
This Article  
 
Share  
Bibliographic References  
Add to:  
Digg Furl Spurl Blink Simpy Del.icio.us Y!MyWeb  
Search  
 
2011 26th Annual IEEE Conference on Computational Complexity
Improved Direct Product Theorems for Randomized Query Complexity
San Jose, California USA
June 08June 11
ISBN: 9780769544113
ASCII Text  x  
Andrew Drucker, "Improved Direct Product Theorems for Randomized Query Complexity," 2012 IEEE 27th Conference on Computational Complexity, pp. 111, 2011 26th Annual IEEE Conference on Computational Complexity, 2011.  
BibTex  x  
@article{ 10.1109/CCC.2011.29, author = {Andrew Drucker}, title = {Improved Direct Product Theorems for Randomized Query Complexity}, journal ={2012 IEEE 27th Conference on Computational Complexity}, volume = {0}, year = {2011}, issn = {10930159}, pages = {111}, doi = {http://doi.ieeecomputersociety.org/10.1109/CCC.2011.29}, publisher = {IEEE Computer Society}, address = {Los Alamitos, CA, USA}, }  
RefWorks Procite/RefMan/Endnote  x  
TY  CONF JO  2012 IEEE 27th Conference on Computational Complexity TI  Improved Direct Product Theorems for Randomized Query Complexity SN  10930159 SP1 EP11 A1  Andrew Drucker, PY  2011 KW  direct product theorems KW  query complexity KW  decision trees KW  averagecase complexity KW  hardness amplification VL  0 JA  2012 IEEE 27th Conference on Computational Complexity ER   
DOI Bookmark: http://doi.ieeecomputersociety.org/10.1109/CCC.2011.29
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 algorithmhas success probability at most 1  \eps in computing the Boolean function f on input distribution mu, then for alpha \leq 1, every alpha \eps Tkquery algorithm has success probability at most (2^{\alpha \eps}(1\eps))^k in computing the kfold direct product f^{\otimes k} correctly on k independent inputs from \mu. 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 $\alpha > 0$, the worstcase success probability of any $\alpha R_2(f) k$query randomized algorithm for f^{\otimes k} falls exponentially with k. The best previous statement of this type, due to Klauck, \v{S}palek, and de Wolf, required a query bound of O(bs(f) k). Our proof technique involves defining and analyzing a collection of martingales associated with an algorithm attempting to solve f^{\otimes 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.
Index Terms:
direct product theorems, query complexity, decision trees, averagecase complexity, hardness amplification
Citation:
Andrew Drucker, "Improved Direct Product Theorems for Randomized Query Complexity," ccc, pp.111, 2011 26th Annual IEEE Conference on Computational Complexity, 2011
Usage of this product signifies your acceptance of the Terms of Use.