
OptimalDepth Sorting NetworksDaniel Bundala, Michael Codish, Luis CruzFilipe, Peter SchneiderKamp and Jakub Zavodny
Journal of Computer and System Sciences;
(accepted) (?): ?,
2016
Abstract:We solve a 40yearold open problem on the depth optimality of sorting networks. In 1973, Donald E.~Knuth detailed, in Volume~3 of The Art of Computer Programming, sorting networks of the smallest depth known at the time for n =< 16 inputs, quoting optimality for n =< 8. In 1989, Parberry proved the optimality of the networks with 9 =< n =< 10 inputs. In this article, we present a general technique for obtaining such optimality results, and use it to prove the optimality of the remaining open cases of 11 =< n =< 16 inputs. We show how to exploit symmetry to construct a small set of twolayer networks on $n$ inputs such that if there is a sorting network on $n$ inputs of a given depth, then there is one whose first layers are in this set. For each network in the resulting set, we construct a propositional formula whose satisfiability is necessary for the existence of a sorting network of a given depth. Using an offtheshelf SAT solver we show that the sorting networks listed by Knuth are optimal. For n =< 10 inputs, our algorithm is orders of magnitude faster than the prior ones.Available: bibtex entry Related sites: Michael Codish The Department of Computer Science BenGurion University of the Negev PoB 653, BeerSheva, 84105, Israel mcodish@cs.bgu.ac.il
