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Randomized Boolean decision trees: Several remarks

THEORETICAL COMPUTER SCIENCE 1998年第2期207卷 329-342页

作者： Vereshchagin, NK Moscow MV Lomonosov State Univ Fac Mech & Math Dept Math Log Moscow 119899 Russia

Assume we want to show that (a) the cost of any randomized decision tree computing a given Boolean function is at least c. To this end, it suffices to prove that (b) there is a probability distribution over the set of all assignments to variables of that function with respect to which the average cost of any deterministic decision tree computing that function is at least c. Yao (1977) showed that this method is universal for proving lower bounds for randomized errorless decision trees, that is, that (a) is equivalent to (b), In the present paper we prove that this is the case also for randomized decision trees which are allowed to make errors. This gives the positive answer to the question posed in Yao (1977), In the second part of the paper we exhibit an example when randomized directional decision trees (defined in Yao (1977)) to evaluate read once formulae are not optimal. We construct a formula F-n of n Boolean variables such that the cost of the optimal directional decision tree computing F-n is Omega(n(alpha)) and there is an undirectional randomized decision tree computing F-n of cost O(n(beta)) for some beta < alpha. (C) 1998-Elsevier Science B,V. All rights reserved.

关键词： Boolean decision trees randomized algorithm directional algorithms undirectional algorithms

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