Improved Bounds for the Randomized Decision Tree Complexity of Recursive Majority

We consider the randomized decision tree complexity of the recursive 3-majority function. We prove a lower bound of (1/2-delta) . 2.57143(h) for the two-sided-error randomized decision tree complexity of evaluating height h formulae with error delta is an element of [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) . 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) . 2.64944(h). The previous best known algorithm achieved complexity (1.004) . 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. (C) 2015 Wiley Periodicals, Inc.