1. 1 OPTICAL MULTICAST ROUTING

2. 2 Outlines Introduction Multicast Routing Problem Node Architecture OMMP Multicast Routing Problem

3. 3 Introduction Traditional communication models have been –one-to-one or unicast, and –one-to-all or broadcast. Between these two extremes lies multicast, –the targeting of a data stream to a selected set of receivers. This model is used to characterize the communication patterns in a wide spectrum of applications such as –replicated databases, –command and control systems, –distributed games, –audio/video conferencing, and –distributed interactive simulation.

4. 4 Introduction The following are some applications that make use of multicast communication. –Multimedia: A number of users "tune in" to a video or audio transmission from a multimedia source station. –Teleconferencing: A group of workstations forms a multicast group so that a transmission from any member is received by all other group members. –Databases: All copies of a replicated file or database are updated at the same time. –Distributed computation: Intermediate results are sent to all participants in a distributed computation. –Real-time workgroups: Files, graphics, and messages are exchanged among active group members in real time.

5. 5 Introduction A good communication network which provides a multicast transmission system should have properties such as –high probability of delivery of information, –low delay between source and destinations, and –Information hiding from intermediate routers. These properties can be achieved by using optical signals instead of electrical signals to transfer the information. The optical medium will also provide enormous bandwidth (tens of terahertz). It is very difficult to exploit such a high bandwidth as a single channel. Hence the complete bandwidth is channelized, with different wavelengths using WDM.

6. 6 Introduction Among the WDM optical networks, wavelength routed networks are becoming popular for wide area networks. To support multicasting, in wavelength routed networks, the routing nodes should have the capability of optical splitting. If it is assumed that every node in the network has optical splitting capability and wavelength conversion capability, then the problem of multicast routing boils down to the problem of multicast routing in electronic networks. However, split-capable nodes are costlier than wavelength routing nodes without split capability. Hence it is suggested that only a few nodes in the network be allowed to have splitting capability. The multicast routing problem in wavelength routed networks is addressed in this chapter.

7. 7 Multicast Routing Problem A WDM network employing wavelength routing consists of optical wavelength routing nodes interconnected by point to point fiber links in an arbitrary topology. The optical routing nodes do have the capability of switching a wavelength individually. A wavelength routing node may have the capability to tap a small amount of optical power from the wavelength channel which is forwarded by that node. The tapped optical power may be used by the local node. This type of node is called as a drop and continue node (DaC node) (Tap and Continuous) [5], [197].

8. 8 Conventional v.s. WDM To support multicasting, in a conventional network (electronic network), all nodes are assumed to have the capability of buffering an incoming message and transmitting it onto more than one output link. To support multicasting in a WDM network, nodes in the network need to have light (optical) splitting capability. A node with splitting capability can forward an incoming message to more than one output link. If a network has splitting capability at all nodes, then it is referred to as a network with full splitting capability. In a network with full splitting capability, a single tree can be generated to include all the destinations of a multicast session, as in a conventional electronic network [124], [151].

9. 9 sparse splitting capability A tree refers to a set of destinations connected together with a source as the root. The source needs to transmit a message only once to communicate to all the destinations belonging to the same tree. A split-capable node is very expensive due to its complex architecture [197]. Hence only a subset of the nodes in a network are assumed to be split-capable nodes. A network with a few split-capable nodes is called a network with sparse splitting capability [92]. In a network with sparse splitting capability, it may not be possible to include all destinations of a session in one multicast tree. Hence a set of trees is constructed to include all destinations of a multicast session. This means that the source needs to transmit the multicast data onto more than one channel, maybe on different fibers or on different wavelengths. The set of trees corresponding to a single multicast session is called a multicast forest [92], [159], [197].

10. 10

11. 11 Multicast Routing Problem A network with sparse wavelength conversion and sparse splitting capability consists of nodes with different capabilities. A node may have splitting capability and/or wavelength conversion capability. In general, a node with split capability is called a multicast-capable node (MC node). A node with only splitting capability is called a split node, and A DaC node with wavelength conversion capability is termed a wavelength conversion node (WC node). A node having both splitting and wavelength conversion capabilities is called a virtual source (VS). A VS node can transmit an incoming message to any number of output links on any wavelength.

12. 12 Multicast Routing Problem A split node and a VS node are used for expanding the tree. A part of the tree which is expanded from a split node or VS node is called a subtree. The difference between a split node and a VS node is that a split node cannot support more than one connection on the same outgoing link, whereas a VS does support by using different wavelengths. This means that the subtree spawned from a VS may use the same physical link which is used by the existing connection but on a different wavelength. A node without splitting capability is called a multicast- incapable node (MI node). –An MI node may have the capability of drop and continue or it may have no such capability. –A node without any capability is an ordinary wavelength router, and it can either drop a message or can switch a message, but not both operations simultaneously. – Hence, for consistency, such a node can be called a drop or continue node (DoC node).

13. 13 Node Architecture of MC nodes

14. 14 Node Architecture of MC nodes An MC node consists of optical power splitters. To support multicast, the input signal needs to be transmitted onto various output connections. These connections may be on different links connected to the node or on different wavelength channels of the same fiber. If a node has n input links with a single fiber per link, and each fiber consists of W number of wavelengths, then to support multicast the input signal needs to be selected by n x W number of switches.

15. 15 MC nodes However, this wavelength conversion stage is an optional one. An MC node need not perform wavelength conversion. Finally, for each output link, one multiplexer is present so that the signals on various wavelengths are multiplexed and transmitted onto the same fiber. In general, for an MC node with n input links, m output links, and W wavelengths per fiber, there are n number of 1 x m splitters, n x m number of 1 x W optical splitters, m x W number of n x 1 SD switches, m x W number of TFs and wavelength converters, and m multiplexers. Power amplification is required to compensate for the power loss due to splitting operation.

16. 16 Architecture of an MI Node A node without splitting capability is called a multicast-incapable node or an MI node. MI nodes may tap (or drop) a small fraction of signal and switch the remaining signal to one of its neighbor nodes. The tapped (dropped) signal may be used by the local station. This type of node which can tap or drop the signal passing through it is called as a DaC node. The architecture of a DaC node is shown in Fig. 8.3. Here, the input signal is demultiplexed and each of these demultiplexed signals is fed as input to the "tap" (drop) module which taps 5% of signal. Then, as in a conventional wavelength routed node, the signal is switched using SD switches.

17. 17

18. 18 MULTICAST TREE GENERATION Multicasting is the process of transmitting data by a source to a set of destinations. Instead of transmitting packets from a sender to each receiver, the routes between source and receivers can share some links. In conventional networks, multicast route determination is traditionally formulated as a problem related to tree construction [161].

19. 19 Tree construction The reasons for adapting tree structure for multicast communication are listed below. –The source need not send a packet to individual destinations. –The packets are transmitted in parallel to various destinations. –The tree structure minimizes data replication, since the packet is replicated optically by routers only at branch points in the tree.

20. 20 Overview In a network with full splitting capability, a single tree can be generated to include all the destinations of a multicast session, as in a conventional electronic network [124], [151], [152]. Hence, in a network with full splitting capability, the constraints to generate a multicast tree would be minimizing the number of transceivers and wavelengths in a fiber. In a network with sparse splitting capability, it may not be possible to include all the destinations of a session in one multicast tree. Hence, a set of trees is constructed to include all the destinations of a multicast session. This means that the source needs to transmit the multicast data onto more than one channel and maybe on different fibers or on different wavelengths. The set of trees corresponding to a single session is called a multicast forest.

21. 21 DaC network In a densely connected network, the number of splitters required in a node is high. Hence it is difficult and also expensive to fabricate such a node. Also, the optical power splitting causes loss in the optical signal and hence optical amplifiers are required. In [5], a new tree generation algorithm is proposed which uses only DaC capability of a node. First, a directed graph is constructed for the given network. Then, an optimal trail, which starts from the source of a multicast session and visits all destinations, is computed.

22. 22 multiple-destination minimum- cost trail [MDMCT]) A trail is a path where nodes are allowed to be visited more than once. A node, if it is a destination, taps a small amount of optical signal. In [5], it is stated that constructing a trail with an objective of minimizing the number of directed edges (also referred to as multiple-destination minimum-cost trail [MDMCT]) is an NP-complete problem. Hence a heuristic is presented in which first a Steiner tree is constructed using a minimum path heuristic [168], then a trail is computed around the Steiner tree. Even though this method of constructing a multicast tree avoids using optical split at the nodes, it requires more number of wavelength channels. This is because every link in the Steiner tree is traversed twice, once in each direction.

23. 23 Multicast Tree Generation Full Splitting Capability: all nodes in the network have split capability. Hence, a single tree can be generated to route multicast traffic. The tree generation is similar to the tree generation in conventional networks. Apart from finding a path, in a conventional network it is necessary to allocate bandwidth and buffers. But in optical networks, the constraints are number of transceivers and wavelengths. Hence, the tree generation algorithm should consider the availability of these resources while generating a multicast tree.

24. 24 Light-tree In [151], the problem of minimizing the number of transmitters and receivers that are required to generate a multicast tree is considered. It introduces the concept of a light-tree. A light-tree is a point-to-multipoint optical path established in the network created by allocating the same wavelength on every link of the tree. The concept of light-trees can be implemented by incorporating optical multicasting (splitting) capability at all nodes of a network in order to reduce the average packet hop distance and the total number of transceivers in a network. Thus, a light-tree provides single-hop communication between a source node and a set of destination nodes. A solution is provided in [151] for routing unicast traffic and broadcast traffic using light-trees. To carry unicast traffic, the virtual topology design problem is formulated based on the light-tree concept.

25. 25 The proposed optimization problem has one of the following objective functions: –Minimize the network-wide average packet hop distance. –Minimize the total number of transceivers in the network. In [151] it is demonstrated that the average packet hop distance for a virtual topology based on light-trees is less than that of a virtual topology based on the lightpath concept. It is also demonstrated that the number of transmitters and receivers required for the virtual topology based on light-trees is less than the number of transmitters and receivers required for the virtual topology designed based on the lightpath concept. For broadcast traffic, minimization of the number of transceivers is considered the objective function. The light-tree concept can also be applied to multicast traffic.

26. 26

27. 27

28. 28 Genetic Algorithms for Multiple Multicast Problem on WDM Ring

29. 29 Problem Definition Ring network G(V,E) –V: the set of nodes –E: the set of links –bi-directional link –W wavelengths per link 。

30. 30 Problem Definition r groups of multicasts ， –M i ={s i, D i } ， i=1, 2, …, r, 1 ≦ k i ≦ n ； where –D i ={d 1i, d 2i, …, d kii } be the destination –s i :source –For each multicast M i ={s i, D i } ， a multicast tree MT i is need –Construct a multicast forest MF=U i=1,2,…r MT i 。 –Construct MF with wavelength continuity constraint, such the number of used wavelengths is minimized 。

31. 31 OMMP Optimal multiple multicast problem, OMMP 給定一個 WDM 網路與 r 個多點傳送的需求所 成的集合 M={Mi={si, Di} ， i=1, 2, …, r, 1 ≦ ki ≦ n} ，建立一個多點傳送樹林，並決定每 一個多點傳送樹之波長通道指派，使的所需求 的波長通道為最少。 OMMP is a NP-hard problem Since RWA(NP-hard) is a special case of OMMP RWA on Ring is a NP-hard problem.

32. 32 Example

33. 33 Possible Assignment of Example

34. 34 Observation Each MT i can be constructed by ： – 建立一個順時針方向的路徑： P c (s i, d l-1i ) – 建立一個逆時針方向的路徑： P r (s i, d l+1i ) – 建立兩個路徑，一個順時針與逆時針之路徑 P r (s i, d l’i ) 與 P c (s i, d l’i ) ，對某一個 l’  D 。

35. 35 Model

36. 36 Genetic Algorithm Begin Initialize population; while (not terminal condition) do Begin choose parents from population; /* Selection */ construct offspring by combining parents; /* Crossover */ optimize (offspring); /* Mutation */ if suited (offspring) then replace worst fit (population) with better offspring; /* Survival of the fittest */ End; End.

37. 37 Genetic Algorithm Chromosome Encoding Objective Function Penalty Function Crossover Mutation Selection

38. 38 Chromosome Encoding routing gene MG i ={mg i k, i=1,...,r; k=1,2} AG i ={ag i k, i=1,...,r; k=1,2} r: number of connections. r=4 7127...38122234

39. 39 Example of chromosome encoding 1 2 3 4 5 8 7 6

40. 40 Wavelength gene

41. 41 Objective Function Objective function The assignment represented by the connection may not constraint-satisfy, thus, a penalty function should be included in objective function.

42. 42 Penalty Function Assume both connections c 1 =(1,2) and c2=(1,4) are assigned to wavelength 1 with clockwise direction, then conflict occurred. Penalty should be defined. How to detect the conflict in a connection gene? A conflict-detection algorithm should be developed. O(M 2 ) pairs of connections should be examined. The conflict between two connections can be detected in constant time O(1).

43. 43 Graph AA 2 3 1 5 4 c 1 =(1,4), c 2 =(2,4), c 3 =(1,2), c 4 =(5,2) c1c1 c1c1 c2c2 c2c2 c3c3 c3c3 c4c4 c4c4 c1c1 c2c2 c3c3 c4c4

44. 44 Crossover Operators Single point crossover (SPC) Single point wavelength crossover (SPWC) Single point routing path crossover (SPWC) Single assigning wavelength exchanging operator (SAWEO) Wavelength exchanging operator (WEO)

45. 45 Mutations Single Routing Path Mutations (SRPM) Multiple Routing Paths Mutations (MRPM) Single wavelength assignment mutation (SWAM) Multiple wavelength assignment mutation (MWAM) Multicast assignment mutation (MAM)

46. 46 Heuristic Algorithms 2-phase algorithm –Routing phase Maximal-Gap Routing Minimal Load Routing –Assignment Phase Greedy Wavelength Assign

47. 47 Extended Genetic Algorithm Produce only feasible solutions No need for the penalty function The wavelength assigned to the multicast is determined by the greedy wavelength assign algorithm.

48. 48 Chromosome Encoding routing gene MG i ={mg i k, i=1,...,r; k=1,2} 7127...38122234

49. 49 Objective Function Objective function

50. 50 Operators Crossover: –single point crossover Mutation: –Single routing path mutation (SRPM) –Multiple routing paths mutation (MRPM)

51. 51 Hybrid Genetic Algorithm To speed up algorithm Heuristic routing algorithms are used. HGA1 =heuristics + SGA HGA2 =heuristics + EGA

52. 52 Experiments Run on PC with a Pentium III 1GHz CPU and 512MB RAM. For nodes n=100, 200, 300 Two sets of multicast requests are randomly generated. –Specific –Random MAXM={5, 10} : the maximal number destinations in D.

53. 53 Specific Set 1 n n/5 n/5+1 2n/5+1 2n/5 3n/5+1 3n/5 4n/5+1 4n/5

54. 54 Specific Set Ranges A i = { j | n*(i-1)/5+1 ≦ j ≦ n*i/5 } The source and destination nodes of multicast M i, i=1,2,...,r are randomly selected from nodes in A i and two of which are n*(i-1)/5+1 and n*i/5. The lower bound of the minimal used wavelengths of the set M specific is r/5.

55. 55 Specific n=100 (MAXM =5 or 10)

56. 56 Specific n=200 (MAXM =5 or 10)

57. 57 Specific n=300 (MAXM =5 or 10)

58. 58

59. 59 Random n=100 (MAXM =5 or 10)

60. 60 Random n=200 (MAXM =5 or 10)

61. 61 Random n=300 (MAXM =5 or 10)

62. 62

63. 63 More Improvement

64. 64 More Improvement

65. 65 Conclusion and Further Research Proposed –Mathematic Model for multiple multicast problem on WDM ring –Several Heuristic Algorithms –Genetic Algorithms Further Research in the problem –Lower bound proof –CPLEX package to found optimal solution –Other Soft-computing method Simulated Annealing, Tabu search, Ant algorithm, Scatter search

66. 66 Multicast Problem on WDM Multicast is a point to multipoint communication, by which a source node sends messages to multiple destination nodes. A light-tree, as a point to multipoint extension of a light-path, is a tree in the physical topology and occupies the same wavelength in all fiber links in the tree. Given an multicast request in a WDM network system, compute a set of routing trees and assign wavelengths to them such the cost is minimized.

67. 67 WDM switch sd1d1 d2d2 w1w1 w2w2 Multicast Capable d3d3 continuous drop

68. 68 Assumptions Source node be the uniquely one has light-splitting function (multicast capable). Each node of the tree is a multicast-Incapable optical switch (MI node) (no light-splitting function). Each node can perform drop and continue function. Single-hop WDM network All Optical Network Static Traffic

69. 69 Introduction The problem is formalized as follows: –Given a multicast request in a WDM network system, compute a set of routing paths (light-trees) and assign wavelengths to them such that the total cost in minimized. The objective function has two components – the number of used wavelengths –the cost of light-tree The objective is minimize cost of light-tree plus the cost α*(number of used wavelengths).

70. 70 System Models WDM network –Connected and undirected graph G(V, E, c, w) –V: vertex-set –E: edge-set –Each edge e in E is associated with two weight functions c(e): communication cost –W: the number of wavelengths provided by WDM network.

71. 71 System Models Cost of path P(u,v): A multicast request in the system are given, denoted by r (s, D) –source s –destination: D

72. 72 System Models T k (s, D k ) be the routing tree for request r (s, D) in wavelength k, where k<W. T= ∪ k=1,2,...,W T k (s, D k ); D= ∪ k=1,2,...,W D k ; T is the light-forest, D i ∩D j = ,for i≠j. The light signal is forwarded to (continuous) the output port leading to its child, which then transmit the signal to its child until all nodes in the D k receive it.

73. 73 Objective The cost of the light-tree T k (s, D k ) The cost of the light-forest T is defined:

74. 74 Total cost where y j =1 if wavelength j is used; y j =0, otherwise Special case: –One objective of the multicast routing is to construct a routing tree (or forest) which has the minimal cost. The problem is regarded as the minimum Steiner tree problem, which was proved to be NP-hard. –Another objective is to minimize the number of wavelengths used in the system.

75. 75 s 791012 4 5 8 13 14 1 2 11 3 15 1617 6 2 3 5 4 6 1 9 4 10 7 11 5 8 2 2 3 5 6 31 4 4 2 6 7 3 1 3 2 destination node 3 Example Light-splitting

76. 76 Genetic Algorithm Basic idea: modified the GA of R-H Whang et al. to WDM network p i is between 1 and R i, i=1,2,...,|D|, where R i is the number of candidate path from s to d i p1p1 p2p2 p3p3 p4p4 pipi P |D|

77. 77 p1p1 p2p2 p3p3 p4p4 pipi P |D| Chromosome Encoding

78. 78 Light-Forest Construct Algorithm(LFCA) Path by path construct –Integrated the path and wavelength in single phase Step 1: Sort paths in increasing order according to the cost of each path O(|D| log |D|) time. Assume that p 1,p 2,...., p |D| be the new index. Step 2: p1 is assigned to wavelength 1,w=1, T 1 ={p 1 }, T 2 =...=T k =ø. O(n)

79. 79 Light-Forest Construct Algorithm Step 3: For i= 2 to |D] do Begin –j=1 – while j ≦ w do { if pi is not conflict with Tj –then »{assigned pi to T j »T j =T j ∪ p i »flag=TRUE} –else j=j+1 } if flag is not TRUE –then »w=w+1 »Tw=Tw ∪ pi –End Time complexity: O(|D| 2 *n)

80. 80 s 791012 4 5 8 13 141 2113 15 16176 6 3 5 4 6 1 9 4 10 7 11 5 8 2 2 3 5 6 31 4 4 12 6 7 3 1 3 2 Example p 1 =s  7  1 (10) p 2 =s  7  14  2 (13) p 3 =s  9  13  3 (15) p 4 =s  10  4 (8) p 5 =s  10  4  5 (12) p 6 =s  9  13  5  6 (26) cost=8+10+4+15+13+26+2*α

81. 81 Conflict Test light-tree is represented by a directed tree root at s. O(n) time: add path into a directed tree, then test the out-degree of the visited vertex, if the out-degree >1 then conflict occurred.

82. 82 The light-forest construct a feasible solution of the WDM network, thus, there is no need for the penalty function Minimized Transform to maximization form where C max denotes the maximum value observer so far of the cost function in the population. Fitness Function Fitness = C max - Cost

83. 83 Crossover Operator single point crossover multiple point crossover

84. 84 Single point Crossover 2 31413 1 22321 2 31421 1 22313 After crossover, the light-forest should be reconstructed

85. 85 Multiple point Crossover 2 31413 1 22321 2 32313 1 21421 After crossover, the light-forest should be reconstructed

86. 86 Mutation Operator single point mutation heuristic mutation

87. 87 Single point mutation After single point mutation, the light-forest may be changed. The old path is traversed backward from di to s –The edge we traversed are removed If the use(e)=1 until the following saturations occurred, reach s reach destination node dl in D which p l is assigned to the same wavelength reach a node with out-degree > 1.

88. 88 Example of single point mutation s 791012 4 5 8 13 141 2113 15 16176 6 3 5 4 6 1 9 4 10 7 11 5 8 2 2 3 5 6 31 4 4 12 6 7 3 1 3 2 p 1 =s  7  1 (10) p 3 =s  9  13  3 (15) p 4 =s  10  4 (8) p 5 =s  10  4  5 (12)

89. 89 Example of single point mutation s 791012 4 5 8 13 141 2113 15 16176 6 3 5 4 6 1 9 4 10 7 11 5 8 2 2 3 5 6 31 4 4 12 6 7 3 1 3 2 p 1 =s  7  1 (10) p 3 =s  9  13  3 (15) p 4 =s  10  4 (8) p 5 =s  10  4  5 (12) if p5 is mutated to p5=s  8  5 then the old path 4  5 is removed and new path is tested whether is conflict to current light-tree or not. if no then assign new path to current wavelength. otherwise, another light-tree of different wavelength is tested and selected to assign.

90. 90 Example of single point mutation s 791012 4 5 8 13 141 2113 15 16176 6 3 5 4 6 1 9 4 10 7 11 5 8 2 2 3 5 6 31 4 4 12 6 7 3 1 3 2 p 1 =s  7  1 (10) p 3 =s  9  13  3 (15) p 4 =s  10  4 (8) p 5 =s  10  4  5 (12) if p4 is mutated to p4=s  10  12  4 then the old path 4  5 is not removed and new path is tested whether is conflict to current light-tree or not. if no then assign new path to current wavelength. otherwise, another light-tree of different wavelength is tested and selected to assign.

91. 91 Example of mutation s 791012 4 5 8 13 141 2113 15 16176 6 3 5 4 6 1 9 4 10 7 11 5 8 2 2 3 5 6 31 4 4 12 6 7 3 1 3 2

92. 92 Heuristic Initialization Farthest First –To improve the performance of GA. Notations –Edge(P(s i,d i )) The set of edges that in path P(s,d i ) or edges that at least one of its endpoints (not s) on P(s, d i ). –G’=G-Edge(P(s i,d i ))

93. 93 Example Edge(P(s,2))={(s,7), (7,14), (14,2), (1,7), (1,14), (3,14), (2,11), (2,15), (2,16)} s 791012 4 5 8 13 14 1 2 11 3 15 1617 6 2 3 5 4 6 1 9 4 10 7 11 5 8 2 2 3 5 6 31 4 4 2 6 7 3 1 3 2 destination node 3

94. 94 Sub-trees Let degree(s) be the degree of the source node s on tree P (minimal cost tree). PT(v i ) be the sub-tree rooted at node v i. s 791012 4 5 8 13 14 1 2 11 3 15 1617 6 2 3 5 4 6 1 9 4 10 7 11 5 8 2 2 3 5 6 31 4 4 2 6 7 3 1 3 2 destination node 3 PT(7) PT(8) PT(10)

95. 95 Properties of sub-tree If there are more than one destination on the leaves in a sub-tree PT(v i ), than violated the light-splitting constraint. Thus, only one destination node on leaves can be chosen to route by this sub-tree and the others should be re-routed. To determined the rerouting paths of the destinations in the leaves, some heuristics are proposed.

96. 96 Algorithm: Farthest-First For each sub-tree PT(v i ), only the farthest destination is routed by the path P(s, d i ). s 791012 4 5 8 13 14 1 2 11 3 15 1617 6 2 3 5 4 6 1 9 4 10 7 11 5 8 2 2 3 5 6 31 4 4 2 6 7 3 1 3 2 destination node 3 10 7 6 Reroute

97. 97 Re-route How to re-route? The routed paths and nodes used by the farthest path in each sub-tree cannot be used twice. The re-routing paths together with the exist paths can not perform cycle(s) or violated the light-splitting constraint. EDGE(P(s,v)) in each sub-trees should be removed.

98. 98 s 912 13 1 11 3 15 1617 3 10 11 2 31 3 1 destination node Remove EDGE(P(s,v)) s 791012 4 5 8 13 14 1 2 11 3 15 1617 6 2 3 5 4 6 1 9 4 10 7 11 5 8 2 2 3 5 6 31 4 4 2 6 7 3 1 3 2 destination node 3 PT(7) PT(8) PT(10) UNREACH={ 3, 1 } G’

99. 99 s 9 12 13 1 11 3 15 1617 3 10 11 2 31 3 1 s 12 1 11 15 1617 3 10 11 2 3 3 1

100. 100 s 791012 4 5 8 13 14 1 2 11 3 15 1617 6 2 3 5 4 6 1 9 4 10 7 11 5 8 2 2 3 5 6 31 4 4 2 6 7 3 1 3 2 destination node 3

101. 101

102. 102 Experimental PC Pentium III 1GMHz, 512MB RAM. Borland C++. Windows 2000.

103. 103 Result n|D||D|αIAGAreduce 2041012311110.81% 206101341238.94% 2081016714713.61% 4041016714415.97% 4061018416412.20% 408102081956.67% 604102011943.61% 606102372217.24% 608102562435.35% average9.38%

104. 104 Conclusions A genetic algorithm is proposed to solve the Multicast routing on WDM network.