Optimal Sorting Networks

We settle the optimality of sorting networks given in The Art of Computer Programming vol. 3 more than 40 years ago. The book lists efficient sorting networks with n <= 16 inputs. In this work we give general combinatorial arguments showing that if a sorting network with smaller depth exists then there exists one with a special form. We then construct propositional formulas whose satisfiability is necessary for the existence of such a network. Using a SAT solver we conclude that the listed networks have optimal depth. For n <= 10 inputs where optimality was known previously, our algorithm is four orders of magnitude faster than those in prior work. This is joint work with Jakub Zavodny.