1. 1 Wide-Sense Nonblocking Multicast in a Class of Regular Optical Networks From: C. Zhou and Y. Yang, IEEE Transactions on communications, vol. 50, No. 1, Jan. 2002 From: C. Zhou and Y. Yang, IEEE Transactions on communications, vol. 50, No. 1, Jan. 2002

2. 2 Abstract Study multicast communication in a class of optical WDM networks with regular topologies. Study multicast communication in a class of optical WDM networks with regular topologies. Derive the necessary and sufficient conditions on the minimum number of wavelengths required for a WDM network to be wide-sense nonblocking for multicast communication under some commonly used routing algorithm. Derive the necessary and sufficient conditions on the minimum number of wavelengths required for a WDM network to be wide-sense nonblocking for multicast communication under some commonly used routing algorithm.

3. 3 Outline Introduction Introduction Linear Arrays Linear Arrays Rings (unidirectional, bidirectional) Rings (unidirectional, bidirectional) Meshes and Tori Meshes and Tori Hypercubes Hypercubes Conclusions Conclusions

4. 4 Definitions Wavelength-division multiplexing (WDM): multiple wavelength channels from different end-users can be multiplexed on the same fiber. Wavelength-division multiplexing (WDM): multiple wavelength channels from different end-users can be multiplexed on the same fiber. A connection or a lightpath is an ordered pair of nodes ( x, y ) corresponding to transmission of a packet from source x to destination y. A connection or a lightpath is an ordered pair of nodes ( x, y ) corresponding to transmission of a packet from source x to destination y.

5. 5 Definitions Multicast communication: transmitting information from a single source node to multiple destination nodes. Multicast communication: transmitting information from a single source node to multiple destination nodes. A multicast assignment is a mapping from a set of source nodes to a maximum set of destination nodes with no overlapping allowed among the destination nodes of different source nodes. A multicast assignment is a mapping from a set of source nodes to a maximum set of destination nodes with no overlapping allowed among the destination nodes of different source nodes.

6. 6 Examples of multicast assignments in a 4-node network There are a total of N connections in any multicast assignment. There are a total of N connections in any multicast assignment. An arbitrary multicast communication pattern can be decomposed into several multicast assignments. An arbitrary multicast communication pattern can be decomposed into several multicast assignments.

7. 7 Nonblocking Strictly Nonblocking (SNB): For any legitimate connection request, it is always possible to provide a connection path without disturbing existing connections. Strictly Nonblocking (SNB): For any legitimate connection request, it is always possible to provide a connection path without disturbing existing connections. Wide-sense Nonblocking (WSNB): If the path selection must follow a routing algorithm to maintain the nonblocking connecting capacity. Wide-sense Nonblocking (WSNB): If the path selection must follow a routing algorithm to maintain the nonblocking connecting capacity. Rearrangeable Nonblocking (RNB) Rearrangeable Nonblocking (RNB)

8. 8 Focus of this paper To determine the minimum number of wavelengths required for a WDM network to be wide-sense nonblocking for arbitrary multicast assignments. To determine the minimum number of wavelengths required for a WDM network to be wide-sense nonblocking for arbitrary multicast assignments. In other words, to determine the condition on which any multicast assignment can be embedded in a WDM network on-line under the routing algorithm. In other words, to determine the condition on which any multicast assignment can be embedded in a WDM network on-line under the routing algorithm.

9. 9 Assumption Each link in the network is bidirectional. Each link in the network is bidirectional. No wavelength converter facility is available in the network. Thus, a connection must use the same wavelength throughout its path. No wavelength converter facility is available in the network. Thus, a connection must use the same wavelength throughout its path. No light splitters are equipped at each routing nodes. No light splitters are equipped at each routing nodes.

10. 10 Outline Introduction Introduction Linear Arrays Linear Arrays Rings (unidirectional, bidirectional) Rings (unidirectional, bidirectional) Meshes and Tori Meshes and Tori Hypercubes Hypercubes Conclusions Conclusions

11. 11 Linear array There are only two possible directions for any connection in a linear array and the routing algorithm is unique.

12. 12 Theorem 1 The necessary and sufficient condition for a WDM linear array with N nodes to be wide-sense nonblocking for any multicast assignment is the number of wavelengths w w =N － 1. The necessary and sufficient condition for a WDM linear array with N nodes to be wide-sense nonblocking for any multicast assignment is the number of wavelengths w w =N － 1.

13. 13 Proof of Theorem 1 Sufficiency ( w w  N － 1): Sufficiency ( w w  N － 1): –The connections on the same link in different directions may use the same wavelength. –There are at most N － 1 connections in the same direction in a multicast assignment.

14. 14 Proof of Theorem 1 Necessity ( w w  N － 1): Necessity ( w w  N － 1):

15. 15 Wavelength assignment algorithm in a linear array T R : the wavelengths used for rightward connections. T L : the wavelengths used for leftward connections. T W : available wavelengths Step 1: = =Φ, || = N － 1. Step 1: T L = T R =Φ, |T W | = N － 1. Step 2: When a new rightward (leftward) connection is requested, assign it a wavelength in － ( － ), and add this wavelength to (). Step 2: When a new rightward (leftward) connection is requested, assign it a wavelength in T W － T R (T W － T L ), and add this wavelength to T R (T L ). Step 3: When an existing rightward (leftward) connection is released, delete it from (). Step 3: When an existing rightward (leftward) connection is released, delete it from T R (T L ).

16. 16 Outline Introduction Introduction Linear Arrays Linear Arrays Rings (unidirectional, bidirectional) Rings (unidirectional, bidirectional) Meshes and Tori Meshes and Tori Hypercubes Hypercubes Conclusions Conclusions

17. 17 Unidirectional Rings The routing algorithm is unique. The routing algorithm is unique. Assume the directional of a ring is counter-clockwise. Assume the directional of a ring is counter-clockwise.

18. 18 Theorem 2 The necessary and sufficient condition for a unidirectional WDM ring with N nodes to be wide-sense nonblocking for any multicast assignment is the number of wavelengths w w = N. The necessary and sufficient condition for a unidirectional WDM ring with N nodes to be wide-sense nonblocking for any multicast assignment is the number of wavelengths w w = N.

19. 19 Conflict graph Given a collection of connections Given a collection of connections G=(V,E) : an undirected graph, where V={ v : v is a connection in the network} E={ ab : a and b share a physical fiber link} ( a, b can’t use the same wavelength.) G=(V,E) : an undirected graph, where V={ v : v is a connection in the network} E={ ab : a and b share a physical fiber link} ( a, b can’t use the same wavelength.) The chromatic number  (G) of G is the minimum number of wavelengths required for the corresponding connections. The chromatic number  (G) of G is the minimum number of wavelengths required for the corresponding connections.

20. 20 Conflict graph (contd.) Find  (G) is a NP-complete problem. Find  (G) is a NP-complete problem.  (G) can be efficiently determined for conflict graphs for multicast communication in most of the networks.  (G) can be efficiently determined for conflict graphs for multicast communication in most of the networks.

21. 21 Proof of Theorem 2 Sufficiency ( w w  N ) : Since there are a total of N connections in any multicast assignment. Sufficiency ( w w  N ) : Since there are a total of N connections in any multicast assignment. Necessity ( w w  N ) : Consider the multicast assignment:  N = { i  ( i  1) mod N : 0  i  N  1} The conflict graph of  N is K N. Therefore w w   (K N )= N. Necessity ( w w  N ) : Consider the multicast assignment:  N = { i  ( i  1) mod N : 0  i  N  1} The conflict graph of  N is K N. Therefore w w   (K N )= N.

22. 22 6666 Conflict graph=K 6

23. 23 T u : currently used wavelengths T n : available wavelengths

24. 24 Bidirectional Rings There are two possible paths for a connection between any two nodes: clockwise or counter-clockwise. The shortest path routing algorithm is adopted.

25. 25

26. 26 Theorem 3 w w =  N /2 . The necessary and sufficient condition for a bidirectional WDM ring with N nodes to be wide-sense nonblocking for any multicast assignment under shortest path routing is the number of wavelengths w w =  N /2 .

27. 27 Proof of Theorem 3 Sufficiency ( w w   N /2  ) : Sufficiency ( w w   N /2  ) : – –In a multicast assignment, there are at most N connections. –  N /2  –We can divide the N connections in a multicast assignment into  N /2  pairs with the connections in each pair using the same wavelength.

28. 28

29. 29 Proof of Theorem 3 Necessity ( w w   N /2  ) : Necessity ( w w   N /2  ) :

30. 30 (0,2) (1,3) (2,4)(3,0) (4,1) Proof of Theorem 3 Necessity ( w w   N /2 , N is odd ) :  N = { i  ( i+ ( N  1)/2) mod N : 0  i  N  1} Necessity ( w w   N /2 , N is odd ) :  N = { i  ( i+ ( N  1)/2) mod N : 0  i  N  1}

31. 31

32. 32 Outline Introduction Introduction Linear Arrays Linear Arrays Rings (unidirectional, bidirectional) Rings (unidirectional, bidirectional) Meshes and Tori Meshes and Tori Hypercubes Hypercubes Conclusions Conclusions

33. 33 Meshes and Tori Definition 1: Under the row-major shortest path routing, for a connection request (( x 0, y 0 ), ( x 1, y 1 )) in a mesh or a torus, the path is deterministically from node ( x 0, y 0 ) to node ( x 1, y 1 ) in row x 0 along the shortest path first, then to node ( x 1, y 1 ) in column y 1 along the shortest path. Definition 1: Under the row-major shortest path routing, for a connection request (( x 0, y 0 ), ( x 1, y 1 )) in a mesh or a torus, the path is deterministically from node ( x 0, y 0 ) to node ( x 1, y 1 ) in row x 0 along the shortest path first, then to node ( x 1, y 1 ) in column y 1 along the shortest path.

34. 34 Meshes Definition 2: For a connection in a mesh under row- major shortest path routing, if the connection goes right at the first step from the source, we refer to it as a rightward connection. If the connection goes left at the first step, we refer to it as a leftward connection. Otherwise, we refer to it as a straight connection. Definition 2: For a connection in a mesh under row- major shortest path routing, if the connection goes right at the first step from the source, we refer to it as a rightward connection. If the connection goes left at the first step, we refer to it as a leftward connection. Otherwise, we refer to it as a straight connection.

35. 35 Theorem 4 The necessary and sufficient condition for a WDM mesh with p rows and q columns to be wide-sense nonblocking for any multicast assignment under row-major shortest path routing is the number of wavelengths w w = p  ( q  1) The necessary and sufficient condition for a WDM mesh with p rows and q columns to be wide-sense nonblocking for any multicast assignment under row-major shortest path routing is the number of wavelengths w w = p  ( q  1)

36. 36 Proof of Theorem 4 Sufficiency ( w w  p  ( q  1)) : Sufficiency ( w w  p  ( q  1)) : –Wavelengths: w 0, w 1, …, w p  ( q  1)  1 –Let R i ={ w i  ( q  1), w i  ( q  1)+1, …, w (i+1)  ( q  1)  1 }, 0  i  p － 1. (|R i |= q  1) –Let the connections destined to row i use the wavelengths within range R i. –The connections destined to the same column will use different wavelengths.

37. 37 Proof of Theorem 4

38. 38 Proof of Theorem 4 Sufficiency ( w w  p  ( q  1)) (continued … ) Sufficiency ( w w  p  ( q  1)) (continued … ) –Among all the connections to the same row, if the sources of two connections are in different rows, they can use the same wavelength. (by row-major routing) –Consider those connections originated from the same row and destined to the same row. Since each row can be considered as a linear array with q nodes. By Thm 1, q  1 wavelengths are sufficient for WSNB. Hence w w  p  ( q  1).

39. 39 Necessity ( w w  p  ( q  1)) : Necessity ( w w  p  ( q  1) ) : p  (q  1) Node (0,0) is the source of rightward p  (q  1) connections. p  (q  1) They must share the link (0,0)  (0,1). So p  (q  1) wavelengths are required.

40. 40 T R i : currently used wavelengths for rightward connections to row i. T L i : currently used wavelengths for leftward connections to row i.

41. 41 Torus Definition: A torus network is a mesh with wrap- around connections in both the x and y directions. This allowed the most distant processors to communicate in 2 hops. Definition: A torus network is a mesh with wrap- around connections in both the x and y directions. This allowed the most distant processors to communicate in 2 hops.

42. 42

43. 43 Theorem 5 The necessary and sufficient condition for a WDM torus with p rows and q columns to be wide-sense nonblocking for any multicast assignment under row-major shortest path routing is the number of wavelengths w w = p  q /2 . The necessary and sufficient condition for a WDM torus with p rows and q columns to be wide-sense nonblocking for any multicast assignment under row-major shortest path routing is the number of wavelengths w w = p  q /2 .

44. 44 Proof of Theorem 5 Sufficiency ( w w  p  q /2  ) : Sufficiency ( w w  p  q /2  ) : –(Similar to meshes) Divide the wavelengths to sets R 0 ~R p  1, and let the connections destined to row i use the wavelengths within set R i. –We need only to consider those connections originated from the same row and destined to the same row. Since each row can be considered as a bidirectional ring with q nodes. By Thm 3,  q /2  wavelengths are sufficient for WSNB. Hence w w  p  q /2 .

45. 45 Necessity ( w w  p  q /2  ) when q is even: Consider the connection (0,0)  ( i, j ), where 1  i  p  1, 0  j  q -1. Necessity ( w w  p  q /2  ) when q is even: Consider the connection (0,0)  ( i, j ), where 1  i  p  1, 0  j  q -1. q =6 There are p  ( q / 2) There are p  ( q / 2) connections passed through the link (0,0)  (0,1) or (0,0)  (0, q -1).

46. 46 Necessity ( w w  p  q /2  ) when q is odd: Consider the connection (0, j )  ( i, j +( q  1)/2 mod q ), where 0  i  p  1, 0  j  q -1. Necessity ( w w  p  q /2  ) when q is odd: Consider the connection (0, j )  ( i, j +( q  1)/2 mod q ), where 0  i  p  1, 0  j  q -1. q =5 The connection (0, j )  ( i, j +( q  1)/2 mod q ) must pass through the node (0, j +( q  1)/2 mod q ). So there are p connections pass through the link (0, j )  (0, j +( q  1)/2 mod q ). By Thm 3, a ring with q nodes need ( q  1)/2+1 wavelengths. Hence p  q /2  wavelengths is necessary.

47. 47 ( )

48. 48 Outline Introduction Introduction Linear Arrays Linear Arrays Rings (unidirectional, bidirectional) Rings (unidirectional, bidirectional) Meshes and Tori Meshes and Tori Hypercubes Hypercubes Conclusions Conclusions

49. 49 Hypercubes Definition: (e-cube routing) In an n -cube with N =2 n nodes, let each node b be binary-coded as b = b n b n  1 …b 2 b 1, where the i st bit corresponds to the i th dimension. In e-cube routing, a route from node s = s n s n  1 …s 2 s 1 to node d = d n d n  1 …d 2 d 1 is uniquely determined as follows: Definition: (e-cube routing) In an n -cube with N =2 n nodes, let each node b be binary-coded as b = b n b n  1 …b 2 b 1, where the i st bit corresponds to the i th dimension. In e-cube routing, a route from node s = s n s n  1 …s 2 s 1 to node d = d n d n  1 …d 2 d 1 is uniquely determined as follows:

50. 50 Hypercubes Definition: (e-cube routing) (contd.) s = s n s n  1 …s 2 s 1  s n s n  1 …s 2 d 1  s n s n  1 …d 2 d 1  …  s n d n  1 …d 2 d 1  d n d n  1 …d 2 d 1 = d Note that if s i = d i, no routing is needed along dimension i. Definition: (e-cube routing) (contd.) s = s n s n  1 …s 2 s 1  s n s n  1 …s 2 d 1  s n s n  1 …d 2 d 1  …  s n d n  1 …d 2 d 1  d n d n  1 …d 2 d 1 = d Note that if s i = d i, no routing is needed along dimension i.

51. 51 Example of e-cube routing

52. 52 Definitions A hypercube can be divided into two subcubes: 0-subcube={ b n b n  1 …b 2 0: b i  {0,1}} 1-subcube={ b n b n  1 …b 2 1: b i  {0,1}} A hypercube can be divided into two subcubes: 0-subcube={ b n b n  1 …b 2 0: b i  {0,1}} 1-subcube={ b n b n  1 …b 2 1: b i  {0,1}} For a connection in a hypercube, if the destination of this connection is in the 0-subcube (1-subcube), it is referred to as a 0-connection (1-connection). For a connection in a hypercube, if the destination of this connection is in the 0-subcube (1-subcube), it is referred to as a 0-connection (1-connection).

53. 53 Theorem 6 The necessary and sufficient condition for a WDM hypercube with N =2 n nodes to be wide-sense nonblocking for any multicast assignment under e-cube routing is the number of wavelengths w w = N/ 2 = 2 n  1. The necessary and sufficient condition for a WDM hypercube with N =2 n nodes to be wide-sense nonblocking for any multicast assignment under e-cube routing is the number of wavelengths w w = N/ 2 = 2 n  1.

54. 54 Proof of Theorem 6 Sufficiency ( w w  N/ 2 ) : Sufficiency ( w w  N/ 2 ) : –At most N /2 connections go from the 1(or 0)-subcube to the 0(or 1)-subcube. (The first step must go to the other subcube by e-cube routing.) –At most N/ 2 0/1-connections are in the 0/1-subcube. –Any 0-connection and any 1-connection cannot interfere with each other since they are in different directions or in different subcubes.  N/ 2 wavelengths are sufficient.

55. 55 Proof of Theorem 6 Necessity ( w w  N /2): Consider the case { 00 … 0  b n b n  1 … b 2 1 and 11 … 1  b n b n-1 …b 2 0 :  b i  {0,1}} Necessity ( w w  N /2): Consider the case { 00 … 0  b n b n  1 … b 2 1 and 11 … 1  b n b n-1 …b 2 0 :  b i  {0,1} }

56. 56 T 0 : used wavelengths for 0-connections T 1 : used wavelengths for 1-connections T w : available wavelengths

57. 57 Conclusion Summary:

58. 58 Conclusion Future work: Future work: –Generalize the approach developed in this paper to other routing algorithms and other network topologies including irregular networks. –Determine nonblocking conditions for multicast assignments in WDM networks with wavelength converters and/or light splitting switches.