Improved bounds for the randomized decision tree complexity

We consider the randomized decision tree complexity of the recursive 3-majority function. We prove a lower bound of (1/2 − δ) · 2.57143 for the two-sided-error randomized decision tree complexity of evaluating height h formulae with error δ ∈ [0, 1/2). This improves the lower bound of (1 − 2δ)(7/3) given by Jayram, Kumar, and Sivakumar (STOC’03), and the one of (1 − 2δ) · 2.55 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) · 2.64944. The previous best known algorithm achieved complexity (1.004) · 2.65622. 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.