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Lecture 4 Sorting Networks

Lecture 4 Sorting Networks. Comparator. comparator. A Sorting Network. 2. 2. 9. 5. 6. 5. 9. 5. 2. 5. 6. 2. 6. 9. 6. 9. A sorting network is a comparison network which output monotone nondecreasing sequence for every input. Depth. 2. 2. 9. 5. 6. 5. 9. 5. 2. 5. 6.

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Lecture 4 Sorting Networks

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  1. Lecture 4Sorting Networks

  2. Comparator comparator

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

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

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

  6. Insertion Sort

  7. key

  8. Sorting network constructed from insertion sort.

  9. How to construct a sortingnetwork 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 0 0 0 0 bitonic clean 1 0 1 0 bitonic 1 1 0 0 bitonic 1 0 0 1

  22. The half-cleaner 0 0 0 0 bitonic 1 1 1 0 bitonic 1 1 1 1 bitonic clean 1 1 0 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) 0 0 1 0 1 1 0 0 0

  25. Proof (case 2) 0 0 1 0 1 0 0 1 0

  26. Proof (case 3) 0 0 1 0 1 1 1 0 0 0 1

  27. Proof (case 4) 0 0 1 1 1 0 0 1 1 0 0

  28. Proof (case 5) 1 1 0 0 1 1 1 1 0 1 1

  29. Proof (case 6) 1 1 0 1 1 1 0 1 1 0 1

  30. Proof (case 7) 1 0 1 1 0 0 1 1 0 0 1

  31. Proof (case 8) 1 0 1 0 1 0 1 0 0 1 1

  32. bitonic sorted Half cleaners

  33. 0 0 0 0 sorted 0 0 1 0 sorted 0 0 0 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

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