TwentyFive Comparators is Optimal when Sorting Nine Inputs (and TwentyNine for Ten)
Abstract
This paper describes a computerassisted nonexistence proof of nineinput sorting networks consisting of 24 comparators, hence showing that the 25comparator sorting network found by Floyd in 1964 is optimal. As a corollary, we obtain that the 29comparator 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.
 Publication:

arXiv eprints
 Pub Date:
 May 2014
 arXiv:
 arXiv:1405.5754
 Bibcode:
 2014arXiv1405.5754C
 Keywords:

 Computer Science  Discrete Mathematics;
 Computer Science  Data Structures and Algorithms
 EPrint:
 18 pages