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With high probability

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With high probability

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. it can be made as close as desired to 1 by making n big enough.

Applications

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, 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 algorithms in which 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. QMA and QIP are related complexity class.
• 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.

References

• 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.
• "Principles of Distributed Computing (lecture 7)" (PDF). ETH Zurich. Retrieved 21 February 2015.
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