Computing majority with low-fan-in majority queries
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In this paper we examine the problem of computing majority function MAJ_n on n bits by depth-two formula, where each gate is a majority function on at most k inputs. We present such formula that gives the first nontrivial upper bound for this problem, with k = 2/3 n + 4. This answers an open question in [Kulikov, Podolskii, 2017]. We also look at this problem in adaptive setting - when we are allowed to query for value of MAJ_k on any subset, and wish to minimize the number of such queries. We give a simple lower bound for this setting with n/k queries, and we present two algorithms for this model: the first one makes ≈ 2n/k k queries in the case when we are limited to the standard majority functions, and the second one makes n/k k queries when we are allowed to change the threshold of majority function.
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