# Constant-Depth Sorting Networks

## Abstract

We consider 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. 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=n/2. 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. The talk is based on joint work with Natalia Dobrokhotova-Maikova and Alexander Kozachinskiy: https://drops.dagstuhl.de/opus/volltexte/2023/17546/pdf/LIPIcs-ITCS- 2023-43.pdf

Bio

Vladimir Podolskii is a theoretical computer scientist working in computational complexity. He is currently an associate professor at the Tufts CS Department. Prior to that he has worked at Steklov Mathematical Institute in Moscow since 2009 as research scientist. From 2014 till 2022 he also worked at HSE University in Moscow as an associate professor (part-time). From 2022 till 2023 we worked as visiting associate professor at Courant Institute, NYU. He received his PhD from Moscow State University in 2009.