Towards Simpler Sorting Networks and Monotone Circuits for Majority

Abstract: In this paper, we study the problem of computing the majority function by low-depth monotone circuits and a related problem of constructing low-depth sorting networks. We consider both the classical setting with elementary operations of arity $2$ and the generalized setting with operations of arity $k$, where $k$ is a parameter. For both problems and both settings, there are various constructions known, the minimal known depth being logarithmic. However, there is currently no known construction that simultaneously achieves sub-log-squared depth, effective constructability, simplicity, and has a potential to be used in practice. In this paper we make progress towards resolution of this problem. For computing majority by standard monotone circuits (gates of arity 2) we provide an explicit monotone circuit of depth $O(\log_2^{5/3} n)$. The construction is a combination of several known and not too complicated ideas. For arbitrary arity of gates $k$ we provide a new sorting network architecture inspired by representation of inputs as a high-dimensional cube. As a result we provide a simple construction that improves previous upper bound of $4 \log_k^2 n$ to $2 \log_k^2 n$. We prove the similar bound for the depth of the circuit computing majority of $n$ bits consisting of gates computing majority of $k$ bits. Note, that for both problems there is an explicit construction of depth $O(\log_k n)$ known, but the construction is complicated and the constant hidden in $O$-notation is huge.