1. Batcher Sorting Network, n = 4

2. Batcher Sorting Network, n = 8

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

4. 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

5. 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

6. Lemma 3 1 1 1 x 1 E x 1 1 O x

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

8. Merge Network
Merge[4]
sorted
sorted
sorted

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

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

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

12. 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

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

14. Batcher Sorting Network, n = 4
Merge[4]

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

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