[Theory] UC Theory Seminar: a reminder

[Alexander Razborov]

Departments of Mathematics & Computer Science
Combinatorics & Theory Seminar

Tuesday, November 29, 3:30pm
Location Kent 107

SPEAKER: Vladimir Podolskii (Courant Institute & Steklov Institute)

TITLE: 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://eccc.weizmann.ac.il/report/2022/116/