Optimal-Depth Sorting Networks

TL;DR

This paper proves the optimal depth of sorting networks for 11 to 16 inputs by combining symmetry exploitation and SAT solving, resolving a 40-year-old open problem in the field.

Contribution

It introduces a general technique leveraging symmetry and propositional formulas to establish depth optimality of sorting networks for 11 to 16 inputs.

Findings

Proved optimality of sorting networks for 11-16 inputs.

Demonstrated significant speed improvements over previous methods for n ≤ 10.

Validated existing networks' optimality using SAT solvers.

Abstract

We solve a 40-year-old 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 two-layer 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 off-the-shelf 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.