Batcher Sorting Network, n = 4

Batcher Sorting Network, n = 8

Lemma 1 Any subsequence of a sorted sequence is a sorted sequence. sorted sorted 1 1 1 1 1 1 1 1

Lemma 2 For a sorted sequence, the number of 0’s in the even subsequence is either equal to, or one greater than, the number of 0’s in the odd subsequence. sorted 1 1 1 1 1 1 even odd

Lemma 3 For two sorted sequences and : denotes the the number of 0’s in denotes the even subsequence of denotes the odd subsequence of

Lemma 3 1 1 1 x ¢ 1 E x ¢ 1 1 O x ¢

Lemma 3 For two sorted sequences and : (by Lemma 2) (by Lemma 2)

Merge Network Merge[4] sorted sorted sorted

Merge Network (pf.) sorted sorted sorted sorted Merge[4] (by Lemma 1)

Merge Network (pf.) sorted sorted Merge[4] By Lemma 3 and differ by at most 1 By Lemma 3 sorted

Merge Network (pf.) Merge[4] sorted and differ by at most 1 By Lemma 3

Merge Network (pf.) 1 1 1 1 1 1 Merge[4] By Lemma 3 and Merge[4] 1 1 and differ by at most 1 By Lemma 3 1 1 1 1

Batcher Sorting Network Merge[8] sorted Sort[4]

Batcher Sorting Network, n = 4 Merge[4]

Batcher Sorting Network, n = 8 Merge[8] Sort[4]

Sorting Networks AKS (Ajtai, Komlós, Szemerédi) Network: based on expander graphs. AKS (Chvátal) Batcher AKS better for