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Pré-Publication, Document De Travail Année : 2006

On the Probabilistic Query Complexity of Transitively Symmetric Problems

Résumé

We obtain optimal lower bounds on the nonadaptive probabilistic query complexity of a class of problems defined by a rather weak symmetry condition. In fact, for each problem in this class, given a number T of queries we compute exactly the performance (i.e., the probability of success on the worst instance) of the best nonadaptive probabilistic algorithm that makes T queries. We show that this optimal performance is given by a minimax formula involving certain probability distributions. Moreover, we identify two classes of problems for which adaptivity does not help. We illustrate these results on a few natural examples, including unordered search, Simon's problem, distinguishing one-to-one functions from two-to-one functions, and hidden translation. For these last three examples, which are of particular interest in quantum computing, the recent theorems of Aaronson, of Laplante and Magniez, and of Bar-Yossef, Kumar and Sivakumar on the probabilistic complexity of black-box problems do not yield any nonconstant lower bound.
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Dates et versions

hal-00120934 , version 1 (18-12-2006)
hal-00120934 , version 2 (19-12-2006)

Identifiants

  • HAL Id : hal-00120934 , version 2

Citer

Pascal Koiran, Vincent Nesme, Natacha Portier. On the Probabilistic Query Complexity of Transitively Symmetric Problems. 2006. ⟨hal-00120934v2⟩
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