Sorting nine inputs requires twenty-five comparisons

• Twenty-five comparators are minimal to sort nine inputs.
• Twenty-nine comparators are minimal to sort ten inputs.
• New symmetry-breaking results control the growth of the search space.
• Optimized and parallelized algorithms for generating size-optimal sorting networks.
• Use of SAT-solving to speed up the last part of the computation.

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, 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.