Improved Direct Product Theorems for Randomized Query Complexity

The "direct product problem" is a fundamental question in complexity theory which seeks to understand how the difficulty of computing a function on each of k independent inputs scales with k. We prove the following direct product theorem (DPT) for query complexity: if every T-query algorit...

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Vydané v:2011 IEEE 26th Annual Conference on Computational Complexity s. 1 - 11
Hlavný autor: Drucker, A.
Médium: Konferenčný príspevok..
Jazyk:English
Vydavateľské údaje: IEEE 01.06.2011
Predmet:
average-case complexity
Boolean functions
Complexity theory
Computational modeling
Decision trees
direct product theorems
hardness amplification
Integrated circuit modeling
query complexity
Random variables
ISBN:9781457701795, 1457701790
ISSN:1093-0159
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Popis
Shrnutí:The "direct product problem" is a fundamental question in complexity theory which seeks to understand how the difficulty of computing a function on each of k independent inputs scales with k. We prove the following direct product theorem (DPT) for query complexity: if every T-query algorithm has success probability at most 1 - ε in computing the Boolean function f on input distribution μ, then for α ≤ 1, every αεTk-query algorithm has success probability at most (2 αε (1 - ε)) k in computing the fc-fold direct product f ⊗k correctly on k independent inputs from μ. In light of examples due to Shaltiel, this statement gives an essentially optimal tradeoff between the query bound and the error probability. Using this DPT, we show that for an absolute constant α >; 0, the worst-case success probability of any αR 2 (f)k-query randomized algorithm for f ⊗k falls exponentially with k. The best previous statement of this type, due to Klauck, Spalek, and de Wolf, required a query bound of O(bs(f)k). Our proof technique involves defining and analyzing a collec tion of martingales associated with an algorithm attempting to solve f ⊗k . Our method is quite general and yields a new XOR lemma and threshold DPT for the query model, as well as DPTs for the query complexity of learning tasks, search problems, and tasks involving interaction with dynamic entities. We also give a version of our DPT in which decision tree size is the resource of interest.
ISBN:9781457701795
1457701790
ISSN:1093-0159
DOI:10.1109/CCC.2011.29