Computing majority with low-fan-in majority queries

TL;DR

This paper introduces a new depth-two majority formula with low fan-in that efficiently computes the majority function, providing the first nontrivial upper bound for this problem and analyzing adaptive query strategies.

Contribution

It presents the first nontrivial upper bound for computing majority with low fan-in formulas and explores adaptive query algorithms with different thresholds.

Findings

First nontrivial upper bound with $k = rac{2}{3} n + 4$

Lower bound of $oxed{ ext{ceil}(n/k)}$ queries in adaptive setting

Algorithms achieving $2rac{n}{k} ext{log} k$ and $rac{n}{k} ext{log} k$ queries under different conditions

Abstract

In this paper we examine the problem of computing majority function $\mathrm{MAJ}_n$ 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 $k = \frac{2}{3} n + 4$. 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 $\mathrm{MAJ}_k$ on any subset, and wish to minimize the number of such queries. We give a simple lower bound for this setting with $\lceil n/k \rceil$ queries, and we present two algorithms for this model: the first one makes $\approx 2\frac{n}{k} \log k$ queries in the case when we are limited to the standard majority functions, and the second one makes $\frac{n}{k} \log k$ queries when we are allowed to change the threshold of majority function.