List of sorting networks

Bert Dobbelaere

Below are the best performing sorting networks known by the author for network sizes of up to 32 inputs, in terms of two metrics:

Provided is a minimum set of networks that each can be the best choice depending on how both criteria are weighted.

If no source is mentioned, the network was generated by the SorterHunter program, which in many cases matches or outperforms pre-existing networks.

For a number of input sizes (n), the used method was able to reduce the upper bound for the minimal network size S(n) compared to the results reported by V. Valsalam & R. Miikkulainen [VM13].
This was the case for 18…23 inputs. For 24…29, 33…48 and 54…56 inputs, a reduction was achieved vs. the Batcher odd-even and Van Voorhis (4,4) merges of smaller networks.

For 25 and 26 inputs, a reduction in depth was achieved vs. the Batcher odd-even merge of smaller networks, lowering the optimal depth upper bound to 13 layers. For 34 inputs, a similar reduction in depth to 15 layers was achieved, and for 42…44 inputs a reduction to 17 layers.

For questions, remarks, or to contribute improved results please contact bert.o.dobbelaere[at]telenet[dot]be.

Sorting network for 2 inputs, 1 CE, 1 layer:

[(0,1)]

Trivial sorting network for two inputs. The only simpler "network" is the one for one input, which uses no CE at all and is represented by a horizontal line. Auto generated Sorting network for 3 inputs, 3 CEs, 3 layers:

[(0,2)]
[(0,1)]
[(1,2)]

Auto generated Sorting network for 4 inputs, 5 CEs, 3 layers:

[(0,2),(1,3)]
[(0,1),(2,3)]
[(1,2)]

Auto generated Sorting network for 5 inputs, 9 CEs, 5 layers:

[(0,3),(1,4)]
[(0,2),(1,3)]
[(0,1),(2,4)]
[(1,2),(3,4)]
[(2,3)]

Auto generated Sorting network for 6 inputs, 12 CEs, 5 layers:

[(0,5),(1,3),(2,4)]
[(1,2),(3,4)]
[(0,3),(2,5)]
[(0,1),(2,3),(4,5)]
[(1,2),(3,4)]

Auto generated Sorting network for 7 inputs, 16 CEs, 6 layers:

[(0,6),(2,3),(4,5)]
[(0,2),(1,4),(3,6)]
[(0,1),(2,5),(3,4)]
[(1,2),(4,6)]
[(2,3),(4,5)]
[(1,2),(3,4),(5,6)]

Auto generated Sorting network for 8 inputs, 19 CEs, 6 layers:

[(0,2),(1,3),(4,6),(5,7)]
[(0,4),(1,5),(2,6),(3,7)]
[(0,1),(2,3),(4,5),(6,7)]
[(2,4),(3,5)]
[(1,4),(3,6)]
[(1,2),(3,4),(5,6)]

Auto generated Sorting network for 9 inputs, 25 CEs, 7 layers:

[(0,3),(1,7),(2,5),(4,8)]
[(0,7),(2,4),(3,8),(5,6)]
[(0,2),(1,3),(4,5),(7,8)]
[(1,4),(3,6),(5,7)]
[(0,1),(2,4),(3,5),(6,8)]
[(2,3),(4,5),(6,7)]
[(1,2),(3,4),(5,6)]

Auto generated Sorting network for 10 inputs, 29 CEs, 8 layers:

[(0,8),(1,9),(2,7),(3,5),(4,6)]
[(0,2),(1,4),(5,8),(7,9)]
[(0,3),(2,4),(5,7),(6,9)]
[(0,1),(3,6),(8,9)]
[(1,5),(2,3),(4,8),(6,7)]
[(1,2),(3,5),(4,6),(7,8)]
[(2,3),(4,5),(6,7)]
[(3,4),(5,6)]

Auto generated Sorting network for 10 inputs, 31 CEs, 7 layers:

[(0,1),(2,5),(3,6),(4,7),(8,9)]
[(0,6),(1,8),(2,4),(3,9),(5,7)]
[(0,2),(1,3),(4,5),(6,8),(7,9)]
[(0,1),(2,7),(3,5),(4,6),(8,9)]
[(1,2),(3,4),(5,6),(7,8)]
[(1,3),(2,4),(5,7),(6,8)]
[(2,3),(4,5),(6,7)]

31 is the optimal size for 7 layers [Fon18] Auto generated Sorting network for 11 inputs, 35 CEs, 8 layers:

[(0,9),(1,6),(2,4),(3,7),(5,8)]
[(0,1),(3,5),(4,10),(6,9),(7,8)]
[(1,3),(2,5),(4,7),(8,10)]
[(0,4),(1,2),(3,7),(5,9),(6,8)]
[(0,1),(2,6),(4,5),(7,8),(9,10)]
[(2,4),(3,6),(5,7),(8,9)]
[(1,2),(3,4),(5,6),(7,8)]
[(2,3),(4,5),(6,7)]

35 is the optimal size [Harder19] (see also [Fon18] for 8 or 9 layers). Auto generated Sorting network for 12 inputs, 39 CEs, 9 layers:

[(0,8),(1,7),(2,6),(3,11),(4,10),(5,9)]
[(0,1),(2,5),(3,4),(6,9),(7,8),(10,11)]
[(0,2),(1,6),(5,10),(9,11)]
[(0,3),(1,2),(4,6),(5,7),(8,11),(9,10)]
[(1,4),(3,5),(6,8),(7,10)]
[(1,3),(2,5),(6,9),(8,10)]
[(2,3),(4,5),(6,7),(8,9)]
[(4,6),(5,7)]
[(3,4),(5,6),(7,8)]

39 is the optimal size [Harder19] (see also [Fon18] for 9 layers). Auto generated Sorting network for 12 inputs, 40 CEs, 8 layers:

[(0,8),(1,7),(2,6),(3,11),(4,10),(5,9)]
[(0,2),(1,4),(3,5),(6,8),(7,10),(9,11)]
[(0,1),(2,9),(4,7),(5,6),(10,11)]
[(1,3),(2,7),(4,9),(8,10)]
[(0,1),(2,3),(4,5),(6,7),(8,9),(10,11)]
[(1,2),(3,5),(6,8),(9,10)]
[(2,4),(3,6),(5,8),(7,9)]
[(1,2),(3,4),(5,6),(7,8),(9,10)]

40 is the optimal size for 8 layers [Fon18]. Auto generated Sorting network for 13 inputs, 45 CEs, 10 layers:

[(0,12),(1,10),(2,9),(3,7),(5,11),(6,8)]
[(1,6),(2,3),(4,11),(7,9),(8,10)]
[(0,4),(1,2),(3,6),(7,8),(9,10),(11,12)]
[(4,6),(5,9),(8,11),(10,12)]
[(0,5),(3,8),(4,7),(6,11),(9,10)]
[(0,1),(2,5),(6,9),(7,8),(10,11)]
[(1,3),(2,4),(5,6),(9,10)]
[(1,2),(3,4),(5,7),(6,8)]
[(2,3),(4,5),(6,7),(8,9)]
[(3,4),(5,6)]

Auto generated Sorting network for 13 inputs, 46 CEs, 9 layers:

[(0,11),(1,7),(2,4),(3,5),(8,9),(10,12)]
[(0,2),(3,6),(4,12),(5,7),(8,10)]
[(0,8),(1,3),(2,5),(4,9),(6,11),(7,12)]
[(0,1),(2,10),(3,8),(4,6),(9,11)]
[(1,3),(2,4),(5,10),(6,8),(7,9),(11,12)]
[(1,2),(3,4),(5,8),(6,9),(7,10)]
[(2,3),(4,7),(5,6),(8,11),(9,10)]
[(4,5),(6,7),(8,9),(10,11)]
[(3,4),(5,6),(7,8),(9,10)]

Auto generated Sorting network for 14 inputs, 51 CEs, 10 layers:

[(0,1),(2,3),(4,5),(6,7),(8,9),(10,11),(12,13)]
[(0,2),(1,3),(4,8),(5,9),(10,12),(11,13)]
[(0,4),(1,2),(3,7),(5,8),(6,10),(9,13),(11,12)]
[(0,6),(1,5),(3,9),(4,10),(7,13),(8,12)]
[(2,10),(3,11),(4,6),(7,9)]
[(1,3),(2,8),(5,11),(6,7),(10,12)]
[(1,4),(2,6),(3,5),(7,11),(8,10),(9,12)]
[(2,4),(3,6),(5,8),(7,10),(9,11)]
[(3,4),(5,6),(7,8),(9,10)]
[(6,7)]

Front-concentrated, only single element in last layer Auto generated Sorting network for 14 inputs, 52 CEs, 9 layers:

[(0,1),(2,3),(4,5),(6,7),(8,9),(10,11),(12,13)]
[(0,2),(1,3),(4,8),(5,9),(10,12),(11,13)]
[(0,10),(1,6),(2,11),(3,13),(5,8),(7,12)]
[(1,4),(2,8),(3,6),(5,11),(7,10),(9,12)]
[(0,1),(3,9),(4,10),(5,7),(6,8),(12,13)]
[(1,5),(2,4),(3,7),(6,10),(8,12),(9,11)]
[(1,2),(3,5),(4,6),(7,9),(8,10),(11,12)]
[(2,3),(4,5),(6,7),(8,9),(10,11)]
[(3,4),(5,6),(7,8),(9,10)]

Symmetric variant replacing asymmetric predecessor, was surprisingly hard to find using SorterHunter. Auto generated Sorting network for 15 inputs, 56 CEs, 10 layers:

[(1,2),(3,10),(4,14),(5,8),(6,13),(7,12),(9,11)]
[(0,14),(1,5),(2,8),(3,7),(6,9),(10,12),(11,13)]
[(0,7),(1,6),(2,9),(4,10),(5,11),(8,13),(12,14)]
[(0,6),(2,4),(3,5),(7,11),(8,10),(9,12),(13,14)]
[(0,3),(1,2),(4,7),(5,9),(6,8),(10,11),(12,13)]
[(0,1),(2,3),(4,6),(7,9),(10,12),(11,13)]
[(1,2),(3,5),(8,10),(11,12)]
[(3,4),(5,6),(7,8),(9,10)]
[(2,3),(4,5),(6,7),(8,9),(10,11)]
[(5,6),(7,8)]

Auto generated Sorting network for 15 inputs, 57 CEs, 9 layers:

[(0,6),(1,10),(2,14),(3,9),(4,12),(5,13),(7,11)]
[(0,7),(2,5),(3,4),(6,11),(8,10),(9,12),(13,14)]
[(1,13),(2,3),(4,6),(5,9),(7,8),(10,14),(11,12)]
[(0,3),(1,4),(5,7),(6,13),(8,9),(10,11),(12,14)]
[(0,2),(1,5),(3,8),(4,6),(7,10),(9,11),(12,13)]
[(0,1),(2,5),(3,10),(4,8),(6,7),(9,12),(11,13)]
[(1,2),(3,4),(5,6),(7,9),(8,10),(11,12)]
[(3,5),(4,6),(7,8),(9,10)]
[(2,3),(4,5),(6,7),(8,9),(10,11)]

Auto generated Sorting network for 16 inputs, 60 CEs, 10 layers:

[(0,13),(1,12),(2,15),(3,14),(4,8),(5,6),(7,11),(9,10)]
[(0,5),(1,7),(2,9),(3,4),(6,13),(8,14),(10,15),(11,12)]
[(0,1),(2,3),(4,5),(6,8),(7,9),(10,11),(12,13),(14,15)]
[(0,2),(1,3),(4,10),(5,11),(6,7),(8,9),(12,14),(13,15)]
[(1,2),(3,12),(4,6),(5,7),(8,10),(9,11),(13,14)]
[(1,4),(2,6),(5,8),(7,10),(9,13),(11,14)]
[(2,4),(3,6),(9,12),(11,13)]
[(3,5),(6,8),(7,9),(10,12)]
[(3,4),(5,6),(7,8),(9,10),(11,12)]
[(6,7),(8,9)]

Size and depth match the handcrafted network of M.W. Green, 1969 [TAOCPv3]. It is remarkable that the last three layers (found by independent computer search) are identical to those in Green's network. Auto generated Sorting network for 16 inputs, 61 CEs, 9 layers:

[(0,5),(1,4),(2,12),(3,13),(6,7),(8,9),(10,15),(11,14)]
[(0,2),(1,10),(3,6),(4,7),(5,14),(8,11),(9,12),(13,15)]
[(0,8),(1,3),(2,11),(4,13),(5,9),(6,10),(7,15),(12,14)]
[(0,1),(2,4),(3,8),(5,6),(7,12),(9,10),(11,13),(14,15)]
[(1,3),(2,5),(4,8),(6,9),(7,11),(10,13),(12,14)]
[(1,2),(3,5),(4,11),(6,8),(7,9),(10,12),(13,14)]
[(2,3),(4,5),(6,7),(8,9),(10,11),(12,13)]
[(4,6),(5,7),(8,10),(9,11)]
[(3,4),(5,6),(7,8),(9,10),(11,12)]

Auto generated Sorting network for 17 inputs, 71 CEs, 12 layers:

[(0,11),(1,15),(2,10),(3,5),(4,6),(8,12),(9,16),(13,14)]
[(0,6),(1,13),(2,8),(4,14),(5,15),(7,11)]
[(0,8),(3,7),(4,9),(6,16),(10,11),(12,14)]
[(0,2),(1,4),(5,6),(7,13),(8,9),(10,12),(11,14),(15,16)]
[(0,3),(2,5),(6,11),(7,10),(9,13),(12,15),(14,16)]
[(0,1),(3,4),(5,10),(6,9),(7,8),(11,15),(13,14)]
[(1,2),(3,7),(4,8),(6,12),(11,13),(14,15)]
[(1,3),(2,7),(4,5),(9,11),(10,12),(13,14)]
[(2,3),(4,6),(5,7),(8,10)]
[(3,4),(6,8),(7,9),(10,12)]
[(5,6),(7,8),(9,10),(11,12)]
[(4,5),(6,7),(8,9),(10,11),(12,13)]

Size matches result from Valsalam & Miikkulainen [VM13]. Depth reduced from 17 to 12. Auto generated Sorting network for 17 inputs, 72 CEs, 11 layers:

[(0,8),(1,3),(2,6),(4,15),(5,13),(9,16),(10,11),(12,14)]
[(0,2),(1,4),(3,15),(5,8),(6,14),(7,9),(13,16)]
[(2,13),(3,11),(5,10),(6,9),(7,12),(8,15),(14,16)]
[(0,7),(1,5),(3,6),(4,10),(8,14),(9,11),(12,13),(15,16)]
[(0,1),(2,6),(4,12),(5,7),(9,14),(10,13),(11,15)]
[(1,4),(2,3),(6,11),(7,12),(8,9),(13,14),(15,16)]
[(2,5),(3,8),(7,10),(9,12),(11,13),(14,15)]
[(1,2),(3,7),(4,5),(6,9),(8,10),(13,14)]
[(3,4),(5,7),(6,8),(9,11),(10,12)]
[(2,4),(5,6),(7,8),(9,10),(11,12)]
[(2,3),(4,5),(6,7),(8,9),(10,11),(12,13)]

Auto generated Sorting network for 17 inputs, 74 CEs, 10 layers:

[(1,2),(3,4),(5,6),(7,8),(9,10),(11,12),(13,14),(15,16)]
[(1,3),(2,4),(5,7),(6,8),(9,11),(10,12),(13,15),(14,16)]
[(1,5),(2,6),(3,7),(4,8),(9,13),(10,14),(11,15),(12,16)]
[(0,3),(1,13),(2,10),(4,7),(5,11),(6,12),(8,9),(14,15)]
[(0,13),(1,8),(2,5),(3,6),(4,14),(7,15),(9,16),(10,11)]
[(0,1),(2,8),(3,4),(5,10),(6,13),(7,11),(12,14)]
[(1,5),(3,8),(4,10),(6,7),(9,12),(11,13)]
[(1,2),(4,6),(5,8),(7,10),(9,11),(12,14),(13,15)]
[(2,3),(4,5),(6,8),(7,9),(10,11),(12,13),(14,15)]
[(3,4),(5,6),(7,8),(9,10),(11,12),(13,14),(15,16)]

This network was obtained by performing a size optimization of the network found by Ehlers&Müller (see [EM14]) Auto generated Sorting network for 18 inputs, 77 CEs, 12 layers:

[(0,1),(2,3),(4,5),(6,7),(8,9),(10,11),(12,13),(14,15),(16,17)]
[(0,2),(1,3),(4,12),(5,13),(6,8),(9,11),(14,16),(15,17)]
[(0,14),(1,16),(2,15),(3,17)]
[(0,6),(1,10),(2,9),(7,16),(8,15),(11,17)]
[(1,4),(3,9),(5,7),(8,14),(10,12),(13,16)]
[(0,1),(2,5),(3,13),(4,14),(7,9),(8,10),(12,15),(16,17)]
[(1,2),(3,5),(4,6),(11,13),(12,14),(15,16)]
[(4,8),(5,12),(6,10),(7,11),(9,13)]
[(1,4),(2,8),(3,6),(5,7),(9,15),(10,12),(11,14),(13,16)]
[(2,4),(5,8),(6,10),(7,11),(9,12),(13,15)]
[(3,5),(6,8),(7,10),(9,11),(12,14)]
[(3,4),(5,6),(7,8),(9,10),(11,12),(13,14)]

Improved size over result (78) reported in [VM13]. Auto generated Sorting network for 18 inputs, 78 CEs, 11 layers:

[(0,6),(1,10),(2,15),(3,5),(4,9),(7,16),(8,13),(11,17),(12,14)]
[(0,12),(1,4),(3,11),(5,17),(6,14),(7,8),(9,10),(13,16)]
[(1,13),(2,7),(4,16),(6,9),(8,11),(10,15)]
[(0,1),(2,3),(4,12),(5,13),(7,9),(8,10),(14,15),(16,17)]
[(0,2),(1,11),(3,4),(5,7),(6,16),(10,12),(13,14),(15,17)]
[(1,8),(4,10),(5,6),(7,13),(9,16),(11,12)]
[(1,3),(2,5),(4,7),(6,8),(9,11),(10,13),(12,15),(14,16)]
[(1,2),(3,5),(4,6),(7,9),(8,10),(11,13),(12,14),(15,16)]
[(2,3),(5,8),(6,7),(9,12),(10,11),(14,15)]
[(3,4),(5,6),(7,8),(9,10),(11,12),(13,14)]
[(4,5),(6,7),(8,9),(10,11),(12,13)]

Matches depth of 11 layers reported in [Baddar09] while reducing size from 84 to 78. Auto generated Sorting network for 19 inputs, 85 CEs, 12 layers:

[(0,12),(1,4),(2,8),(3,5),(6,17),(7,11),(9,14),(10,13),(15,16)]
[(0,2),(1,7),(3,6),(4,11),(5,17),(8,12),(10,15),(13,16),(14,18)]
[(3,10),(4,14),(5,15),(6,13),(7,9),(11,17),(16,18)]
[(0,7),(1,10),(4,6),(9,15),(11,16),(12,17),(13,14)]
[(0,3),(2,6),(5,7),(8,11),(12,16)]
[(1,8),(2,9),(3,4),(6,15),(7,13),(10,11),(12,18)]
[(1,3),(2,5),(6,9),(7,12),(8,10),(11,14),(17,18)]
[(0,1),(2,3),(4,8),(6,10),(9,12),(14,15),(16,17)]
[(1,2),(5,8),(6,7),(9,11),(10,13),(14,16),(15,17)]
[(3,6),(4,5),(7,9),(8,10),(11,12),(13,14),(15,16)]
[(3,4),(5,6),(7,8),(9,10),(11,13),(12,14)]
[(2,3),(4,5),(6,7),(8,9),(10,11),(12,13),(14,15)]

Improved size over result (86) reported by Valsalam & Miikkulainen [VM13] Auto generated Sorting network for 19 inputs, 87 CEs, 11 layers:

[(0,1),(2,3),(4,5),(6,7),(8,10),(11,12),(13,14),(15,16),(17,18)]
[(0,2),(1,3),(4,6),(5,7),(8,9),(11,13),(12,14),(15,17),(16,18)]
[(0,4),(1,5),(2,6),(3,7),(9,10),(11,15),(12,16),(13,17),(14,18)]
[(0,11),(1,8),(2,13),(3,17),(4,10),(5,6),(9,16),(12,15)]
[(1,2),(3,13),(4,12),(5,14),(6,16),(7,10),(8,15)]
[(0,1),(2,11),(3,9),(5,12),(6,15),(7,13),(10,18),(14,17)]
[(1,4),(3,8),(5,11),(6,9),(7,12),(10,13),(14,15),(16,17)]
[(2,4),(3,5),(6,7),(8,11),(9,12),(10,14),(13,15)]
[(2,3),(4,5),(6,8),(7,9),(10,11),(12,14),(13,16),(15,17)]
[(1,2),(4,6),(5,8),(7,10),(9,11),(12,13),(14,16)]
[(3,4),(5,6),(7,8),(9,10),(11,12),(13,14),(15,16)]

Auto generated Sorting network for 20 inputs, 91 CEs, 12 layers:

[(0,3),(1,7),(2,5),(4,8),(6,9),(10,13),(11,15),(12,18),(14,17),(16,19)]
[(0,14),(1,11),(2,16),(3,17),(4,12),(5,19),(6,10),(7,15),(8,18),(9,13)]
[(0,4),(1,2),(3,8),(5,7),(11,16),(12,14),(15,19),(17,18)]
[(1,6),(2,12),(3,5),(4,11),(7,17),(8,15),(13,18),(14,16)]
[(0,1),(2,6),(7,10),(9,12),(13,17),(18,19)]
[(1,6),(5,9),(7,11),(8,12),(10,14),(13,18)]
[(3,5),(4,7),(8,10),(9,11),(12,15),(14,16)]
[(1,3),(2,4),(5,7),(6,10),(9,13),(12,14),(15,17),(16,18)]
[(1,2),(3,4),(6,7),(8,9),(10,11),(12,13),(15,16),(17,18)]
[(2,3),(4,6),(5,8),(7,9),(10,12),(11,14),(13,15),(16,17)]
[(4,5),(6,8),(7,10),(9,12),(11,13),(14,15)]
[(3,4),(5,6),(7,8),(9,10),(11,12),(13,14),(15,16)]

Improved size over result (92) reported by Valsalam & Miikkulainen [VM13] Auto generated Sorting network for 20 inputs, 93 CEs, 11 layers:

[(0,12),(1,13),(2,14),(3,15),(4,16),(5,17),(6,18),(7,19),(8,10),(9,11)]
[(0,2),(1,3),(4,6),(5,7),(8,9),(10,11),(12,14),(13,15),(16,18),(17,19)]
[(0,1),(2,3),(4,5),(6,7),(12,13),(14,15),(16,17),(18,19)]
[(0,4),(1,12),(2,16),(3,17),(5,8),(6,9),(7,18),(10,13),(11,14),(15,19)]
[(1,6),(3,10),(4,5),(7,11),(8,12),(9,16),(13,18),(14,15)]
[(0,4),(2,8),(3,9),(6,7),(10,16),(11,17),(12,13),(15,19)]
[(1,4),(3,6),(5,8),(7,10),(9,12),(11,14),(13,16),(15,18)]
[(2,3),(4,5),(6,8),(7,9),(10,12),(11,13),(14,15),(16,17)]
[(2,4),(3,6),(5,7),(8,10),(9,11),(12,14),(13,16),(15,17)]
[(1,2),(3,5),(6,7),(8,9),(10,11),(12,13),(14,16),(17,18)]
[(3,4),(5,6),(7,8),(9,10),(11,12),(13,14),(15,16)]

First 3 layers inspired by [EM14] Auto generated Sorting network for 21 inputs, 99 CEs, 15 layers:

[(0,1),(2,3),(4,5),(6,7),(8,9),(10,11),(12,13),(14,15),(16,17),(18,19)]
[(0,2),(1,3),(4,6),(5,7),(8,10),(9,11),(12,14),(13,15),(16,18),(17,19)]
[(0,8),(1,9),(2,10),(3,11),(4,12),(5,13),(6,14),(7,15)]
[(0,4),(1,5),(3,7),(6,20),(8,12),(9,13),(10,14),(15,19)]
[(2,6),(3,18),(7,20)]
[(2,16),(3,6),(5,18),(7,17),(11,20)]
[(0,2),(3,8),(6,12),(7,10),(9,16),(11,15),(13,17),(14,18),(19,20)]
[(1,7),(2,3),(4,9),(10,11),(13,16),(15,18),(17,19)]
[(1,4),(5,10),(6,13),(7,8),(11,14),(12,16),(15,17),(18,19)]
[(1,2),(3,4),(5,6),(10,12),(11,13),(14,16),(17,18)]
[(2,3),(4,5),(6,9),(10,11),(12,13),(14,15),(16,17)]
[(6,7),(8,9),(15,16)]
[(4,6),(7,8),(9,12),(13,15)]
[(3,4),(5,7),(8,10),(9,11),(12,14)]
[(5,6),(7,8),(9,10),(11,12),(13,14)]

Improved size over result (102) reported by Valsalam & Miikkulainen [VM13], breaking 100 element barrier. Auto generated Sorting network for 21 inputs, 100 CEs, 12 layers:

[(0,7),(1,10),(3,5),(4,8),(6,13),(9,19),(11,14),(12,17),(15,16),(18,20)]
[(0,11),(1,15),(2,12),(3,4),(5,8),(6,9),(7,14),(10,16),(13,19),(17,20)]
[(0,6),(1,3),(2,18),(4,15),(5,10),(8,16),(11,17),(12,13),(14,20)]
[(2,6),(5,12),(7,18),(8,14),(9,11),(10,17),(13,19),(16,20)]
[(1,2),(4,7),(5,9),(6,17),(10,13),(11,12),(14,19),(15,18)]
[(0,2),(3,6),(4,5),(7,10),(8,11),(9,15),(12,16),(13,18),(14,17),(19,20)]
[(0,1),(2,3),(5,9),(6,12),(7,8),(11,14),(13,15),(16,19),(17,18)]
[(1,2),(3,9),(6,13),(10,11),(12,15),(16,17),(18,19)]
[(1,4),(2,5),(3,7),(6,10),(8,9),(11,12),(13,14),(17,18)]
[(2,4),(5,6),(7,8),(9,11),(10,13),(12,15),(14,16)]
[(3,4),(5,7),(6,8),(9,10),(11,13),(12,14),(15,16)]
[(4,5),(6,7),(8,9),(10,11),(12,13),(14,15),(16,17)]

Auto generated Sorting network for 22 inputs, 106 CEs, 13 layers:

[(0,1),(2,3),(4,5),(6,7),(8,9),(10,11),(12,13),(14,15),(16,17),(18,19),(20,21)]
[(0,2),(1,3),(4,6),(5,7),(8,12),(9,13),(14,16),(15,17),(18,20),(19,21)]
[(0,4),(1,5),(2,6),(3,7),(8,10),(9,12),(11,13),(14,18),(15,19),(16,20),(17,21)]
[(0,14),(1,15),(2,18),(3,19),(4,16),(5,17),(6,20),(7,21),(9,11),(10,12)]
[(2,8),(3,11),(6,9),(10,18),(12,15),(13,19)]
[(0,2),(1,10),(3,16),(5,18),(6,14),(7,15),(8,12),(9,13),(11,20),(19,21)]
[(2,6),(3,10),(4,8),(5,12),(9,16),(11,18),(13,17),(15,19)]
[(1,4),(7,13),(8,14),(9,12),(17,20)]
[(1,2),(3,8),(4,6),(7,11),(10,14),(13,18),(15,17),(19,20)]
[(2,4),(5,10),(7,9),(11,16),(12,14),(17,19)]
[(5,6),(7,8),(9,11),(10,12),(13,14),(15,16)]
[(3,5),(6,7),(8,10),(9,12),(11,13),(14,15),(16,18)]
[(3,4),(5,6),(7,8),(9,10),(11,12),(13,14),(15,16),(17,18)]

Improved size over result (108) from Valsalam & Miikkulainen [VM13]. Auto generated Sorting network for 22 inputs, 107 CEs, 12 layers:

[(0,1),(2,3),(4,5),(6,7),(8,9),(10,11),(12,13),(14,15),(16,17),(18,19),(20,21)]
[(0,2),(1,3),(4,6),(5,7),(8,10),(11,13),(14,16),(15,17),(18,20),(19,21)]
[(0,4),(1,5),(2,6),(3,7),(8,12),(9,13),(14,18),(15,19),(16,20),(17,21)]
[(0,14),(1,15),(2,18),(3,19),(4,16),(5,17),(6,20),(7,21),(9,11),(10,12)]
[(0,8),(2,10),(4,14),(5,12),(6,15),(7,17),(9,16),(11,19),(13,21)]
[(1,9),(2,4),(3,16),(5,18),(6,10),(7,13),(8,14),(11,15),(12,20),(17,19)]
[(1,8),(3,11),(4,5),(7,12),(9,14),(10,18),(13,20),(16,17)]
[(1,2),(3,5),(4,8),(6,9),(7,11),(10,14),(12,15),(13,17),(16,18),(19,20)]
[(2,4),(3,6),(5,9),(7,10),(11,14),(12,16),(15,18),(17,19)]
[(3,4),(5,7),(6,8),(9,11),(10,12),(13,15),(14,16),(17,18)]
[(5,6),(7,8),(9,10),(11,12),(13,14),(15,16)]
[(4,5),(6,7),(8,9),(10,11),(12,13),(14,15),(16,17)]

Auto generated Sorting network for 23 inputs, 114 CEs, 14 layers:

[(0,1),(2,3),(4,5),(6,7),(8,9),(10,11),(12,13),(14,15),(16,17),(18,19),(20,21)]
[(0,2),(1,3),(4,6),(5,7),(8,10),(9,11),(12,14),(13,15),(16,18),(17,19),(21,22)]
[(0,4),(1,5),(2,6),(3,7),(8,12),(9,13),(10,14),(11,15),(17,21),(18,20),(19,22)]
[(0,8),(1,9),(2,10),(3,11),(4,12),(5,13),(6,14),(7,15)]
[(1,2),(5,18),(7,19),(9,16),(10,21),(12,20),(15,22)]
[(5,9),(6,7),(10,18),(11,21),(12,17),(13,20),(14,15)]
[(3,17),(6,16),(7,14),(8,12),(15,19),(20,21)]
[(3,4),(5,8),(6,10),(9,12),(13,16),(14,15),(17,18),(19,21)]
[(0,5),(1,8),(2,12),(3,9),(4,10),(7,13),(11,17),(14,16),(18,20)]
[(2,6),(3,5),(4,8),(7,11),(10,12),(13,18),(14,17),(15,20)]
[(1,3),(2,5),(6,9),(7,10),(11,13),(12,14),(15,18),(16,17),(19,20)]
[(2,3),(4,6),(8,9),(11,12),(13,14),(15,16),(17,19)]
[(3,4),(5,6),(7,8),(9,10),(12,13),(14,15),(17,18)]
[(4,5),(6,7),(8,9),(10,11),(16,17)]

Reduction in size vs. 118 reported in [Baddar09] Auto generated Sorting network for 23 inputs, 115 CEs, 13 layers:

[(0,20),(1,12),(2,16),(4,6),(5,10),(7,21),(8,14),(9,15),(11,22),(13,18),(17,19)]
[(0,3),(1,11),(2,7),(4,17),(5,13),(6,19),(8,9),(10,18),(12,22),(14,15),(16,21)]
[(0,1),(2,4),(3,12),(5,8),(6,9),(7,10),(11,20),(13,16),(14,17),(15,18),(19,21)]
[(2,5),(4,8),(6,11),(7,14),(9,16),(12,17),(15,19),(18,21)]
[(1,8),(3,14),(4,7),(9,20),(10,12),(11,13),(15,22),(16,19)]
[(0,7),(1,5),(3,4),(6,11),(8,15),(9,14),(10,13),(12,17),(18,22),(19,20)]
[(0,2),(1,6),(4,7),(5,9),(8,10),(13,15),(14,18),(16,19),(17,22),(20,21)]
[(2,3),(4,5),(6,8),(7,9),(10,11),(12,13),(14,16),(15,17),(18,19),(21,22)]
[(1,2),(3,6),(4,10),(7,8),(9,11),(12,14),(13,19),(15,16),(17,20)]
[(2,3),(5,10),(6,7),(8,9),(13,18),(14,15),(16,17),(20,21)]
[(3,4),(5,7),(10,12),(11,13),(16,18),(19,20)]
[(4,6),(8,10),(9,12),(11,14),(13,15),(17,19)]
[(5,6),(7,8),(9,10),(11,12),(13,14),(15,16),(17,18)]

Obtained by removing highest input from 24 inputs case with 13 layers. Auto generated Sorting network for 23 inputs, 116 CEs, 12 layers:

[(0,1),(2,3),(4,5),(6,7),(8,9),(10,11),(12,13),(14,15),(16,17),(18,19),(20,21)]
[(0,2),(1,3),(4,6),(5,7),(8,10),(9,11),(12,14),(13,15),(17,19),(18,20),(21,22)]
[(0,4),(1,5),(2,6),(3,7),(8,12),(9,13),(10,14),(11,15),(16,21),(17,22)]
[(1,10),(2,9),(3,11),(6,19),(12,17),(14,22),(16,18),(20,21)]
[(0,16),(1,2),(3,21),(4,17),(5,14),(6,13),(7,22),(9,18),(10,20),(15,19)]
[(1,10),(2,9),(3,17),(4,12),(5,18),(6,20),(7,15),(8,16),(11,14),(13,21),(19,22)]
[(0,8),(1,4),(2,10),(3,9),(5,6),(11,21),(12,16),(13,20),(14,15),(17,18)]
[(2,8),(3,5),(4,12),(6,9),(7,11),(10,16),(13,17),(15,21),(18,20)]
[(1,2),(4,8),(5,10),(6,12),(7,13),(9,16),(11,18),(14,17),(15,19)]
[(2,4),(3,5),(6,8),(7,9),(10,12),(11,13),(14,16),(15,20),(17,18),(19,21)]
[(3,6),(5,8),(7,10),(9,12),(11,14),(13,16),(15,17),(18,20)]
[(3,4),(5,6),(7,8),(9,10),(11,12),(13,14),(15,16),(17,18),(19,20)]

Auto generated Sorting network for 24 inputs, 120 CEs, 13 layers:

[(0,20),(1,12),(2,16),(3,23),(4,6),(5,10),(7,21),(8,14),(9,15),(11,22),(13,18),(17,19)]
[(0,3),(1,11),(2,7),(4,17),(5,13),(6,19),(8,9),(10,18),(12,22),(14,15),(16,21),(20,23)]
[(0,1),(2,4),(3,12),(5,8),(6,9),(7,10),(11,20),(13,16),(14,17),(15,18),(19,21),(22,23)]
[(2,5),(4,8),(6,11),(7,14),(9,16),(12,17),(15,19),(18,21)]
[(1,8),(3,14),(4,7),(9,20),(10,12),(11,13),(15,22),(16,19)]
[(0,7),(1,5),(3,4),(6,11),(8,15),(9,14),(10,13),(12,17),(16,23),(18,22),(19,20)]
[(0,2),(1,6),(4,7),(5,9),(8,10),(13,15),(14,18),(16,19),(17,22),(21,23)]
[(2,3),(4,5),(6,8),(7,9),(10,11),(12,13),(14,16),(15,17),(18,19),(20,21)]
[(1,2),(3,6),(4,10),(7,8),(9,11),(12,14),(13,19),(15,16),(17,20),(21,22)]
[(2,3),(5,10),(6,7),(8,9),(13,18),(14,15),(16,17),(20,21)]
[(3,4),(5,7),(10,12),(11,13),(16,18),(19,20)]
[(4,6),(8,10),(9,12),(11,14),(13,15),(17,19)]
[(5,6),(7,8),(9,10),(11,12),(13,14),(15,16),(17,18)]

Reduction in size vs. 123 reported in [Baddar09] Auto generated Sorting network for 24 inputs, 122 CEs, 12 layers:

[(0,1),(2,3),(4,5),(6,7),(8,9),(10,11),(12,13),(14,15),(16,17),(18,19),(20,21),(22,23)]
[(0,2),(1,3),(4,6),(5,7),(8,10),(9,11),(12,14),(13,15),(16,18),(17,19),(20,22),(21,23)]
[(0,4),(1,5),(2,6),(3,7),(8,12),(9,13),(10,14),(11,15),(16,20),(17,21),(18,22),(19,23)]
[(0,16),(1,18),(2,17),(3,19),(4,20),(5,22),(6,21),(7,23),(9,10),(13,14)]
[(2,10),(3,11),(5,18),(6,14),(7,15),(8,16),(9,17),(12,20),(13,21)]
[(0,8),(1,9),(2,12),(3,20),(4,16),(5,13),(6,17),(7,19),(10,18),(11,21),(14,22),(15,23)]
[(1,8),(3,16),(4,12),(5,10),(6,9),(7,20),(11,19),(13,18),(14,17),(15,22)]
[(2,4),(3,5),(7,13),(9,12),(10,16),(11,14),(18,20),(19,21)]
[(1,2),(4,8),(5,9),(6,10),(7,11),(12,16),(13,17),(14,18),(15,19),(21,22)]
[(2,4),(3,8),(5,6),(7,9),(10,12),(11,13),(14,16),(15,20),(17,18),(19,21)]
[(3,5),(6,8),(7,10),(9,12),(11,14),(13,16),(15,17),(18,20)]
[(3,4),(5,6),(7,8),(9,10),(11,12),(13,14),(15,16),(17,18),(19,20)]

Auto generated Sorting network for 25 inputs, 130 CEs, 15 layers:

[(0,1),(2,3),(4,5),(6,7),(8,9),(10,11),(12,13),(14,15),(16,17),(18,19),(20,21),(22,23)]
[(0,2),(1,3),(4,6),(5,7),(8,10),(9,11),(12,14),(13,15),(16,18),(17,19),(21,22),(23,24)]
[(0,4),(1,5),(2,6),(3,7),(8,12),(9,13),(10,14),(11,15),(18,21),(20,23),(22,24)]
[(0,8),(1,9),(2,10),(3,11),(4,12),(5,13),(6,14),(7,15),(16,20),(17,22),(19,24),(21,23)]
[(1,18),(3,21),(5,23),(6,19),(11,14),(15,24)]
[(1,16),(3,17),(6,9),(7,11),(13,19),(14,23)]
[(0,1),(2,16),(3,8),(7,20),(10,13),(11,22),(15,23)]
[(1,2),(5,10),(7,18),(11,21),(15,20),(19,22)]
[(4,7),(5,6),(9,18),(10,17),(11,12),(13,21),(14,15),(19,20),(22,23)]
[(3,4),(7,8),(9,10),(11,16),(12,17),(13,18),(19,21),(20,22)]
[(1,3),(2,4),(5,11),(6,16),(7,9),(8,10),(12,13),(14,19),(15,18)]
[(2,3),(5,7),(6,9),(8,11),(10,16),(12,14),(15,17)]
[(3,5),(4,6),(7,8),(9,11),(10,12),(13,14),(15,16),(17,18)]
[(4,7),(6,8),(9,10),(11,12),(13,15),(14,16),(17,19),(18,21)]
[(4,5),(6,7),(8,9),(10,11),(12,13),(14,15),(16,17),(18,19),(20,21)]

Improved size vs. Batcher odd-even merge. Auto generated Sorting network for 25 inputs, 131 CEs, 13 layers:

[(0,1),(2,3),(4,5),(6,7),(8,9),(10,11),(12,13),(14,15),(16,17),(18,19),(20,21),(22,23)]
[(0,2),(1,3),(4,6),(5,7),(8,10),(9,11),(12,14),(13,15),(16,18),(17,19),(20,22),(21,24)]
[(0,4),(1,5),(2,6),(3,7),(8,12),(9,13),(10,14),(11,15),(16,20),(21,22),(23,24)]
[(0,8),(1,12),(2,10),(3,14),(4,9),(5,13),(6,11),(7,15),(17,22),(18,21),(19,24)]
[(1,18),(3,9),(5,17),(6,20),(7,13),(11,14),(12,22),(15,24),(21,23)]
[(1,16),(3,12),(5,21),(6,18),(7,11),(10,17),(14,23),(19,20)]
[(0,1),(2,5),(4,16),(6,8),(7,18),(9,21),(10,14),(11,13),(12,19),(15,23),(20,22)]
[(1,2),(3,5),(4,6),(7,9),(8,12),(10,16),(11,20),(13,22),(14,17),(15,18),(19,21)]
[(1,4),(2,6),(3,7),(5,9),(8,10),(11,14),(12,16),(13,17),(15,19),(18,20),(22,23)]
[(2,4),(3,8),(5,10),(7,12),(9,16),(11,15),(13,19),(14,21),(17,18),(20,22)]
[(3,4),(5,8),(6,7),(9,12),(10,11),(13,16),(14,15),(17,19),(18,21)]
[(5,6),(7,8),(9,10),(11,12),(13,14),(15,16),(17,18),(20,21)]
[(4,5),(6,7),(8,9),(10,11),(12,13),(14,15),(16,17),(18,19)]

Improved size and depth over obtainable metrics by Batcher odd-even merge. The existence of a 13 layer network was conjectured in the conclusion of [Ehlers17]. Auto generated Sorting network for 26 inputs, 138 CEs, 15 layers:

[(0,1),(2,3),(4,5),(6,7),(8,9),(10,11),(12,13),(14,15),(16,17),(18,19),(20,21),(22,23),(24,25)]
[(0,2),(1,3),(4,6),(5,7),(8,10),(9,11),(14,16),(15,17),(18,20),(19,21),(22,24),(23,25)]
[(0,4),(1,6),(2,5),(3,7),(8,14),(9,16),(10,15),(11,17),(18,22),(19,24),(20,23),(21,25)]
[(0,18),(1,19),(2,20),(3,21),(4,22),(5,23),(6,24),(7,25),(9,12),(13,16)]
[(3,11),(8,9),(10,13),(12,15),(14,22),(16,17)]
[(0,8),(1,9),(2,14),(6,12),(7,15),(10,18),(11,23),(13,19),(16,24),(17,25)]
[(1,2),(3,18),(4,8),(7,22),(17,21),(23,24)]
[(3,14),(4,10),(5,18),(7,20),(8,13),(11,22),(12,17),(15,21)]
[(1,4),(5,6),(7,9),(8,10),(15,17),(16,18),(19,20),(21,24)]
[(2,5),(3,10),(6,14),(9,13),(11,19),(12,16),(15,22),(20,23)]
[(2,8),(5,7),(6,9),(11,12),(13,14),(16,19),(17,23),(18,20)]
[(2,4),(3,5),(6,11),(7,10),(9,16),(12,13),(14,19),(15,18),(20,22),(21,23)]
[(3,4),(5,8),(6,7),(9,11),(10,12),(13,15),(14,16),(17,20),(18,19),(21,22)]
[(5,6),(7,8),(9,10),(11,12),(13,14),(15,16),(17,18),(19,20)]
[(4,5),(6,7),(8,9),(10,11),(12,13),(14,15),(16,17),(18,19),(20,21)]

Improved size over Batcher odd-even merge Auto generated Sorting network for 26 inputs, 139 CEs, 14 layers:

[(0,1),(2,3),(4,21),(5,6),(7,8),(9,10),(11,12),(13,14),(15,16),(17,18),(19,20),(22,23),(24,25)]
[(0,24),(1,25),(3,21),(4,22),(5,7),(6,8),(9,11),(10,12),(13,15),(14,16),(17,19),(18,20)]
[(0,4),(2,22),(3,23),(5,9),(6,10),(7,11),(8,12),(13,17),(14,18),(15,19),(16,20),(21,25)]
[(0,2),(1,3),(4,21),(5,13),(6,14),(7,15),(8,16),(9,17),(10,18),(11,19),(12,20),(22,24),(23,25)]
[(0,5),(1,7),(2,4),(3,10),(15,22),(18,24),(20,25),(21,23)]
[(2,9),(3,7),(4,14),(5,13),(6,15),(10,19),(11,21),(12,20),(16,23),(18,22)]
[(1,5),(2,6),(3,15),(7,17),(8,18),(10,22),(11,13),(12,14),(19,23),(20,24)]
[(1,2),(4,7),(5,11),(8,9),(10,12),(13,15),(14,20),(16,17),(18,21),(23,24)]
[(3,8),(4,6),(7,16),(9,18),(10,13),(12,15),(17,22),(19,21)]
[(4,5),(6,11),(7,9),(14,19),(16,18),(20,21)]
[(2,4),(3,6),(7,10),(8,11),(9,12),(13,16),(14,17),(15,18),(19,22),(21,23)]
[(3,4),(5,8),(6,7),(9,10),(11,13),(12,14),(15,16),(17,20),(18,19),(21,22)]
[(5,6),(7,8),(9,11),(10,13),(12,15),(14,16),(17,18),(19,20)]
[(4,5),(6,7),(8,9),(10,11),(12,13),(14,15),(16,17),(18,19),(20,21)]

Auto generated Sorting network for 26 inputs, 141 CEs, 13 layers:

[(0,1),(2,3),(4,5),(6,7),(8,9),(10,11),(12,13),(14,15),(16,17),(18,19),(20,21),(22,23),(24,25)]
[(0,2),(1,3),(4,6),(5,7),(8,10),(9,11),(14,16),(15,17),(18,20),(19,21),(22,24),(23,25)]
[(0,4),(1,5),(2,6),(3,7),(8,14),(9,16),(10,13),(11,17),(12,15),(18,22),(19,23),(20,24),(21,25)]
[(0,18),(1,19),(2,20),(3,21),(4,22),(5,23),(6,24),(7,25),(8,12),(10,14),(11,15),(13,17)]
[(1,18),(2,10),(3,20),(4,8),(5,22),(6,14),(7,24),(9,12),(11,19),(13,16),(15,23),(17,21)]
[(0,4),(1,9),(3,13),(5,15),(6,18),(7,19),(8,11),(10,20),(12,22),(14,17),(16,24),(21,25)]
[(2,4),(3,11),(5,9),(10,12),(13,15),(14,22),(16,20),(21,23)]
[(1,4),(3,8),(6,10),(7,13),(9,11),(12,18),(14,16),(15,19),(17,22),(21,24)]
[(1,2),(3,6),(4,5),(7,12),(8,10),(9,14),(11,16),(13,18),(15,17),(19,22),(20,21),(23,24)]
[(2,3),(4,6),(5,10),(7,9),(11,13),(12,14),(15,20),(16,18),(19,21),(22,23)]
[(3,4),(5,6),(7,8),(9,10),(11,12),(13,14),(15,16),(17,18),(19,20),(21,22)]
[(5,7),(6,8),(9,11),(10,12),(13,15),(14,16),(17,19),(18,20)]
[(4,5),(6,7),(8,9),(10,11),(12,13),(14,15),(16,17),(18,19),(20,21)]

Improved depth from 14 that could be achieved by Batcher odd-even merge Auto generated Sorting network for 27 inputs, 147 CEs, 16 layers:

[(0,1),(2,3),(4,5),(6,7),(8,9),(10,11),(12,14),(15,16),(17,18),(19,20),(21,22),(23,24),(25,26)]
[(0,2),(1,3),(4,6),(5,7),(8,10),(9,11),(12,13),(15,17),(16,18),(19,21),(20,22),(23,25),(24,26)]
[(0,23),(1,24),(2,25),(3,26),(4,8),(5,9),(6,10),(7,11),(13,14),(15,19),(16,20),(17,21),(18,22)]
[(0,4),(1,6),(2,19),(3,20),(5,13),(9,21),(11,14),(12,16),(17,23),(18,24),(22,26)]
[(5,17),(6,16),(7,22),(9,25),(10,24),(12,15),(13,20),(14,26)]
[(1,12),(4,15),(7,23),(10,19),(11,16),(13,18),(20,24),(22,25)]
[(0,1),(6,12),(8,11),(9,15),(10,17),(14,24),(16,21),(18,19)]
[(1,4),(2,8),(3,11),(12,15),(14,20),(16,22),(21,25)]
[(2,5),(3,17),(8,13),(11,23),(21,22),(24,25)]
[(1,2),(3,10),(5,6),(7,13),(11,15),(14,21),(18,23),(20,22)]
[(4,5),(6,9),(7,8),(13,17),(14,16),(19,23),(22,24)]
[(2,4),(3,6),(5,7),(8,12),(9,10),(11,13),(14,18),(15,17),(16,19),(21,23)]
[(3,5),(6,8),(7,9),(10,12),(11,14),(13,16),(15,18),(17,19),(20,21),(22,23)]
[(5,6),(8,11),(9,10),(12,14),(13,15),(17,18),(19,21)]
[(4,5),(6,7),(8,9),(10,11),(12,13),(14,15),(16,17),(18,20),(21,22)]
[(3,4),(5,6),(7,8),(9,10),(11,12),(13,14),(15,16),(17,18),(19,20)]

Improved size compared to 150 that could be obtained by Batcher odd-even merge Auto generated Sorting network for 27 inputs, 148 CEs, 14 layers:

[(0,17),(1,20),(2,23),(3,24),(4,9),(5,18),(6,16),(7,13),(8,15),(10,14),(11,21),(19,26),(22,25)]
[(0,4),(1,8),(2,19),(3,7),(5,6),(9,17),(10,11),(12,25),(13,24),(14,21),(15,20),(16,18),(23,26)]
[(0,2),(1,3),(4,23),(5,10),(6,14),(7,8),(9,19),(11,16),(12,22),(13,15),(17,26),(18,21),(20,24)]
[(1,5),(2,12),(3,6),(4,9),(7,11),(8,18),(10,13),(14,15),(16,20),(19,23),(21,24),(25,26)]
[(0,2),(3,7),(4,22),(5,10),(8,13),(9,25),(11,14),(15,18),(20,21)]
[(0,1),(2,3),(4,5),(6,14),(7,10),(9,12),(11,17),(19,22),(23,25)]
[(1,2),(3,6),(5,11),(7,9),(8,19),(12,18),(14,26),(16,23),(17,22),(20,25)]
[(3,5),(6,16),(9,14),(10,12),(13,17),(15,23),(18,25),(19,20),(21,22),(24,26)]
[(2,10),(5,9),(6,8),(11,13),(12,14),(15,16),(17,21),(22,24)]
[(2,7),(3,6),(8,10),(9,11),(12,15),(13,19),(14,16),(17,20),(18,21),(22,23),(24,25)]
[(4,7),(5,8),(6,9),(10,11),(12,13),(14,17),(15,19),(16,20),(18,22),(23,24)]
[(1,4),(3,7),(5,6),(8,10),(9,12),(11,13),(14,15),(16,17),(18,19),(20,21)]
[(2,4),(6,7),(8,9),(10,12),(11,14),(13,15),(16,18),(17,19),(20,22),(21,23)]
[(3,4),(5,6),(7,8),(9,10),(11,12),(13,14),(15,16),(17,18),(19,20),(21,22)]

Auto generated Sorting network for 27 inputs, 153 CEs, 13 layers:

[(0,19),(1,26),(2,25),(3,24),(4,23),(5,22),(6,21),(7,20),(9,10),(11,17),(12,15),(13,14),(16,18)]
[(0,1),(2,3),(4,5),(6,7),(8,9),(11,16),(12,14),(13,15),(17,18),(19,26),(20,21),(22,23),(24,25)]
[(0,2),(1,3),(4,6),(5,7),(8,18),(9,14),(10,12),(13,17),(15,16),(19,24),(20,22),(21,23),(25,26)]
[(0,4),(1,5),(2,20),(3,21),(6,19),(7,25),(8,13),(9,17),(10,11),(12,15),(14,18),(22,24),(23,26)]
[(1,2),(3,19),(4,6),(5,22),(7,20),(8,10),(9,12),(11,13),(14,16),(15,17),(21,23),(24,25)]
[(0,8),(1,4),(2,6),(3,10),(5,7),(9,11),(12,13),(14,15),(16,17),(18,19),(20,22),(21,24),(23,25)]
[(1,9),(2,13),(4,8),(5,12),(6,10),(7,20),(14,24),(15,22),(17,25),(18,21),(23,26)]
[(3,4),(6,14),(7,11),(8,15),(9,18),(10,17),(12,23),(13,21),(16,20),(19,26)]
[(1,3),(2,4),(5,6),(7,8),(10,13),(11,15),(12,16),(14,18),(19,24),(20,23),(21,22),(25,26)]
[(2,7),(4,8),(6,9),(10,11),(12,14),(13,15),(16,18),(17,21),(19,20),(23,24)]
[(2,3),(4,7),(5,6),(8,10),(9,12),(11,16),(13,14),(15,17),(18,19),(20,23),(21,22),(24,25)]
[(4,5),(6,7),(8,9),(10,12),(11,13),(14,16),(15,18),(17,19),(20,21),(22,23)]
[(3,4),(5,6),(7,8),(9,10),(11,12),(13,14),(15,16),(17,18),(19,20),(21,22),(23,24)]

Obtained by reduction of network for 28 inputs, L=159, D=13 by Chengu Wang Auto generated Sorting network for 28 inputs, 155 CEs, 14 layers:

[(0,1),(2,3),(4,5),(6,7),(8,9),(10,11),(12,13),(14,15),(16,17),(18,19),(20,21),(22,23),(24,25),(26,27)]
[(0,2),(1,3),(4,6),(5,7),(8,10),(9,11),(12,14),(13,15),(16,18),(17,19),(20,22),(21,23),(24,26),(25,27)]
[(0,4),(1,5),(2,6),(3,7),(8,12),(9,13),(14,18),(15,19),(20,24),(21,25),(22,26),(23,27)]
[(0,20),(1,21),(2,22),(3,23),(4,24),(5,25),(6,26),(7,27),(9,17),(10,18),(11,15),(12,16)]
[(1,2),(4,20),(5,6),(7,23),(8,12),(9,16),(10,14),(11,18),(13,17),(15,19),(21,22),(25,26)]
[(0,8),(1,9),(2,12),(3,5),(4,10),(6,16),(7,13),(11,21),(14,20),(15,25),(17,23),(18,26),(19,27),(22,24)]
[(2,4),(3,7),(5,17),(8,14),(9,11),(10,22),(13,19),(16,18),(20,24),(23,25)]
[(1,8),(3,9),(5,11),(6,10),(7,15),(12,20),(16,22),(17,21),(18,24),(19,26)]
[(1,2),(4,6),(5,9),(10,16),(11,17),(12,14),(13,15),(18,22),(21,23),(25,26)]
[(4,8),(6,12),(7,11),(10,14),(13,17),(15,21),(16,20),(19,23)]
[(2,4),(6,8),(7,16),(9,14),(10,12),(11,20),(13,18),(15,17),(19,21),(23,25)]
[(3,10),(5,12),(7,9),(11,13),(14,16),(15,22),(17,24),(18,20)]
[(3,6),(5,8),(7,10),(9,12),(11,14),(13,16),(15,18),(17,20),(19,22),(21,24)]
[(3,4),(5,6),(7,8),(9,10),(11,12),(13,14),(15,16),(17,18),(19,20),(21,22),(23,24)]

Improved size over 156 that could be obtained by Batcher odd-even merge Auto generated Sorting network for 28 inputs, 159 CEs, 13 layers:

[(0,27),(1,26),(2,25),(3,24),(4,23),(5,22),(6,21),(7,20),(8,9),(10,11),(12,15),(13,14),(16,17),(18,19)]
[(0,1),(2,3),(4,5),(6,7),(8,10),(9,11),(12,14),(13,15),(16,18),(17,19),(20,21),(22,23),(24,25),(26,27)]
[(0,2),(1,3),(4,6),(5,7),(8,19),(9,12),(10,14),(11,16),(13,17),(15,18),(20,22),(21,23),(24,26),(25,27)]
[(0,4),(1,5),(2,20),(3,21),(6,24),(7,25),(8,13),(9,11),(10,17),(12,15),(14,19),(16,18),(22,26),(23,27)]
[(1,2),(3,24),(4,6),(5,22),(7,20),(8,9),(10,12),(11,13),(14,16),(15,17),(18,19),(21,23),(25,26)]
[(0,8),(1,4),(2,6),(3,9),(5,7),(10,11),(12,13),(14,15),(16,17),(18,24),(19,27),(20,22),(21,25),(23,26)]
[(1,10),(2,13),(4,8),(5,12),(6,9),(7,20),(14,25),(15,22),(17,26),(18,21),(19,23)]
[(3,4),(6,14),(7,11),(8,15),(9,17),(10,18),(12,19),(13,21),(16,20),(23,24)]
[(1,3),(2,4),(5,6),(7,8),(9,13),(11,15),(12,16),(14,18),(19,20),(21,22),(23,25),(24,26)]
[(2,7),(4,8),(6,10),(9,11),(12,14),(13,15),(16,18),(17,21),(19,23),(20,25)]
[(2,3),(4,7),(5,6),(8,9),(10,12),(11,16),(13,14),(15,17),(18,19),(20,23),(21,22),(24,25)]
[(4,5),(6,7),(8,10),(9,12),(11,13),(14,16),(15,18),(17,19),(20,21),(22,23)]
[(3,4),(5,6),(7,8),(9,10),(11,12),(13,14),(15,16),(17,18),(19,20),(21,22),(23,24)]

Network found by Chengu Wang, improved depth. Auto generated Sorting network for 29 inputs, 164 CEs, 15 layers:

[(0,1),(2,3),(4,5),(6,7),(8,9),(10,11),(12,13),(14,15),(16,28),(17,26),(18,25),(19,23),(21,27),(22,24)]
[(0,2),(1,3),(4,6),(5,7),(8,10),(9,11),(12,14),(13,15),(17,22),(18,19),(20,27),(23,25),(24,26)]
[(0,4),(1,5),(2,6),(3,7),(8,12),(9,13),(10,14),(11,15),(16,20),(17,18),(19,22),(23,24),(25,26),(27,28)]
[(0,8),(1,9),(2,10),(3,11),(4,12),(5,13),(6,14),(7,15),(16,18),(20,22),(21,25),(24,27),(26,28)]
[(1,8),(2,24),(4,19),(5,20),(6,21),(7,27),(9,18),(10,23),(11,26),(13,22),(14,25),(15,28),(16,17)]
[(0,6),(2,4),(3,24),(5,10),(12,19),(13,18),(14,21),(15,25),(20,23),(26,27)]
[(0,16),(1,6),(3,12),(4,8),(5,17),(7,24),(14,20),(15,26),(18,21),(19,23),(25,27)]
[(1,5),(2,16),(3,10),(6,9),(7,18),(8,17),(11,19),(13,14),(15,22),(21,23),(25,26)]
[(1,2),(3,5),(4,8),(6,16),(7,11),(9,17),(10,12),(14,20),(15,18),(19,24),(22,27)]
[(4,6),(9,16),(10,13),(11,19),(12,14),(20,21),(22,26),(23,24)]
[(2,4),(3,6),(7,16),(8,9),(11,17),(15,19),(18,23),(24,25)]
[(3,4),(5,9),(7,10),(11,13),(12,16),(14,17),(15,20),(19,21),(22,24)]
[(5,8),(6,7),(9,12),(10,11),(13,14),(15,16),(17,20),(18,19),(21,23),(24,25)]
[(5,6),(7,8),(9,10),(11,12),(13,15),(14,16),(17,18),(19,20),(21,22)]
[(4,5),(6,7),(8,9),(10,11),(12,13),(14,15),(16,17),(18,19),(20,21),(22,23)]

Improved size over 165 that could be obtained by Batcher odd-even merge Auto generated Sorting network for 29 inputs, 166 CEs, 14 layers:

[(0,1),(2,3),(4,5),(6,7),(8,9),(10,11),(12,13),(14,15),(16,17),(18,19),(20,21),(22,23),(24,25),(26,27)]
[(0,2),(1,3),(4,6),(5,7),(8,10),(9,11),(12,14),(13,15),(16,18),(17,19),(20,22),(21,23),(24,26),(25,27)]
[(0,4),(1,5),(2,6),(3,7),(8,12),(9,13),(10,14),(11,15),(16,20),(17,21),(18,22),(19,23),(24,28)]
[(0,8),(1,9),(2,10),(3,11),(4,12),(5,13),(6,14),(7,15),(16,24),(17,25),(18,26),(19,27),(20,28)]
[(0,16),(1,8),(2,4),(3,12),(5,10),(6,9),(7,14),(11,13),(17,24),(18,20),(19,28),(21,26),(22,25),(23,27)]
[(1,2),(3,5),(4,8),(6,22),(7,11),(9,25),(10,12),(13,14),(17,18),(19,21),(20,24),(26,28)]
[(1,17),(2,18),(3,19),(4,20),(5,10),(7,23),(8,24),(11,27),(12,28),(13,25),(21,26)]
[(3,17),(4,16),(5,21),(6,18),(7,9),(8,20),(10,26),(11,23),(14,28),(15,27),(22,24)]
[(1,4),(3,8),(5,16),(7,17),(9,21),(10,22),(11,19),(12,20),(14,24),(15,26),(23,28)]
[(2,5),(7,8),(9,18),(11,17),(12,16),(13,22),(14,20),(15,19),(23,24)]
[(2,4),(6,12),(9,16),(10,11),(13,17),(14,18),(15,22),(19,25),(20,21)]
[(5,6),(8,12),(9,10),(11,13),(14,16),(15,17),(18,20),(19,23),(21,22),(25,26)]
[(3,5),(6,7),(8,9),(10,12),(11,14),(13,16),(15,18),(17,20),(19,21),(22,23),(24,25),(26,28)]
[(3,4),(5,6),(7,8),(9,10),(11,12),(13,14),(15,16),(17,18),(19,20),(21,22),(23,24),(25,26),(27,28)]

Obtained by removing 3 inputs from the 32 input network with 185 elements and 14 layers. Auto generated Sorting network for 30 inputs, 172 CEs, 14 layers:

[(0,1),(2,3),(4,5),(6,7),(8,9),(10,11),(12,13),(14,15),(16,17),(18,19),(20,21),(22,23),(24,25),(26,27),(28,29)]
[(0,2),(1,3),(4,6),(5,7),(8,10),(9,11),(13,15),(14,16),(18,20),(19,21),(22,24),(23,25),(26,28),(27,29)]
[(0,4),(1,5),(2,6),(3,7),(8,14),(9,17),(10,16),(12,20),(13,19),(15,21),(22,26),(23,27),(24,28),(25,29)]
[(0,8),(1,9),(2,14),(3,17),(4,10),(5,11),(6,16),(12,26),(13,23),(15,27),(18,24),(19,25),(20,28),(21,29)]
[(1,13),(2,12),(3,15),(4,18),(5,19),(6,20),(7,21),(8,22),(9,23),(10,24),(11,25),(14,26),(16,28),(17,27)]
[(0,4),(2,8),(3,13),(5,9),(6,22),(7,23),(10,12),(11,15),(14,18),(16,26),(17,19),(20,24),(21,27),(25,29)]
[(0,2),(1,14),(3,5),(4,8),(9,13),(11,17),(12,18),(15,28),(16,20),(21,25),(24,26),(27,29)]
[(2,4),(5,9),(6,14),(7,13),(8,10),(15,23),(16,22),(19,21),(20,24),(25,27)]
[(6,8),(7,11),(10,14),(12,16),(13,17),(15,19),(18,22),(21,23)]
[(4,6),(7,9),(8,10),(11,13),(12,14),(15,17),(16,18),(19,21),(20,22),(23,25)]
[(1,8),(3,18),(5,20),(7,22),(9,24),(10,12),(11,26),(13,15),(14,16),(17,19),(21,28)]
[(1,2),(3,10),(5,12),(7,14),(9,16),(11,18),(13,20),(15,22),(17,24),(19,26),(27,28)]
[(2,4),(3,6),(5,8),(7,10),(9,12),(11,14),(13,16),(15,18),(17,20),(19,22),(21,24),(23,26),(25,27)]
[(3,4),(5,6),(7,8),(9,10),(11,12),(13,14),(15,16),(17,18),(19,20),(21,22),(23,24),(25,26)]

Combines best known size and depth. An interesting pattern emerged through evolution in the last 4 layers: each layer "specializes" in elements bridging vertical distances of respectively 15, 7, 3 and 1, equally spaced with a stride of 2. This pattern is also typical for "pairwise" sorting networks [Par92]. Auto generated Sorting network for 31 inputs, 180 CEs, 14 layers:

[(0,1),(2,3),(4,5),(6,7),(8,9),(10,11),(12,13),(14,15),(16,17),(18,19),(20,21),(22,23),(24,25),(26,27),(28,29)]
[(0,2),(1,3),(4,6),(5,7),(8,10),(9,11),(12,14),(13,15),(16,18),(17,19),(20,22),(21,23),(24,26),(25,27),(28,30)]
[(0,4),(1,5),(2,6),(3,7),(8,12),(9,13),(10,14),(11,15),(16,20),(17,21),(18,22),(19,23),(24,28),(25,29),(26,30)]
[(0,8),(1,9),(2,10),(3,11),(4,12),(5,13),(6,14),(7,15),(16,24),(17,25),(18,26),(19,27),(20,28),(21,29),(22,30)]
[(0,16),(1,8),(2,4),(3,12),(5,10),(6,9),(7,14),(11,13),(17,24),(18,20),(19,28),(21,26),(22,25),(23,30),(27,29)]
[(1,2),(3,5),(4,8),(6,22),(7,11),(9,25),(10,12),(13,14),(17,18),(19,21),(20,24),(23,27),(26,28),(29,30)]
[(1,17),(2,18),(3,19),(4,20),(5,10),(7,23),(8,24),(11,27),(12,28),(13,29),(14,30),(21,26)]
[(3,17),(4,16),(5,21),(6,18),(7,9),(8,20),(10,26),(11,23),(13,25),(14,28),(15,27),(22,24)]
[(1,4),(3,8),(5,16),(7,17),(9,21),(10,22),(11,19),(12,20),(14,24),(15,26),(23,28),(27,30)]
[(2,5),(7,8),(9,18),(11,17),(12,16),(13,22),(14,20),(15,19),(23,24),(26,29)]
[(2,4),(6,12),(9,16),(10,11),(13,17),(14,18),(15,22),(19,25),(20,21),(27,29)]
[(5,6),(8,12),(9,10),(11,13),(14,16),(15,17),(18,20),(19,23),(21,22),(25,26)]
[(3,5),(6,7),(8,9),(10,12),(11,14),(13,16),(15,18),(17,20),(19,21),(22,23),(24,25),(26,28)]
[(3,4),(5,6),(7,8),(9,10),(11,12),(13,14),(15,16),(17,18),(19,20),(21,22),(23,24),(25,26),(27,28)]

Obtained by removing one input from the network with 32 inputs, 185 elements and 14 layers. Auto generated Sorting network for 32 inputs, 185 CEs, 14 layers:

[(0,1),(2,3),(4,5),(6,7),(8,9),(10,11),(12,13),(14,15),(16,17),(18,19),(20,21),(22,23),(24,25),(26,27),(28,29),(30,31)]
[(0,2),(1,3),(4,6),(5,7),(8,10),(9,11),(12,14),(13,15),(16,18),(17,19),(20,22),(21,23),(24,26),(25,27),(28,30),(29,31)]
[(0,4),(1,5),(2,6),(3,7),(8,12),(9,13),(10,14),(11,15),(16,20),(17,21),(18,22),(19,23),(24,28),(25,29),(26,30),(27,31)]
[(0,8),(1,9),(2,10),(3,11),(4,12),(5,13),(6,14),(7,15),(16,24),(17,25),(18,26),(19,27),(20,28),(21,29),(22,30),(23,31)]
[(0,16),(1,8),(2,4),(3,12),(5,10),(6,9),(7,14),(11,13),(15,31),(17,24),(18,20),(19,28),(21,26),(22,25),(23,30),(27,29)]
[(1,2),(3,5),(4,8),(6,22),(7,11),(9,25),(10,12),(13,14),(17,18),(19,21),(20,24),(23,27),(26,28),(29,30)]
[(1,17),(2,18),(3,19),(4,20),(5,10),(7,23),(8,24),(11,27),(12,28),(13,29),(14,30),(21,26)]
[(3,17),(4,16),(5,21),(6,18),(7,9),(8,20),(10,26),(11,23),(13,25),(14,28),(15,27),(22,24)]
[(1,4),(3,8),(5,16),(7,17),(9,21),(10,22),(11,19),(12,20),(14,24),(15,26),(23,28),(27,30)]
[(2,5),(7,8),(9,18),(11,17),(12,16),(13,22),(14,20),(15,19),(23,24),(26,29)]
[(2,4),(6,12),(9,16),(10,11),(13,17),(14,18),(15,22),(19,25),(20,21),(27,29)]
[(5,6),(8,12),(9,10),(11,13),(14,16),(15,17),(18,20),(19,23),(21,22),(25,26)]
[(3,5),(6,7),(8,9),(10,12),(11,14),(13,16),(15,18),(17,20),(19,21),(22,23),(24,25),(26,28)]
[(3,4),(5,6),(7,8),(9,10),(11,12),(13,14),(15,16),(17,18),(19,20),(21,22),(23,24),(25,26),(27,28)]

This network was obtained by evolution using a fixed prefix in its first 4 layers. Its size and depth combine the best results that can be achieved using a Batcher odd-even merge of either twice the 16 input network with 60 CEs and 10 layers or twice the 16 input network with 61 CEs and 9 layers. Auto generated