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

TL;DR

This paper proves that the minimal sorting network for nine inputs uses 25 comparators and for ten inputs uses 29 comparators, closing longstanding open problems using a combination of symmetry analysis and SAT encoding.

Contribution

It introduces a computer-assisted proof method combining symmetry exploitation and SAT solving to establish the optimal size of small sorting networks.

Findings

Proves 25 comparators are optimal for nine-input sorting networks.

Establishes 29 comparators as optimal for ten-input sorting networks.

Closes the smallest open cases of the optimal sorting network problem since 1966.

Abstract

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.