1. 1 Distributed Computing Optical networks: switching cost and traffic grooming Shmuel Zaks zaks@cs.technion.ac.il ©

2. 2 Outline Outline  Optical networks  Model  The Min ADM Problem  The Traffic Grooming Problem  Algorithm GROOMBYSC

3. 3 the fiber serves as a transmission medium Electronic switch Optic fiber Optical networks - 1 st generation

4. 4 Routing in the optical domain Two complementing technologies: - Wavelength Division Multiplexing (WDM): Transmission of data simultaneously at multiple wavelengths over same fiber - Optical switches: the output port is determined according to the input port and the wavelength Optical networks - 2 nd generation

5. 5 Wavelength Division Multiplexing (WDM) Directed: Symmetric: Undirected: Optic Fiber

6. 6 Optical Switches No two inputs with the same wavelength should be routed on the same edge.

7. 7 Lightpaths ADM Data in electronic form

8. 8 A virtual topology

9. 9 Lightpaths p1 p2 Valid coloring

10. 10 The Routing Problem Input : A graph G=(V,E) A set or sequence of node pairs (a i,b i ) Output: A set or sequence of paths p i =(a i, v 1, …, b i )

11. 11 The Load Given a graph G=(V,E) and a set P of paths on the graph, we define: for any edge e of the graph: the load on this edge l(e)=|P e | The (maximum, minimum, average) load on the network:

12. 12 Wavelength Assignment Problem (WLA) Input: A graph G=(V,E). A set or sequence of paths P. Output: A coloring w of the paths: Constraint:

13. 13 Routing and WLA (RLA/WRA) Input : A graph G=(V,E) A set or sequence of node pairs (a i,b i ) Output: A set or sequence of paths p i =(a i, v 1, …, b i ) A coloring w of the paths: Constraint:

14. 14 Cost Measure: # of colors For any legal coloring w of the paths:

15. 15 Optimization Problems Goal: MINW: Minimize W. or MAXPC: Maximize |Domain(w) | under the constraint W<=W max.

16. 16 Static vs. Dynamic vs. Incremental Static: The input is a set (of pairs or paths), the algorithm calculates its output based on the input. Incremental (Online): The input is a sequence of input elements (pairs or paths). It is supplied to the algorithm one element at a time. The output corresponding to the input element is calculated w/o knowledge of the subsequent input elements. Dynamic: Similar to incremental The sequence may contain deletion requests for previous elements.

17. 17 WLA (A trivial lower bound) For any instance of the WLA problem: W>=L. Proof: Consider an edge e, such that L=l(e). There are L paths p 1, …, p |L| using e, because the paths are simple. Therefore :

18. 18 WLA (A trivial lower bound) For some instances W > L. L=2 W=3

19. 19 Static WLA in Line Graphs The GREEDY algorithm: // The set of integers for i = 1 to |V| do for each path p=(x,i) do for each path p=(i,x) do

20. 20 Static WLA in Line Graphs Correctness, obvious. Optimality: By induction, After node i is processed, the claim is correct, i.e. Where W(i) is the value of after node i is pocessed, and L(i) is the maximum load on the edges processed so far.

21. 21 Outline Outline  Optical networks  Model  The Min ADM Problem  The Traffic Grooming Problem  Algorithm GROOMBYSC

22. 22 Electronic ADM

23. 23 number of wavelengths Switching cost ADM

24. 24 The MIN ADM Problem W=2, ADM=4 W=1, ADM=3

25. 25 The Goal Given a set of lightpaths, find a valid coloring with minimum number of ADMs.

26. 26 Static WLA in Line Graphs Note: After a slight modification, the Greedy algorithm solves optimally the MINADM problem too: At each node, first use the colors added to at this step. It ’ s straigtforward to show that this: Does not harm the optimality w.r. to the MINW prb. Solves the MINADM problem optimally at each node.

27. 27 Static WLA in Line Graphs The GREEDY algorithm: // The set of integers for i = 1 to |V| do for each path p=(x,i) do for each path p=(i,x) do

28. 28 W-ADM tradeoff W=2, ADM=8 W=3, ADM=7

29. 29 ring (Eilam, Moran, Zaks, 2002) reduction from coloring of circular arc graphs. NP-complete Minimizing # of ADMs – Gerstel, Lin, Sasaki, 1998

30. 30 Coloring of Circular arc Graphs Consider: a ring H (the host graph) and A set of paths P in H. The graph G=(P,E) constructed as follows is a circular arc graph: There is an edge (p1,p2) in e if and only if p1 and p2 have a common edge in H. The problem of finding the chromatic number of a circular arc graph is NP-Hard [Tuc 75 ’ ]

31. 31 The reduction The min W problem is exactly the circular arc coloring problem. But we will show NP- hardness even of the special case L=L min. Given an instance C,P where C is the ring and P is the set of paths, we construct an instance C, P ’ (by adding paths to P) such that L min (P ’ )=L(P ’ )=L(P).

32. 32 The reduction (cont ’ d) Claim: P is L-colorable iff P ’ is L-colorable. Claim: ADM(P)=ADM(P ’ ). Therefore w.l.o.g. all the edges have the same load (L).

33. 33 |ADMs|=7=7+0 |ADMs|=9=6+3 |ADMs| = N + |chains| Basic observation N lightpaths cycles chains

34. 34 The reduction (cont ’ d) P ’ is L-colorable iff P ’ can be partitioned into L cycles iff ADM(P ’ )=|P ’ |.

35. 35 R  ALG  2R R  OPT  2R ALG  2 x OPT R: # of lightpaths ALG: # of ADMs used by the algorithm OPT: # of ADMs used by optimal solution Approximation algorithms

36. 36 3/2 - Calinescu, Wan, 2002 10/7+  - Shalom, Z., 2004 10/7 - Epstein, Levin, 2004 ALG  2 x OPT Approximation algorithms

37. 37 Outline Outline  Optical networks  Model  The Min ADM Problem  The Traffic Grooming Problem  Algorithm GROOMBYSC

38. 38 The Traffic Grooming Problem A generalization of the MIN ADM problem. Instead of requests for entire lightpaths, the input contains requests for integer multiples of 1/g of one lighpath’s bandwidth. g is an integer given with the instance.

39. 39 The Traffic Grooming Problem W=2, ADM=8 W=1, ADM=7 g=2

40. 40 The Goal Given a set of requests and a grooming factor g, find a valid coloring with minimum number of ADMs.

41. 41 Notation & Immediate Results R: The # of requests. SOL: The # of ADMs used by a solution. OPT: The # of ADMs used by an optimal solution. R/g  SOL  2R R/g  OPT  2R  SOL = SOL/OPT  2g

42. 42 Outline Outline  Optical networks  Model  The Min ADM Problem  The Traffic Grooming Problem  Algorithm GROOMBYSC

43. 43 Main Result g > 1, Ring Networks: General traffic: An O(log g) approximation algorithm for any fixed g. Can be used in general networks Analysis can be extended to some other topologies.

44. 44 Approximation algorithm (log g) Input: Graph G, set of lightpaths P, g > 0 Step 1 : Choose a parameter k = k(g). Step 2: Consider all subsets of P of size If a subset A is 1-colorable (i.e., any edge is used at most g times) then weight[A]=endpoints(A);

45. 45 Algorithm (cont’d) Step 3: COVER  (an approximation to) the Minimum Weight Set Cover of S[], weight[], using [Chvatal79] Step 4: Convert COVER to a PARTITION PARTITION induces a coloring of the paths

46. 46 Analysis Let, then: If B is 1-colorable then A is 1-colorable (  correctness). Cost(A)  Cost(B). Therefore: …

47. 47 for every set cover SC.

48. 48 Lemma: There is a set cover SC, s.t.: for any set cover SC.

49. 49 Conclusion: For k = g ln g :

50. 50 Proof of Lemma Lemma: There is a set cover SC, s.t.:

51. 51 Proof of Lemma Consider a color of OPT. Consider the set P of paths colored. Consider the set of ADMs operating at wavelength. (i.e. endpoints(P ) ) Divide endpoints(P ) into sets of k consecutive nodes. For simplicity assume |endpoints(P )|=m.k

52. 52 kk k k S 1 S 2 S m M=4 k=6

53. 53 Analysis (cont’d) w/o the assumption we have:

54. 54 Analysis (cont’d) thus Moreover Therefore Is a set cover considered by the algorithm.