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

This paper describes a computer-assisted non-existence proof of 9-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 10 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 8 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-handedly solve smaller instances of the problem, it is their combination that leads to the more efficient solution that scales to handle 9 inputs.