Randomized vs. deterministic decision tree complexity for read-once Boolean functions

The authors consider the deterministic and the randomized decision-tree complexities for Boolean functions, denoted DC(f) and RC(f), respectively. It is well known that RC(f)>or=DC(f)/sup 0.5/ for every Boolean function f (called 0.5-exponent), but no better lower bound is known for all Boolean functions, whereas the best known upper bound is RC(f)= Theta (DC(f)/sup 0//sub .//sup 753 . ./) (or 0.753 . . .-exponent) for some Boolean function f. The present result is a 0.51 lower bound on the exponent for all read-once functions representable by formulae in which each input variable appears exactly once. To obtain it the authors generalize an existing lower bound technique and combine it with restrictions arguments. This result provides a lower bound of n/sup 0.51/ on the number of positions that have to be evaluated by any randomized alpha - beta pruning algorithm computing the value of any two-person zero-sum game tree with n final positions.<<ETX>>

[1]  Marc Snir,et al.  Lower Bounds on Probabilistic Linear Decision Trees , 1985, Theor. Comput. Sci..

[2]  Andrew Chi-Chih Yao,et al.  Probabilistic computations: Toward a unified measure of complexity , 1977, 18th Annual Symposium on Foundations of Computer Science (sfcs 1977).

[3]  Avi Wigderson,et al.  On read-once threshold formulae and their randomized decision tree complexity , 1990, Proceedings Fifth Annual Structure in Complexity Theory Conference.

[4]  Andrew Chi-Chih Yao.  Lower Bounds to Randomized Algorithms for Graph Properties , 1991, J. Comput. Syst. Sci..

[5]  Michael Tarsi,et al.  Optimal Search on Some Game Trees , 1983, JACM.

[6]  Michael E. Saks,et al.  Probabilistic Boolean decision trees and the complexity of evaluating game trees , 1986, 27th Annual Symposium on Foundations of Computer Science (sfcs 1986).

[7]  Leslie G. Valiant,et al.  Short Monotone Formulae for the Majority Function , 1984, J. Algorithms.

[8]  Judea Pearl,et al.  The solution for the branching factor of the alpha-beta pruning algorithm and its optimality , 1982, CACM.

[9]  Valerie King.  Lower bounds on the complexity of graph properties , 1988, STOC '88.

[10]  Péter Hajnal.  On the power of randomness in the decision tree model , 1990, Proceedings Fifth Annual Structure in Complexity Theory Conference.