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Optimal speedup of Las Vegas algorithms. (English) Zbl 0797.68139
Summary: Let $$A$$ be a Las Vegas algorithm, i.e., $$A$$ is a randomized algorithm that always produces the correct answer when it stops but whose running time is a random variable. We consider the problem of minimizing the expected time required to obtain an answer from $$A$$ using strategies which simulate $$A$$ as follows: run $$A$$ for a fixed amount of time $$t_ 1$$, then run $$A$$ independently for a fixed amount of time $$t_ 2$$, etc. The simulation stops if $$A$$ completes its execution during any of the runs. Let $${\mathcal S}= (t_ 1,t_ 2,\dots)$$ be a strategy, and let $$\ell_ A= \inf_{\mathcal S} T(A,{\mathcal S})$$, where $$T(A,{\mathcal S})$$ is the expected value of the running time of the simulation of $$A$$ under strategy $${\mathcal S}$$.
We describe a simple universal strategy $${\mathcal S}^{\text{univ}}$$, with the property that, for any algorithm $$A$$, $$T(A,{\mathcal S}^{\text{univ}})= O(\ell_ A\log (\ell_ A))$$. Furthermore, we show that this is the best performance that can be achieved, up to a constant factor, by any universal strategy.

##### MSC:
 68T15 Theorem proving (deduction, resolution, etc.) (MSC2010) 68Q25 Analysis of algorithms and problem complexity
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##### References:
 [1] Alt, H.; Guibas, L.; Mehlhorn, K.; Karp, R.; Wigderson, A., A method for obtaining randomized algorithms with small tail probabilities, Tech. rept. TR-91-057, (1991), International Computer Science Institute Berkeley · Zbl 0857.68057 [2] Ertel, W., OR-parallel theorem proving with random competition, (), 226-237, Lecture Notes in Artificial Intelligence
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