How can I construct sorting network for $k$ numbers

Question

How can I construct a sorting network for $k$ numbers?

My goal is to implement sorting networks in Java for $k$ in the range $[3,\hspace{-0.03 in}32]$.
To be even more specific, I only want to sort integers.

I found some implementation in this article (pages 2-3), but I don't understand it.

I been trying to convert this problem to SAT. I started with a simple non-optimal network: $[01, 12, \ldots, (n-1)n, 01, 12, \ldots, (n-2)(n-1), \ldots, 01, 12, 01]$ (source). The idea is to convert it to SAT, find the shortest equal-satisfiable SAT formula, and convert it back to a network representation. The problem is that in the network, the order of comparisons is important, so I don't know hot to convert it to SAT. It sounds like some one has already been trying to do something like this, but I don't understand it completely.

Related question.

Answer 1

there is some research angle here dating at least to Knuth's Art of Computer Programming and presumably earlier in finding optimal sorting networks for low $n$. its intractable to find optimal sorting networks for small $n$ but it has been done up to about $n=10$ eg as in this recent notable paper, also using SAT. details about how to reduce the problem to SAT are in the paper. basically a large SAT formula encoding is built that asserts "these boolean variables configure a circuit that sorts all inputs for size $n$". (the more nonresearch angle is to use existing sort algorithms or sorting network configurations as mentioned in the paper by Har-Peled you cite to generate the (nonoptimal) circuits, this is more like a CS/EE exercise.)

Optimal Sorting Networks Daniel Bundala, Jakub Závodný

This paper settles the optimality of sorting networks given in The Art of Computer Programming vol. 3 more than 40 years ago. The book lists efficient sorting networks with n <= 16 inputs. In this paper we give general combinatorial arguments showing that if a sorting network with a given depth exists then there exists one with a special form. We then construct propositional formulas whose satisfiability is necessary for the existence of such a network. Using a SAT solver we conclude that the listed networks have optimal depth. For n <= 10 inputs where optimality was known previously, our algorithm is four orders of magnitude faster than those in prior work.