1. Genetic Algorithm for Multicast in WDM Networks Der-Rong Din

2. Outline Introduction Problem formulation Genetic Algorithm Further Research Problem

3. Introduction There are two types of architectures of WDM optical networks: single-hop systems and multi-hop systems [2]. Single-hop system a communication channel should use the same wavelength throughout the route of the channel Multi-hop system a channel can consist of multiple light-paths and wavelength conversion is allowed at the joint nodes of two light-paths in the channel. In this paper, we consider single-hop systems, since all-optical wavelength conversion is still an immature and expensive technology. (no wavelength conversion)

4. Introduction 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.

5. Introduction Each node of the tree is a multicast- Incapable optical switch (MI node).

6. Introduction The problem is formalized as follows: given an multicast request in a WDM network system, compute a set of routing trees and assign wavelengths to them. The objective is to minimize the (cost + α* # of wavelength) number of distinct wavelengths to be used under the following constraints on each routing tree: the total cost of the tree.

7. System Models WDM network Connected and undirected graph G(V, E, c) V: vertex-set, |V|=n E: edge-set, |E|=m Each edge e in E is associated with a weight function c(e): communication cost

8. 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={d 1, d 2,..., d |D| }

9. System Models This paper assumes an input optical signal can only be forward to an output signal at a switch. T k (s, D k ) be the routing tree for request r (s, D) in wavelength k, where k<K, T= ∪ k=1,2,...,K T k ; D= ∪ k=1,2,...,K D k ; T is the light-forest. The light signal is forwarded to the output port leading to its child, which then transmit the signal to its child until all nodes in the D k receive it.

10. Objective The cost of the tree 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. In a single-hop WDM system, two channels must use different wavelengths if their routes share a common link, which is the wavelength conflict rule.

11. Genetic Algorithm for WDM Multicast Problem (WDMMP) Important components of GA Chromosome encoding Fitness function Penalty function Crossover operation Mutation operation.

12. 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 r(s, {1,2,3,4,5,6}

13. Example of GA since out-degree(s)=4, |D|=6, thus may be 2 wavelengths are need to multicast the request.

14. Genetic Algorithm #1 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|

15. p1p1 p2p2 p3p3 p4p4 pipi Chromosome Encoding

16. Light-Forest Construct Algorithm 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)

17. 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)

18. 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

19. Conflict Test Algorithm for path and Tree 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.

20. Penalty Function The light-forest construct a feasible solution of the WDM network, thus, there is no need for the penalty function.

21. 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 Algorithm

22. Crossover Operator single point crossover multiple point crossover

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

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

25. Mutation Operator single point mutation heuristic mutation

26. 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.

27. 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)

28. 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.

29. 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.

30. 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

31. Heuristic Mutations Wavelength reduced mutation try to reduced the number of wavelengths used by the mutlicast request Cost reduced mutation try to reduced the cost of each light-tree of different wavelengths used by the mutlicast request

32. Wavelength reduced mutation Let number dest(w i ) be the number of destination nodes in the wavelength wi. Find out the minimal dest(wi) of paths. Wavelength reduced mutation is reassigned the destination in this wavelength to another. Local optimal steategry.

33. Wavelength reduced mutation algorithm For the destination di which is selected to be assigned to another wavelength, choose wavelength wk, k is initially set to be 1. Remove the current light-tree in wavelength wk and form the graph G’, find a minimal cost path form s to G’, find minimal paths from dl to di, where dl is the destination node in wavelength wk and is a leaf node, Find the minimal cost of these paths resulted from 1 and 2. Reassign the wavelength of path pi to wk, Change the chromosome encoding in pi field to corresponding index.

34. Data structure The operation of the “Change the chromosome encoding in pi field to corresponding index” may cause some problem The new search path from s to di may not included in the rating table Ri. The searching time of path is long. To avoid the duplicated in the Ri, the operation should check whether or not the new path has been included in the Ri, if yes then return the corresponding index if no, then new path should be inserted into the Routing Table Ri of di, If the data structure of the routing table do not well-designed then the time spent for the heuristic mutation will long.

35. Data structure Operation: Given a index pi, return the path from s to di. Given a path, check that whether this is path is in the Ri, if yes return the index of pi; otherwise, insert this path into Ri, and return the new index of pi. Data structure Index array (IA) Depth search tree (DST) Double Links between DST and IA

36. DST For each destination di, Find k-shortest path for the di from s to di on G. 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 some paths from s to 6 s  7  14  2  16  17  6 s  7  14  2  15  6 s  7  14  2  15  5  6 s  7  14  2  11  3  13  5  6 s  7  14  2  11  3  9  8  5  6 s  7  14  2  11  3  13  1  9  8  5  6 s  10  4  5  6 s  10  12  5  6 s  10  4  12  5  6

37. DST some paths from s to 6 s  7  14  2  16  17  6 s  7  14  2  15  6 s  7  14  2  15  5  6 s  7  14  2  11  3  13  5  6 s  7  14  2  11  3  9  8  5  6 s  7  14  2  11  3  13  1  9  8  5  6 s  10  4  5  6 s  10  12  5  6 s  10  4  12  5  6 S 7 14 2 15 1611 175 3 13 5 5 1 9 9 8 8 10 4 12 5 5 5 5

38. IA +DST S 7 14 2 15 1611 175 3 13 5 5 1 9 9 8 8 10 4 12 5 5 5 5 123456789123456789

39. Cost reduced mutation For each wavelength (each ligth-tree), if dest(wi) >1 then fine the longest path in this light-tree, try to find another shorter path to replaced it. That is: find a minimal cost path form s to G’, find minimal paths from dl to di, where dl is the destination node in wavelength wk and is a leaf node, Find the minimal cost of these paths resulted from 1 and 2. Reassign the wavelength of path pi to wk, Change the chromosome encoding in pi field to corresponding index.

40. Notice The IA and DST structure were established during the initial phase.

41. Some Problem The set of paths should be used to construct a tree of forest on WDM network to satisfy the wavelength constraint. An tree constructing algorithm is needed. About O(|D|*n) An wavelength assignment is needed. About O(e) time. An integrated algorithm can be proposed to combine two algorithms.

42. Time complexity analysis Random generated a population path- oriented gene without wavelength assignment. Determine the result WDM-forest by applying integrated algorithm. Time complexity: O(e + |D|n)* population_size * generation_size.

43. Paper Figure IP router WDM switch s d1d1 d2d2 IP router WDM switch s d1d1 d2d2

44. 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

45. 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

46. 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

47. 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

48. 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

49. Pair (s,d i )pathCost P 1 =(s,1) s  7  1 10 P 2 =(s,2) s  7  14  2 13 P 3 =(s,3) s  9  13  3 15 P 4 =(s,4) s  10  4 8 P 5 =(s,5) s  10  4  5 12 P 6 =(s,6) s  9  13  5  6 26

50. Example 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 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)

51. 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

52. 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

53. 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

54. 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

55. 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

56. 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 s->