Sorting Networks: to the End and Back Again

TL;DR

This paper explores new properties of the ends of sorting networks to improve bounds on their optimality, leading to the discovery of faster networks for 17 to 20 inputs, including an optimal network for 17 inputs.

Contribution

It introduces out-side-in properties of sorting networks, expanding understanding and enabling the construction of more efficient networks than previously known.

Findings

New parallel sorting networks for 17-20 inputs

Faster networks for 17, 19, and 20 inputs

Optimal network for 17 inputs with minimal layers

Abstract

This paper studies new properties of the front and back ends of a sorting network, and illustrates the utility of these in the search for new bounds on optimal sorting networks. Search focuses first on the "outsides" of the network and then on the inner part. All previous works focus only on properties of the front end of networks and on how to apply these to break symmetries in the search. The new, out-side-in, properties help shed understanding on how sorting networks sort, and facilitate the computation of new bounds on optimal sorting networks. We present new parallel sorting networks for 17 to 20 inputs. For 17, 19, and 20 inputs these networks are faster than the previously known best networks. For 17 inputs, the new sorting network is shown optimal in the sense that no sorting network using less layers exists.