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

We investigate the relationship between the complexities of two models of randomized decision trees for read-once Boolean functions; namely, with the directional restriction and without it. A decision tree is directional if it reads a variable in a sub-formula, then it has to evaluate the sub-formula completely before reading another variable that appears in a different sub-formula. It was known that there is a read-once Boolean formula such that an optimal randomized decision tree to evaluate it is not directional. This was first pointed out by Saks and Wigderson [FOCS ’86] and an explicit construction of such a formula was given by Vereshchagin [TCS ’98]. In this paper, 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(nα) and the cost of the optimal randomized decision tree (without the directional restriction) for f is O(nβ ) with α−β > 0.0101. This is the largest known gap so far. ∗An earlier version of this paper [1] has been presented at the 16th CATS symposium (Computing: the Australasian Theory Symposium), Brisbane, Australia, January 2010. †Supported in part by KAKENHI (No. 21500005) from JSPS, Japan.