On the Monte carlo boolean decision tree complexity of read-once formulae

Abstract

In the boolean decision tree model there is at least a linear gap between the Monte Carlo and the Las Vegas complexity of a function depending on the error probability. We prove for a large class of read-once formulae that this trivial speed-up is the best that a Monte Carlo algorithm can achieve. For every formula F belonging to that class we show that the Monte Carlo complexity of F with two-sided error p is (1 − 2p)R(F), and with one-sided error p is (1 − p)R(F), where R(F) denotes the Las Vegas complexity of F. The result follows from a general lower bound that we derive on the Monte Carlo complexity of these formulae. This bound is analogous to the lower bound due to Saks and Wigderson on their Las Vegas complexity.