Proceedings 35th Annual Symposium on Foundations of Computer Science (1994)
Santa Fe, NM, USA
Nov. 20, 1994 to Nov. 22, 1994
N. Nisan , Dept. of Comput. Sci., Hebrew Univ., Jerusalem, Israel
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.
multivariate polynomial, help bits, decision trees, complexity, k-tuple, parallel decision tree algorithms, fixed depth bound, maximum fraction, lower bound, arbitrary binary questions
S. Rudich, M. Saks and N. Nisan, "Products and help bits in decision trees," Proceedings 35th Annual Symposium on Foundations of Computer Science(FOCS), Santa Fe, NM, USA, 1994, pp. 318-329.