An Answer to the Bose-Nelson Sorting Problem for 11 and 12 Channels

TL;DR

This paper establishes the minimal size of optimal sorting networks for 11 and 12 channels, confirming the best known networks are indeed optimal and closing open problems in the Bose-Nelson sorting problem.

Contribution

It generalizes Van Voorhis's result to a broader class of comparator networks and introduces a verified dynamic programming method to compute optimal sorting network sizes.

Findings

11-channel sorting networks require at least 35 comparators

12-channel sorting networks require at least 39 comparators

The bounds confirm the optimality of known networks

Abstract

We show that 11-channel sorting networks have at least 35 comparators and that 12-channel sorting networks have at least 39 comparators. This positively settles the optimality of the corresponding sorting networks given in The Art of Computer Programming vol. 3 and closes the two smallest open instances of the Bose-Nelson sorting problem. We obtain these bounds by generalizing a result of Van Voorhis from sorting networks to a more general class of comparator networks. From this we derive a dynamic programming algorithm that computes the optimal size for a sorting network with a given number of channels. From an execution of this algorithm we construct a certificate containing a derivation of the corresponding lower size bound, which we check using a program formally verified using the Isabelle/HOL proof assistant.