Probabilistic Analysis for Randomized Game Tree Evaluation

Abstract

We giveaprobabilistic analysis fortherandomized game tree evaluation algorithm of Snir. We firstshow thatthere exists an input suchthatthe running time,measured as the number of external nodes read by the algorithm,on that input is maximal in stochastic order among all possible inputs. For this worst case input we identify the exact expectation of the number of external nodes read by the algorithm,give the asymptotic order of the variance including the leading constant,providealimitlawforan appropriatenormalizationas well as atail bound estimating large deviations. Our tail bound improves upon the exponent ofanearlier bound due to Karp and Zhang,where sub-Gaussian tails were shownbasedonan approachusing multi-type branching processes andAzuma’sinequality. Our approach rests on a direct,inductive estimate of the moment generating function.