Formalizing Size-Optimal Sorting Networks: Extracting a Certified Proof Checker

Since the proof of the four color theorem in 1976, computer-generated proofs have become a reality in mathematics and computer science. During the last decade, we have seen formal proofs using verified proof assistants being used to verify the validity of such proofs. In this paper, we describe a formalized theory of size-optimal sorting networks. From this formalization we extract a certified checker that successfully verifies computer-generated proofs of optimality on up to 8 inputs. The checker relies on an untrusted oracle to shortcut the search for witnesses on more than 1.6 million NP-complete subproblems.Comment: IMADA-preprint-c

The Quest for Optimal Sorting Networks: Efficient Generation of Two-Layer Prefixes

Previous work identifying depth-optimal $n$-channel sorting networks for $9\leq n \leq 16$ is based on exploiting symmetries of the first two layers. However, the naive generate-and-test approach typically applied does not scale. This paper revisits the problem of generating two-layer prefixes modulo symmetries. An improved notion of symmetry is provided and a novel technique based on regular languages and graph isomorphism is shown to generate the set of non-symmetric representations. An empirical evaluation demonstrates that the new method outperforms the generate-and-test approach by orders of magnitude and easily scales until $n=40$

Twenty-Five Comparators is Optimal when Sorting Nine Inputs (and Twenty-Nine for Ten)

This paper describes a computer-assisted non-existence proof of nine-input sorting networks consisting of 24 comparators, hence showing that the 25-comparator sorting network found by Floyd in 1964 is optimal. As a corollary, we obtain that the 29-comparator network found by Waksman in 1969 is optimal when sorting ten inputs. This closes the two smallest open instances of the optimal size sorting network problem, which have been open since the results of Floyd and Knuth from 1966 proving optimality for sorting networks of up to eight inputs. The proof involves a combination of two methodologies: one based on exploiting the abundance of symmetries in sorting networks, and the other, based on an encoding of the problem to that of satisfiability of propositional logic. We illustrate that, while each of these can single handed solve smaller instances of the problem, it is their combination which leads to an efficient solution for nine inputs.Comment: 18 page

Faster Sorting Networks for 1717, 1919 and 2020 Inputs

We present new parallel sorting networks for $17$ to $20$ inputs. For $17, 19,$ and $20$ inputs these new networks are faster (i.e., they require less computation steps) than the previously known best networks. Therefore, we improve upon the known upper bounds for minimal depth sorting networks on $17, 19,$ and $20$ channels. The networks were obtained using a combination of hand-crafted first layers and a SAT encoding of sorting networks

Sorting Networks: the End Game

This paper studies properties of the back end of a sorting network and illustrates the utility of these in the search for networks of optimal size or depth. All previous works focus on properties of the front end of networks and on how to apply these to break symmetries in the search. The new properties help shed understanding on how sorting networks sort and speed-up solvers for both optimal size and depth by an order of magnitude