Improved bounds for the randomized decision tree complexity of recursive majority

Authors:
Frédéric Magniez;Ashwin Nayak;Miklos Santha;David Xiao
Affiliations:
LIAFA, Univ. Paris 7, CNRS, Paris, France;C&O and IQC, U. Waterloo and Perimeter Institute, Waterloo, ON, Canada;LIAFA, Univ. Paris 7, CNRS, Paris, France and Centre for Quantum Technologies, National U. of Singapore;LIAFA, Univ. Paris 7, CNRS, Paris, France and Univ. Paris-Sud, Orsay, France
Venue:
ICALP'11 Proceedings of the 38th international colloquim conference on Automata, languages and programming - Volume Part I
Year:
2011

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Abstract

We consider the randomized decision tree complexity of the recursive 3-majority function. For evaluating height h formulae, we prove a lower bound for the δ-two-sided-error randomized decision tree complexity of (1 - 2δ)(5/2)h, improving the lower bound of (1 - 2δ)(7/3)h given by Jayram, Kumar, and Sivakumar (STOC '03). Second, we improve the upper bound by giving a new zero-error randomized decision tree algorithm that has complexity at most (1.007) ċ 2.64946h. The previous best known algorithm achieved complexity (1.004) ċ 2.65622h. 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.