2022/08/17 by Natalia Dobrokhotova-Maikova, Alexander Kozachinskiy, Dobrokhotova-Maikova, Natalia +3 · 1 citation
Computer Science · #Advanced Graph Theory Research #Cellular Automata and Applications #Computational Complexity (cs.CC) #FOS: Computer and information sciences #semigroups and automata theory
paper · pdf · doi:10.48550/arxiv.2208.08394
openalex publication_date 2022/08/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 = Θ(n2/3). This contrasts with the case of majority circuits, in which k = O(n2/3) is achievable already for depth d=3.