Constant-Depth Sorting Networks

TL;DR

This paper investigates the minimal arity of comparators in constant-depth sorting networks, providing new lower bounds for depths up to four and revealing how arity scales with input size.

Contribution

It presents the first lower bounds on the arity of constant-depth sorting networks, showing how minimal arity varies with depth and input size, especially for depths 3 and 4.

Findings

For depth 1 and 2, minimal arity equals the number of inputs.

At depth 3, minimal arity is approximately half the input size.

At depth 4, minimal arity scales as n^{2/3}, which is sublinear.

Abstract

In this paper, we address sorting networks that are constructed from comparators of arity $k > 2$. That is, in our setting the arity of the comparators -- or, in other words, the number of inputs that can be sorted at the unit cost -- is a parameter. We study its relationship with two other parameters -- $n$, the number of inputs, and $d$, the depth. This model received considerable attention. Partly, its motivation is to better understand the structure of sorting networks. In particular, sorting networks with large arity are related to recursive constructions of ordinary sorting networks. Additionally, studies of this model have natural correspondence with a recent line of work on constructing circuits for majority functions from majority gates of lower fan-in. Motivated by these questions, we obtain the first lower bounds on the arity of constant-depth sorting networks. More precisely, we consider sorting networks of depth $d$ up to 4, and determine the minimal $k$ for which there is such a network with comparators of arity $k$. For depths $d=1,2$ we observe that $k=n$. For $d=3$ we show that $k = \lceil \frac n2 \rceil$. For $d=4$ the minimal arity becomes sublinear: $k = \Theta(n^{2/3})$. This contrasts with the case of majority circuits, in which $k = O(n^{2/3})$ is achievable already for depth $d=3$.