1. Routing Permutation in the Baseline Network and in the Omega Network Student : Tzu-hung Chen 陳子鴻 Advisor : Chiuyuan Chen Department of Applied Mathematics National Chiao Tung University

2. Outline  Preliminaries  Previous results  Motivation  Our results  Concluding remarks

3. N × N multistage interconnection network (MIN) P0P0 P1P1 P N-1 N×N MIN O0O0 O1O1 O N-1 Preliminaries

4. 0 1 2 3 4 5 6 7 stage 0 stage 1 stage 2 0 1 2 3 4 5 6 7  The number of processors (inputs/ outputs) :  The number of stages : Preliminaries N = 8, n = 3 InputOutput switching element

5. Preliminaries  A 2 × 2 switching element has only two possible states: straight, cross. (a) straight sub port 0 sub port 1 sub port 0 sub port 1 (b) cross sub port 0 sub port 1 sub port 0 sub port 1

6. Preliminaries (a) 4×4 Baseline network 0 1 2 3 (b ) 4×4 Omega network 0 1 2 3

7. 1 0 2 3 N-2 N-1 N-3 N-4 n-1 stages 1 0 N-2 N-1 (a) N × N Baseline network(b ) N × N Omega network Preliminaries

8. 0 1 2 3 4 5 6 7 (a) 8×8 Baseline network (b ) 8×8 Omega network Preliminaries

9.  Unique path: there is a unique path between each source (input) and each destination (output). Preliminaries  Self routable: a routing in the network only depends on the source and the destination.  Control tag is a sequence of labels that label the successive links on a path.

10.  Input 0 can get to output 6 by using control tag Preliminaries 0 1 2 3 4 5 6 7 stage 0 stage 1 stage 2 0 1 2 3 4 5 6 7 1 0 1

11.  Conflict Preliminaries  Have the same node stage 0 stage 1 stage 2  Have the same link => link-disjoint=> node-disjoint

12. Preliminaries  A permutation of an MIN is one-to-one mapping between the inputs and outputs.  For convenience, let

13. 0 1 2 3 4 5 6 7 1 0 2 3 4 5 7 6 2 1 4 7 3 0 6 5 2 4 7 1 6 3 5 0 stage 0 stage 1stage 2 Preliminaries No conflict occurs in the network. P is an admissible permutation.

14. Preliminaries 0 1 2 3 4 5 6 7 stage 0 stage 1stage 2 Not admissible!Conflict!

15. Preliminaries  A semi-permutation P

16.  Example Preliminaries

17. Previous results

18.  In [11], Shen et al. proposed an O(N logN) algorithm to determine the admissibility of an arbitrary permutation to the Omega network; their results are applicable to Omega-equivalent networks.

19.  In [18], Yang and Wang proposed an algorithm to decompose an arbitrary permutation into two semi-permutations. Previous results

20.  In [17], Yang and Wang used the idea in [18] to prove that an arbitrary permutation can be realized in a Baseline network with node-disjoint paths in four passes.

21. Motivation

22.  Although [11] claimed that their results are applicable to Omega-equivalent networks, an admissible permutation of the Omega network may not be an admissible permutation of the Baseline network.  We propose an algorithm to determine the admissibility of permutations for the Baseline network.

23. 0 1 2 3 4 5 6 7 stage 0 stage 1stage 2 Motivation stage 0 stage 1stage 2 (a) Omega network (b) Baseline network

24.  The algorithm in [11] has one step that can be removed without breaking the correct of the algorithm. Motivation  We propose an algorithm to determine the admissibility of permutations for the Omega network that does not need the step in [11].

25. The motivation of [17]  In [17], Yang and Wang proved that an arbitrary permutation can be realized in a Baseline network with node-disjoint paths in four passes.  In this thesis, we implement the decomposition algorithm in [18] and the algorithm in [17] into a C++ computer program.

26. Our results

27.  Determine the admissibility of permutations for the Baseline network  Determine the admissibility of permutations for the Omega network  We implement the decomposition algorithm in [18] and the algorithm in [17] into a C++ computer program.

28. The Baseline network  stage 0 N×N Baseline network 1 0 2 3 N-2 N-3 N-4 N-1

29. The Baseline network  A permutation P is admissible in a Baseline network if

30. Determine the admissibility of permutations for the Baseline network 0 0 0 0 Input 2i Input 2i+1 Input 2i Input 2i+1 Input 2i Input 2i+1 Input 2i Input 2i+1 stage 0

31. Algorithm Baseline -Admissible

37. Our results

38.  The Omega network

39. 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 stage 0 stage 1stage 2stage 3 0 1 2 3 4 5 6 7 stage 0 stage 1stage 2 (a) 8 × 8 Omega network (b) 16×16 Omega network

40. (i) The upper N/4 switching elements of stage n−1 (the last stage) belong to U and the lower N/4 switching elements of stage n−1 belong to L. (ii) For each switching element of stage ℓ (ℓ = n−2, n−3,..., 1), if this switching element is adjacent to a switching element of stage ℓ+1 which belongs to U (L), then it belongs to U (L). Define sub network U and sub network L

41. 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 stage 0 stage 1stage 2stage 3 (a) 8 × 8 Omega network 0 1 2 3 4 5 6 7 stage 0 stage 1stage 2 (b) 16×16 Omega network sub network U

42. The Omega network  A permutation P is admissible in a Omega network if

43. (a)(b) N = 16 Determine the admissibility of permutations for the Omega network 0 1 2 3 4 5 6 7 0 0 0 0 0 0 0 0 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 stage 0 stage 1stage 2stage 3 stage 0 stage 1stage 2

44. Algorithm Omega -Admissible

50. Our results

51.  We implement the decomposition algorithm in [18] and the algorithm in [17] into a C++ computer program. Our results

52. Our result  The output of our program

55. Concluding Remarks

56.  We propose an algorithm to determine whether a permutation is admissible to the Baseline network.  We also propose an algorithm to determine whether a permutation is admissible to the Omega network.  We have also implemented the decomposition algorithm in [18] and the algorithm in [17] into a C++ computer program.

57. Compare Algorithm [11]Omega network This thesis Baseline network and Omega network( remove one step in [11])

58. Thank you for your attention !

59. Previous results

60. DECOMPOSITION ALGORITHM [18]

62. Example