Balanced Boolean functions that can be evaluated so that every input bit is unlikely to be read
摘要
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}).
引用
@article{arxiv.math/0410282,
title = {Balanced Boolean functions that can be evaluated so that every input bit is unlikely to be read},
author = {Itai Benjamini and Oded Schramm and David B. Wilson},
journal= {arXiv preprint arXiv:math/0410282},
year = {2012}
}
备注
11 pages