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

6. Permutation (ex.) 1 Not link-disjoint Not link-disjoint 2 3 4 5 6 71 2 3 4 5 6 7
stage 0
stage 1
stage 2
Not link-disjoint
Not link-disjoint

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

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 with node-disjoint paths in four passes.

15. Preliminaries

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

17. Combined MIN (ex. Benes)
Benes network
(2n-1)-stage
Baseline network
n-stage
Reverse baseline network
n-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 r0r rs−1.

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

20. Follows the destination tag routing1 2 3 4 5 6 7
stage 0
stage 1
stage 2
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. 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. 1 2 3 4 5 6 7
stage 0
stage 1
stage 2
Stage 3
stage 4
Benes network

22. 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). 1 2 3 4 5 6 7
stage 0
stage 1
stage 2
Stage 3
stage 4
Benes network

23. CF-rearrangeability of optical MINs

24. Dilated benes network 1 2 3 4 5 6 7 Benes network ((2n-1)-stage) 1 2 31 2 3 4 5 6 7
Benes network ((2n-1)-stage) 1 2 3 4 5 6 7
Dilated benes network ((2n-2)-stage)

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

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

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

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

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

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

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

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!