Improved bounds for the randomized decision tree complexity of recursive majority
Résumé
We consider the randomized decision tree complexity of the recursive 3-majority function. For evaluating a height $h$ formulae, we prove a lower bound for the $\delta$-two-sided-error randomized decision tree complexity of $(1-2\delta)(5/2)^h$, improving the lower bound of $(1-2\delta)(7/3)^h$ given by Jayram \etal{} (STOC '03). We also state a conjecture which would further improve the lower bound to $(1-2\delta)2.54355^h$. Second, we improve the upper bound by giving a new zero-error randomized decision tree algorithm that has complexity at most $(1.007) \cdot 2.64946^h$, improving on the previous best known algorithm, which achieved $(1.004) \cdot 2.65622^h$. Our lower bound follows from a better analysis of the base case of the recursion of Jayram \etal{} Our algorithm uses a novel ''interleaving'' of two recursive algorithms.
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