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 in 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 s...
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| Veröffentlicht in: | Computational complexity Jg. 21; H. 2; S. 197 - 244 |
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| 1. Verfasser: | |
| Format: | Journal Article |
| Sprache: | Englisch |
| Veröffentlicht: |
Basel
SP Birkhäuser Verlag Basel
01.06.2012
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| Schlagworte: | |
| ISSN: | 1016-3328, 1420-8954 |
| Online-Zugang: | Volltext |
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| Zusammenfassung: | The “direct product problem” is a fundamental question in complexity theory which seeks to understand how the difficulty in 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
in computing the Boolean function
f
on input distribution
μ
, then for
α
≤ 1, every
-query algorithm has success probability at most
in computing the
k
-fold direct product
correctly on
k
independent inputs from
μ
. In light of examples due to Shaltiel, this statement gives an essentially optimal trade-off 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
α
R
2
(
f
)
k
-query randomized algorithm for
falls exponentially with
k
. The best previous statement of this type, due to Klauck, Špalek, 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
. 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. |
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| ISSN: | 1016-3328 1420-8954 |
| DOI: | 10.1007/s00037-012-0043-7 |