Improved bounds for the randomized decision tree complexity of recursive majority

Abstract: We consider the randomized decision tree complexity of the recursive 3-majority function. We prove a lower bound of $(1/2-\delta) \cdot 2.57143^h$ for the two-sided-error randomized decision tree complexity of evaluating height $h$ formulae with error $\delta \in [0,1/2)$. This improves the lower bound of $(1-2\delta)(7/3)^h$ given by Jayram, Kumar, and Sivakumar (STOC'03), and the one of $(1-2\delta) \cdot 2.55^h$ given by Leonardos (ICALP'13). Second, we improve the upper bound by giving a new zero-error randomized decision tree algorithm that has complexity at most $(1.007) \cdot 2.64944^h$. The previous best known algorithm achieved complexity $(1.004) \cdot 2.65622^h$. The new lower bound follows from a better analysis of the base case of the recursion of Jayram et al. The new algorithm uses a novel "interleaving" of two recursive algorithms.