Proceedings 35th Annual Symposium on Foundations of Computer Science (1994)

Santa Fe, NM, USA

Nov. 20, 1994 to Nov. 22, 1994

ISBN: 0-8186-6580-7

pp: 318-329

N. Nisan , Dept. of Comput. Sci., Hebrew Univ., Jerusalem, Israel

ABSTRACT

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.

INDEX TERMS

multivariate polynomial, help bits, decision trees, complexity, k-tuple, parallel decision tree algorithms, fixed depth bound, maximum fraction, lower bound, arbitrary binary questions

CITATION

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.

doi:10.1109/SFCS.1994.365683

CITATIONS