On Directional vs. Undirectional Randomized Decision Tree Complexity for Read-Once Formulas

We investigate the relationship between the directional and the undirectional complexity of read-once Boolean formulas on the randomized decision tree model. It was known that there is a read-once Boolean formula such that an optimal randomized algorithm to evaluate it is not directional. This was first pointed out by Saks and Wigderson (1986) and an explicit construction of such a formula was given by Vereshchagin (1998). We conduct a systematic search for a certain class of functions and provide an explicit construction of a read-once Boolean formula f on n variables such that the cost of the optimal directional randomized decision tree for f is Ω(n) and the cost of the optimal randomized undirectional decision tree for f is O(n) with α−β > 0.0101. This is the largest known gap so far.