Admissibility Algorithms for Routing Permutations in Network Topologies

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. Preliminaries O0 P0 O1 P1 N×N MIN ON-1 PN-1 N × N multistage interconnection network (MIN)

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

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

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

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

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

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

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

11. Preliminaries • Conflict • Have the same node • Have the same link => link-disjoint => node-disjoint stage 1 stage 0 stage 2 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7

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

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

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

15. Preliminaries • A semi-permutation P

16. Preliminaries • Example

17. Previous results

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

20. Previous results • 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. Motivation • 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. Motivation stage 1 stage 2 stage 0 stage 1 stage 2 stage 0 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 (a) Omega network (b) Baseline network

24. Motivation • The algorithm in [11] has one step that can be removed without breaking the correct of the algorithm. • 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. Our results • 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. 0 1 2 3 N-4 N-3 N-2 N-1 The Baseline network stage 0 N×N Baseline network

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

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

31. Algorithm Baseline -Admissible

32. Algorithm Baseline -Admissible

33. 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 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 stage 1 stage 2 stage 3 stage 0 stage 1 stage 2 stage 0 (a) 8×8 Omega network (b) 16×16 Omega network

40. Define sub network U and sub network L • 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. • 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).

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

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

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

44. Algorithm Omega -Admissible

45. Algorithm Omega -Admissible

46. Algorithm Omega -Admissible

50. Our results