Sharing One Secret vs. Sharing Many Secrets: Tight Bounds for the Max Improvement Ratio
MFCS '01 Proceedings of the 26th International Symposium on Mathematical Foundations of Computer Science
The Computational Power of a Family of Decision Forests
MFCS '01 Proceedings of the 26th International Symposium on Mathematical Foundations of Computer Science
Chernoff-type direct product theorems
CRYPTO'07 Proceedings of the 27th annual international cryptology conference on Advances in cryptology
A strong direct product theorem for disjointness
Proceedings of the forty-second ACM symposium on Theory of computing
Optimal direct sum results for deterministic and randomized decision tree complexity
Information Processing Letters
Constructive proofs of concentration bounds
APPROX/RANDOM'10 Proceedings of the 13th international conference on Approximation, and 14 the International conference on Randomization, and combinatorial optimization: algorithms and techniques
Studies in complexity and cryptography
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We investigate two problems concerning the complexity of evaluating a function f at k-tuple of unrelated inputs by k parallel decision tree algorithms. In the product problem, for some fixed depth bound d, we seek to maximize the fraction of input k-tuples for which all k decision trees are correct. Assume that for a single input to f, the best decision tree algorithm of depth d is correct on a fraction p of inputs. We prove that the maximum fraction of k-tuples on which k depth d algorithms are all correct is at most p/sup k/, which is the trivial lower bound. We show that if we replace the depth d restriction by "expected depth d", then this result fails. In the help-bit problem, we are permitted to ask k-1 arbitrary binary questions about the k-tuple of inputs. For each possible k-1-tuple of answers to these queries we will have a k-tuple of decision trees which are supposed to correctly compute all functions on k-tuples that are consistent with the particular answers. The complexity here is the maximum depth of any of the trees in the algorithm. We show that for all k sufficiently large, this complexity is equal to deg/sup s/(f) which is the minimum degree of a multivariate polynomial whose sign is equal to f. Finally, we give a brief discussion of these problems in the context of other complexity models.