1. Lecture 4 Sorting Networks

2. Comparator
comparator

3. A Sorting Network 9 5 2 2 5 9 6 5 2 2 5 6 6 6 9 9
A sorting network is a comparison network which output
monotone nondecreasing sequence for every input.

4. Depth 9 5 2 2 5 9 6 5 2 2 5 6 6 6 9 9
Depth is the maximum number of comparators on a
path from an input wire to an output wire.

5. Depth = parallel time 9 5 2 2 5 9 6 5 2 2 5 6 6 6 9 9
Depth is the maximum number of comparators on a
path from an input wire to an output wire.

6. Insertion Sort

7. key

8. Sorting network constructed from insertion sort.

9. How to construct a sorting network from merging sort?

10. Divide and Conquer Divide the problem into subproblems.
Conquer the subproblems by solving them recursively.
Combine the solutions to subproblems into the solution for original problem.

11. Merge Sort

12. Procedure

13. Structure
Sorting
network
Merging
network
Sorting
network

14. Construction of Merging Network
0-1 principal.
Bitonic sorter.
Merging network.

15. 0-1 principal

16. Lemma

17. Proof of 0-1 Principal

18. Bitonic Sequence

19. Bitonic 0-1 Sequence

20. Some Properties

21. The half-cleaner
bitonic
clean 1 1
bitonic 1 1
bitonic 1 1

22. The half-cleaner
bitonic 1 1 1
bitonic 1 1 1 1
bitonic
clean 1 1 1

23. Lemma One of two halfs is bitonic clean.
every number in the 1st half ≤ any element in the 2nd half.

24. Proof (case 1) 1 1 1

25. Proof (case 2) 1 1 1

26. Proof (case 3) 1 1 1 1 1

27. Proof (case 4) 1 1 1 1 1

28. Proof (case 5) 1 1 1 1 1 1 1 1

29. Proof (case 6) 1 1 1 1 1 1 1 1

30. Proof (case 7) 1 1 1 1 1 1

31. Proof (case 8) 1 1 1 1 1 1

32. bitonic
sorted
Half cleaners

33. sorted 1
sorted 1
sorted 1 1 1 1
Half cleaners
Merging Network

34. Structure
Sorting
network
Merging
network
Sorting
network

35. Sorting Network
Merging Networks

36. What we learnt in this lecture?
What is sorting network?
Depth = parallel time.
Sorting network from Merge sort.

37. Permutation Network Switching network Rearrangeability
Nework with 2x2 crossbars

38. Crossbar Switch A crossbar switch can realize any matching between
Inputs and outputs.

39. 3-stage Clos Network 1 n n n n m

40. Rearrangeability
Theorem

41. Network with 2x2 crossbars

42. Puzzle