Batcher Sorting Network n 4 Batcher Sorting Network

Batcher Sorting Network, n = 4

Batcher Sorting Network, n = 8 n=4

Lemma 1 Any subsequence of a sorted sequence is a sorted sequence. sorted 0 0 0 1 1 1 sorted

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 0 0 0 0 0 1 1 1 even odd

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

Lemma 3 0 0 0 0 0 1 1 1 0 x¢ x¢E 0 0 1 1 1 x. O¢

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

Merge Network sorted Merge[4]

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

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

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

Merge Network (pf. ) 0 0 Merge[4] 1 1 By Lemma 3 and differ by at most 1 0 0 Merge[4] 0 1 0 0 0 1 1 1

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

Batcher Sorting Network, n = 4 Sort[2] Merge[4]

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

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