduced indefinitely by running the test many times with different randomizations.
* Freivalds' algorithm: a randomized algorithm for verifying matrix multiplication. It runs faster than deterministic algorithms WHP.
* Treap: a randomized binary search tree. Its height is logarithmic WHP. Fusion tree is a related data structure.
* Online codes: randomized codes which allow the user to recover the original message WHP.
* BQP: a complexity class of problems for which there are polynomial-time quantum algorithms which are correct WHP.
* Probably approximately correct learning: A process for machine-learning in which the learned function has low generalization-error WHP.
* Gossip protocols: a communication protocol used in distributed systems to reliably deliver messages to the whole cluster using a constant amount of network resources on each node and ensuring no single point of failure.

## See also

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* Randomized algorithm
* Almost surely

## References

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* Métivier, Y.; Robson, J. M.; Saheb-Djahromi, N.; Zemmari, A. (2010). "An optimal bit complexity randomized distributed MIS algorithm". Distributed Computing. 23 (5–6): 331. doi:10.1007/s00446-010-0121-5.5–6&rft.pages=331&rft.date=2010&rft_id=info:doi/10.1007/s00446-010-0121-5&rft.aulast=Métivier&rft.aufirst=Y.&rft.au=Robson,+J.+M.&rft.au=Saheb-Djahromi,+N.&rft.au=Zemmari,+A.&rfr_id=info:sid/en.wikipedia.org:With+high+probability" class="Z3988">
* "Principles of Distributed Computing (lecture 7)" (PDF). ETH Zurich. Retrieved 21 February 2015.

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