Crosstalk-free Rearrangeability of Optical Multistage Interconnection Networks

1. On the Crosstalk-free Rearrangeability of Combined Optical Multistage Interconnection Networks Student: Chih-Wen Huang Advisor: Chiuyuan Chen Department of Applied Mathematics National Chiao Tung University

2. Outline • Introduction • Preliminaries • CF-rearrangeability of optical MINs • Routing algorithm on baseline networks • Concluding remarks

3. Introduction

4. Multistage interconnection networks (MINs) • N: number of inputs and outputs

5. stage 0 stage 1 stage 2 0 1 0 1 2 3 2 3 4 5 4 5 6 7 6 7 MIN (ex. Baseline network) • n=log2N • Link-disjoint

6. stage 0 stage 1 stage 2 0 1 0 1 2 3 2 3 4 5 4 5 6 7 6 7 Permutation (ex.) Not link-disjoint Not link-disjoint

7. stage 0 stage 1 stage 2 0 1 0 1 2 3 2 3 4 5 4 5 6 7 6 7 Permutation (ex.)

8. Rearrangeable • A permutation is admissible if can be realized on that MIN with link-disjoint paths in one pass. • An MIN is rearrangeable if all N! permutations are admissible. • Theoretically minimum number of stages is 2n−1.

9. Motivation • Das [8] formulated sufficient condition for the rearrangeability of a combined (2n−1)-stage MIN. • Also presented an O(Nn)-time routing algorithm. • Above definition of rearrangeable and results are for electronic MINs.

10. Motivation (cont.) • The purpose of this thesis is to redo the works of Das for optical MINs. • Differences between optical and electronic: • crosstalk problem • node-disjoint paths • maximum number of input-output pairs is N/2

11. Semi-permutation ,where and

12. Crosstalk-free rearrangeable (CF-rearrangeable) • An MIN is crosstalk-free rearrangeable (CF-rearrangeable) if each of the N! permutations can be realized with node-disjoint paths in two passes. • Theoretically minimum number of stages is 2n−2 [3].

13. Theorem [28]: Any permutation can be decomposed into two semi-permutations. Corollary: If any semi-permutation can be implemented in an optical MIN with node-disjoint paths in one pass, then the optical MIN is CF-rearrangeable.

14. Our purposes • Formulate a sufficient condition for a combined (2n−2)-stage (or (2n−1)-stage) optical MIN to be CF-rearrangeable. • Propose an O(Nn)-time permutation routing algorithm. • Propose an algorithm to realize any permutation in a baseline network withnode-disjoint paths in four passes.

15. Preliminaries

16. 0 1 0 1 0 1 0 1 x2 x1 x0 y2 y1 y0 0 0 0 0 0 0 0 0 0 000 000 00 00 00 0 0 1 0 0 1 001 001 2 3 2 3 2 3 2 3 1 1 1 0 1 0 0 1 0 010 010 01 01 01 0 1 1 011 011 0 1 1 4 5 4 5 4 5 4 5 2 2 2 1 0 0 100 1 0 0 100 10 10 10 1 0 1 101 101 1 0 1 6 7 6 7 6 7 6 7 3 3 3 1 1 0 110 110 1 1 0 11 11 11 1 1 1 111 1 1 1 111 Network model stage 0 stage 1 stage 2

17. Baseline network n-stage Reverse baseline network n-stage Combined MIN (ex. Benes) Benes network (2n-1)-stage • Denoted by M(s) ⊕M’(s’).

18. Routing bits • Routing bit rk controls the switch at stage k and if rk = 0 (rk = 1), then a connection is made to sub port 0 (sub port 1). • A path from an input to an output can be described by a sequence r0r1 . . . rs−1.

19. stage 0 stage 1 stage 2 0 1 0 1 2 3 2 3 4 5 4 5 6 7 6 7 Routing bits (ex. Reverse baseline) • Input 0 can get to output 5 by using the routing bits 101. r1=0 r2=1 r0=1 1 0 1

20. stage 0 stage 1 stage 2 0 1 0 1 2 3 2 3 4 5 4 5 6 7 6 7 Follows the destination tag routing • All inputs can get to output 5 by using the routing bits 101 (binary representation of 5). • Reverse baseline network follows the destination tag routing.

21. stage 0 stage 1 stage 2 Stage 3 stage 4 0 1 0 1 2 3 2 3 4 5 4 5 6 7 6 7 AR-bits • In this thesis, we consider the combined s-stage MIN M(s−n+1) ⊕M’(n), where M’(n) follows destination tag routing. Benes network

22. stage 0 stage 1 stage 2 Stage 3 stage 4 0 1 0 1 2 3 2 3 4 5 4 5 6 7 6 7 AR-bits (cont.) • The routing bits for stages k (k =0, 1, . . . , s−n−1) are arbitrary and are referred to as arbitrary routing bits (AR-bits). Benes network

23. CF-rearrangeability of optical MINs

24. 0 1 0 1 0 1 2 3 2 3 2 3 4 5 4 5 4 5 6 7 Benes network ((2n-1)-stage) 6 7 6 7 0 1 2 3 4 5 6 7 Dilated benes network ((2n-2)-stage) Dilated benes network

25. 0 stage 0 stage 1 stage 2 stage 3 stage 4 stage 5 7 2 5 5 13 15 7 8 8 2 11 0 13 x3x2x1r0 r0x3x2r1 r0r1x3r2 r0r1r2r3 r0r2r3r4 15 = = = = = = OS2 OS1 OS0 OS3 OS4 OS5 x3x2x1 r0x3x2 r0r1x3 r0r1r2 r0r2r3 r2r3r4 Optical windows OWk & characteristic string OSk of OWk 101 1011 1011 110 1101 11 1110 111 111 111 111 1111 1111 1111 x3x2x1x0 r2r3r4r5

26. Optical windows (ex.)

27. Each row Rj of OWk is the switch at stage k on the path started from input 2j or 2j +1. • A semi-permutation is crosstalk-free if and only if all rows of each optical window OWk are distinct.

28. Sufficient condition for (2n−2) stages • In a combined (2n − 2)-stage optical MIN M(n−1) ⊕M’(n) in which M’(n) follows destination tag routing, if each AR-bit rk occurs only in each OSℓ for ℓ = k+1, k+2, . . . , 2n−4−k, then the MIN can realize any semi-permutation with node-disjoint paths in one pass and hence is CF-rearrangeable.

29. 0 stage 0 stage 1 stage 2 stage 3 stage 4 stage 5 7 2 5 5 13 15 7 8 8 2 11 0 13 15 = = = = = = OS5 OS0 OS1 OS2 OS4 OS3 Sufficient condition (ex. N=16) 11 x3x2x1 r0 x3x2 r0 x3x2 r0x3x2 r0 x3x2 r0r1x3 r0r1x3 r0r1x3 r0r1x3 r0r1r2 r0r1r2 r0r1r2 r0r2r3 r0r2r3 r0r2r3 r2r3r4

30. Sufficient condition for (2n−1) stages • In a combined (2n − 1)-stage optical MIN M(n) ⊕M’(n) in which M’(n) follows destination tag routing, if each AR-bit rk occurs only in each OSℓ for ℓ = k+1, k+2, . . . , 2n−3−k, then the MIN can realize any semi-permutation with node-disjoint paths in one pass and hence is CF-rearrangeable.

31. 0 1 0 1 2 3 2 3 4 5 4 5 6 7 6 7 8 9 8 9 10 11 10 11 12 13 12 13 14 15 14 15 Routing on dilated Benes network stage 0 stage 1 stage 2 stage 3 stage 4 stage 5

32. Routing on dilated Benes network (cont.)

33. Routing on dilated Benes network (cont.) • The columns in OW1 is given in the order x3x2r0 instead of the order r0x3x2.

34. Routing on dilated Benes network (cont.) • The columns in OW2 is given in the order x3r0r1 instead of the order r0r1x3.

35. Routing on dilated Benes network (cont.) • r2r3r4r5 are the binary representation of the output y3y2y1y0.

36. 0 stage 0 stage 1 stage 2 stage 3 stage 4 stage 5 7 2 5 5 13 15 7 8 8 2 11 0 13 11 15 Routing on dilated Benes network (cont.)

37. Routing algorithm on baseline network

38. Motivation • In [27], Yang and Wang proved that {adm. perm. of baseline} = {adm. perm. of reverse baseline}

39. Motivation (cont.) • In [27], using intermediate destinations of a Benes network propose a recursive routing algorithm on a baseline network with node-disjoint paths in four passes. • Base on the intermediate destinations of a dilated Benes network to propose a non-recursive routing algorithm on a baseline network with node-disjoint paths in four passes.

40. 0 2 stage 0 stage 1 stage 2 stage 3 stage 4 stage 5 7 5 13 5 0 7 13 5 8 15 8 8 11 15 2 2 13 0 7 15 11 11 Routing on baseline network with node-disjoint paths

41. 7 5 13 13 13 0 0 15 5 5 8 8 8 2 15 15 0 2 2 11 7 7 11 11 Routing on baseline network with node-disjoint paths (cont.) stage 0 stage 1 stage 2 stage 3

42. Routing on baseline network with node-disjoint paths (cont.) • The routing bits of first pass can be obtained from the first four column of the result of the last section. • The routing bits of second pass can be obtained from the binary representation of destination.

43. stage 0 stage 0 stage 1 stage 2 stage 3 stage 1 stage 2 stage 3 0 7 13 13 2 5 0 0 5 5 5 13 7 8 8 15 8 15 15 8 11 2 2 2 13 7 7 0 15 11 11 11 The first pass The second pass Routing on baseline network with node-disjoint paths (cont.)

44. Corollary: Each permutation can be realized in a baseline network with node-disjoint paths in four passes, so does a reverse baseline network.

45. Concluding remarks

46. Concluding remarks • Formulate a sufficient condition for a combined (2n−2)-stage (or (2n−1)-stage) optical MIN to be CF-rearrangeable. • Propose an permutation routing algorithm for optical MINs that satisfy the sufficient condition. • Propose an algorithm to realize any permutation in baseline (or reverse baseline) network with node-disjoint paths in four passes.

47. Thanks for your attention!