Batcher Sorting Network, n = 4

1. Batcher Sorting Network, n = 4

2. Batcher Sorting Network, n = 8 n = 4 n = 4

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

4. 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. 0 0 0 0 0 0 0 0 0 0 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. 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 1 0 0 1 1 1 1 1 1 1 1 1 1 1 1 ¢ ¢ ¢ x x x O E Lemma 3

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

8. Merge[4] sorted sorted Merge[4] sorted Merge Network

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

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

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

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

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

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

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

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