Balanced boolean functions that can be evaluated so that every input bit is unlikely to be read

Authors:
Itai Benjamini;Oded Schramm;David B. Wilson
Affiliations:
Weizmann Institute;Microsoft Research;Microsoft Research
Venue:
Proceedings of the thirty-seventh annual ACM symposium on Theory of computing
Year:
2005

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Abstract

A Boolean function of n bits is balanced if it takes the value 1 with probability 1⁄2. We exhibit a balanced Boolean function with a randomized evaluation procedure (with probability 0 of making a mistake) so that on uniformly random inputs, no input bit is read with probability more than Θ(n-1/2√ log n). We construct a balanced monotone Boolean function and a randomized algorithm computing it for which each bit is read with probability Θ(n-1⁄3 log n). We then show that for any randomized algorithm for evaluating a balanced Boolean function, when the input bits are uniformly random, there is some input bit that is read with probability at least Θ(n-1𔊪). For balanced monotone Boolean functions, there is some input bit that is read with probability at least Θ(n-1𔊫).