1. 1 Multicasting in a Class of Multicast-Capable WDM Networks From: Y. Wang and Y. Yang, Journal of Lightwave Technology, vol. 20, No. 3, Mar. 2002 From: Y. Wang and Y. Yang, Journal of Lightwave Technology, vol. 20, No. 3, Mar. 2002

2. 2 Abstract Study multicast communication in a class of multicast-capable WDM networks (i.e., have light splitting switches) with regular topologies under some commonly used routing algorithms. Study multicast communication in a class of multicast-capable WDM networks (i.e., have light splitting switches) with regular topologies under some commonly used routing algorithms. Upper and lower bounds on the minimum number of wavelengths required are determined for a network to be rearrangeable for arbitrary multicast assignments. Upper and lower bounds on the minimum number of wavelengths required are determined for a network to be rearrangeable for arbitrary multicast assignments.

3. 3 Outline Introduction Introduction Multicast-Capable WDM Networks Multicast-Capable WDM Networks Generalization of the conflict graph Generalization of the conflict graph Rings Rings Meshes Meshes Hypercubes Hypercubes Linear Arrays Linear Arrays Comparison and Conclusions Comparison and Conclusions

4. 4 Introduction WDM WDM Multicast Multicast Multicast can be supported more efficiently in optical domain by utilizing the inherent light splitting capacity of optical switches than copying data in electronic domain. Multicast can be supported more efficiently in optical domain by utilizing the inherent light splitting capacity of optical switches than copying data in electronic domain.

5. 5 Multicast-Capable WDM Networks 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. The connections established between one source node and multiple destination nodes ( x, y 1 ), ( x, y 2 ), …, ( x, y n ) are referred to as a multicast connection. The connections established between one source node and multiple destination nodes ( x, y 1 ), ( x, y 2 ), …, ( x, y n ) are referred to as a multicast connection.

6. 6 Multicast-Capable Nodes In a multicast connection, a node may be required to connect an incoming channel to a set of outgoing channels where each outgoing channel is on a different fiber. In a multicast connection, a node may be required to connect an incoming channel to a set of outgoing channels where each outgoing channel is on a different fiber. In order to support multicast efficiently, the routing nodes have to have the capability of splitting and/or broadcasting from input to output. In order to support multicast efficiently, the routing nodes have to have the capability of splitting and/or broadcasting from input to output.

7. 7 Three different multicast model (a) Multicast with same wavelength (MSW) (b) Multicast with same destination wavelength (MSDW) (c) Multicast with any wavelength (MAW)

8. 8 Multicast Assignment 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. An arbitrary multicast communication pattern can be decomposed into several multicast assignments. An arbitrary multicast communication pattern can be decomposed into several multicast assignments.

9. 9 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.

10. 10 FAN-OUT In a multicast assignment, the number of destination nodes from the same source node is referred to as the fan-out of the source node. In a multicast assignment, the number of destination nodes from the same source node is referred to as the fan-out of the source node.

11. 11 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)

12. 12 Focus of this paper To determine bounds on the minimum number of wavelengths (denoted as w r ) required for a multicast-capable WDM network to be rearrangeable for any multicast assignments. To determine bounds on the minimum number of wavelengths (denoted as w r ) required for a multicast-capable WDM network to be rearrangeable for any multicast assignments. In other words, to determine the condition under which any multicast assignment can be embedded in a WDM network offline. In other words, to determine the condition under which any multicast assignment can be embedded in a WDM network offline.

13. 13 Assumption Each link in the network is bidirectional. Each link in the network is bidirectional. Adopt the MSW model but light splitters are available at every routing node. (Wavelength-continuity constraint) Adopt the MSW model but light splitters are available at every routing node. (Wavelength-continuity constraint)

14. 14 Outline Introduction Introduction Multicast-Capable WDM Networks Multicast-Capable WDM Networks Generalization of the conflict graph Generalization of the conflict graph Rings Rings Meshes Meshes Hypercubes Hypercubes Linear Arrays Linear Arrays Comparison and Conclusions Comparison and Conclusions

15. 15 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.)

16. 16 6666 Conflict graph=K 6 Example of Conflict graph

17. 17 In Multicast-capable network Those connections that belong to the same source node do not conflict with each other. In fact, they must utilize the same wavelength under MSW model.  We can treat these connections as a single connection that goes through all the links used by any connection among these connections.

18. 18 Contraction A contraction G : ab of a graph G. (a) The original graph. (b) The contracted graph.

19. 19 Modified Conflict graph For any source node v k that is multicasting in the corresponding conflict graph G, contract all of the vertices a k 1, a k 2, …, a k n that correspond to the multicast connection from source node v k into a single vertex b k. We call the resulting graph a modified conflict graph G ’.

20. 20 Example G G’G’

21. 21 Modified Conflict graph G=(V,E) : an undirected graph, where V={ v : v is a multicast connection in the network} (A source node corresponds to a vertex.) E={ ab : a and b share a physical fiber link} ( a, b can’t use the same wavelength.) (Equivalent definition) G=(V,E) : an undirected graph, where V={ v : v is a multicast connection in the network} (A source node corresponds to a vertex.) E={ ab : a and b share a physical fiber link} ( a, b can’t use the same wavelength.) Wavelength assignment  Graph vertex coloring Wavelength assignment  Graph vertex coloring

22. 22 Outline Introduction Introduction Multicast-Capable WDM Networks Multicast-Capable WDM Networks Generalization of the conflict graph Generalization of the conflict graph Rings Rings Meshes Meshes Hypercubes Hypercubes Linear Arrays Linear Arrays Comparison and Conclusions Comparison and Conclusions

23. 23 Rings - Theorem 1 w r =  N /2 . The necessary and sufficient condition for a bidirectional WDM ring with N nodes to be rearrangeable for any multicast assignment under shortest path routing is the number of wavelengths w r =  N /2 .

24. 24 Meshes – Lemma 1 (Under row-major routing) In a p  q mesh, the degree of a vertex v in the modified conflict graph G’ associated with a source node that has a fan-out of m is  ( q  1)+ m  ( p  2).

25. 25 Proof of Lemma 1 S: source r: the row of S D j (1  j  m): destinations c j : the column of D j In row r, at most q  1 other source nodes. In column c j, at most p-2 destination nodes other than Dj.  deg(v)  (q-1)+m  (p  2).

26. 26 Meshes – Theorem 2 The number of wavelengths required for a p  q mesh to be rearrangeable for any multicast assignment under row-major routing is bounded by the following equation: Proved by Qiao & Mei

27. 27 Lemma G Let G be a simple graph. If, for some integer k, the number of vertices with degree  k is no more than k, then G is k colorable.

28. 28 Proof of Theorem 2 Assume that  at most k vertices in G ’ with degree  k, where 1  k  pq. (We want to find the smallest k.) Assume that  at most k vertices in G ’ with degree  k, where 1  k  pq. (We want to find the smallest k.) Then, for some m  0, q  1+( m  1)  ( p  2) < k  q  1+ m  ( p  2). Then, for some m  0, q  1+( m  1)  ( p  2) < k  q  1+ m  ( p  2). By Lemma 1, if deg( v )  k for some vertex v, the fan-out of v  m. By Lemma 1, if deg( v )  k for some vertex v, the fan-out of v  m.

29. 29 Proof of Theorem 2 (contd.) There are at most pq destinations  km  pq  ( q  1+( m  1)  ( p  2))  m < pq  ( p  2) m 2 + ( q  p  1) m  pq < 0   There are at most pq destinations  km  pq  ( q  1+( m  1)  ( p  2))  m < pq  ( p  2) m 2 + ( q  p  1) m  pq < 0  

30. 30 Proof of Theorem 2 (contd.) By Lemma G, By Lemma G, If p = q = n, the upper bound given here is If p = q = n, the upper bound given here is

31. 31 Outline Introduction Introduction Multicast-Capable WDM Networks Multicast-Capable WDM Networks Generalization of the conflict graph Generalization of the conflict graph Rings Rings Meshes Meshes Hypercubes Hypercubes Linear Arrays Linear Arrays Comparison and Conclusions Comparison and Conclusions

32. 32 e-cube routing Source s  destination d s = s n  1 s n  2 …s 1 s 0  s n  1 s n  2 …s 1 d 0  s n  1 s n  2 …d 1 d 0  …  s n  1 d n  2 …d 1 d 0  d n  1 d n  2 …d 1 d 0 = d Source s  destination d s = s n  1 s n  2 …s 1 s 0  s n  1 s n  2 …s 1 d 0  s n  1 s n  2 …d 1 d 0  …  s n  1 d n  2 …d 1 d 0  d n  1 d n  2 …d 1 d 0 = dwhere s n  1 s n  2 …s i+1 s i d n  1 …d 1 d 0  s n  1 s n  2 …s i+1 d i d n  1 …d 1 d 0 is called an ( i +1)th dimensional link.

33. 33 Lemma 2 In an n -dimensional hypercube under e-cube routing, the number of distinct source nodes that goes through some ( i +1)th dimensional link is 2 i, and the number of distinct destination nodes that goes through some ( i +1)th dimensional link is 2 n-1-i. In an n -dimensional hypercube under e-cube routing, the number of distinct source nodes that goes through some ( i +1)th dimensional link is 2 i, and the number of distinct destination nodes that goes through some ( i +1)th dimensional link is 2 n-1-i.

34. 34 Proof of Lemma 2 s n  1 s n  2 …s i+1 s i d n  1 …d 1 d 0  s n  1 s n  2 …s i+1 d i d n  1 …d 1 d 0 s n  1 s n  2 …s i+1 s i d n  1 …d 1 d 0  s n  1 s n  2 …s i+1 d i d n  1 …d 1 d 0 2 n-1-i 2 i 2 n-1-i 2 i

35. 35 Lemma 3 In an n -dimensional hypercube under e-cube routing, the degree of a vertex v in the modified conflict graph G ’ associated with a source node that has a fan-out of m is less than or equal to In an n -dimensional hypercube under e-cube routing, the degree of a vertex v in the modified conflict graph G ’ associated with a source node that has a fan-out of m is less than or equal to

36. 36 Proof of Lemma 3 Suppose a connection c : s  d in a multicast connection associated with vertex v goes through u links: k 1 +1, k 2 +1, …, k u +1-th dimensional links, where k 1 < k 2 < … < k u. Suppose a connection c : s  d in a multicast connection associated with vertex v goes through u links: k 1 +1, k 2 +1, …, k u +1-th dimensional links, where k 1 < k 2 < … < k u. Let N j ={ w  G’ : the multicast connection of w share the ( k j +1 ) th link with c }, 1  j  u. Let N j ={ w  G’ : the multicast connection of w share the ( k j +1 ) th link with c }, 1  j  u. By Lemma 2, By Lemma 2,

37. 37 Proof of Lemma 3 (contd.) Let i be the integer such that when j  i, and when j > i. Let i be the integer such that when j  i, and when j > i. That is, k j  ( n  1)/2 when j  i, and k j > ( n  1)/2 when j > i. That is, k j  ( n  1)/2 when j  i, and k j > ( n  1)/2 when j > i.

38. 38 Proof of Lemma 3 (contd.) Since k i  ( n  1)/2 =  n /2  1 and k i+1 > ( n  1)/2  n  1  k i+1 ( n  1)/2  n  1  k i+1 < ( n  1)/2 =  n /2  1 Therefore Therefore Each connection in the multicast connection represented by v contributes to the degree of v. Hence … Each connection in the multicast connection represented by v contributes to the degree of v. Hence …

39. 39 Theorem 3 The number of wavelengths required for an n-dimensional hypercube to be rearrangeable for any multicast assignment under e-cube routing is bounded by

40. 40 Proof of Theorem 3 Assume that  at most k vertices in G ’ with degree  k. (We want to find the smallest k.) Assume that  at most k vertices in G ’ with degree  k. (We want to find the smallest k.) Then, for some m  0, Then, for some m  0, By Lemma 3, if deg( v )  k for some vertex v, the fan-out of v  m. By Lemma 3, if deg( v )  k for some vertex v, the fan-out of v  m.

41. 41 Proof of Theorem 3 (contd.) There are at most 2 n connections There are at most 2 n connections Therefore Therefore

42. 42 Outline Introduction Introduction Multicast-Capable WDM Networks Multicast-Capable WDM Networks Generalization of the conflict graph Generalization of the conflict graph Rings Rings Meshes Meshes Hypercubes Hypercubes Linear Arrays Linear Arrays Comparison and Conclusions Comparison and Conclusions

43. 43 Linear array There are only two possible directions for any connection in a linear array and the routing algorithm is unique.

44. 44 Properties of Linear Array Different direction of connections do not have to use the same wavelength. Different direction of connections do not have to use the same wavelength. Revise the modified conflict graph G ’ such that each same-direction multicast connection is represented as a vertex of G ’. Revise the modified conflict graph G ’ such that each same-direction multicast connection is represented as a vertex of G ’. We only need to consider the longest connection among all connections in a same-direction multicast connection. We only need to consider the longest connection among all connections in a same-direction multicast connection.

45. 45 Example Multicast assignment Modified Conflict Graph

46. 46 Theorem 4 The necessary and sufficient condition for a linear array with N nodes to be rearrangeable for any multicast assignment is the number of wavelengths w r =  N /2 . The necessary and sufficient condition for a linear array with N nodes to be rearrangeable for any multicast assignment is the number of wavelengths w r =  N /2 .

47. 47 Proof of Theorem 4 (Necessity: w r   N /2  ) (Necessity: w r   N /2  )

48. 48 Proof of Theorem 4 (contd.) (Sufficiency: w r   N /2  ) Consider the rightward longest multicast connections. (Sufficiency: w r   N /2  ) Consider the rightward longest multicast connections. Let ( i 1, i 2 ), ( i 2, i 3 ), …, ( i k -2, i k-1 ), ( i k -1, i k ) be a maximal collection of multicast connections, called same-wavelength collection (SWC), denoted as ( i 1, i k ). Let ( i 1, i 2 ), ( i 2, i 3 ), …, ( i k -2, i k-1 ), ( i k -1, i k ) be a maximal collection of multicast connections, called same-wavelength collection (SWC), denoted as ( i 1, i k ).

49. 49 Proof of Theorem 4 (contd.) (Sufficiency: w r   N /2  ) (contd.) Each node a of the linear array can belong to only one SWC (i.e., the SWC is ( a, b ) or ( c, a )). (Sufficiency: w r   N /2  ) (contd.) Each node a of the linear array can belong to only one SWC (i.e., the SWC is ( a, b ) or ( c, a )).  There are at most  N /2  SWCs   N /2  wavelengths are sufficient.  There are at most  N /2  SWCs   N /2  wavelengths are sufficient.

50. 50 Summary ( q -1)

51. 51 Conclusion By introducing multicast-capable routing nodes, the number of wavelengths required for embedding an arbitrary multicast assignment is reduced a lot. By introducing multicast-capable routing nodes, the number of wavelengths required for embedding an arbitrary multicast assignment is reduced a lot. Future work: Deriving tighter bounds for n -dim hypercube and p  q mesh Future work: Deriving tighter bounds for n -dim hypercube and p  q mesh