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

Itai Benjamini·Oded Schramm·David B. Wilson
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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 Theta(n^{-1/2} sqrt{log n}). We give a balanced monotone Boolean function for which the corresponding probability is Theta(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 Theta(n^{-1/2}). For balanced monotone Boolean functions, there is some input bit that is read with probability at least Theta(n^{-1/3}).

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Itai Benjamini, Oded Schramm, David B. Wilson. 2004-10-12. Balanced Boolean functions that can be evaluated so that every input bit is unlikely to be read. https://doi.org/10.1145/1060590.1060627

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