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

2. 2 the fiber serves as a transmission medium Electronic switch Optic fiber Optical networks - 1 st generation

3. 3 Optical switch lightpath

4. 4 A virtual topology

5. 5 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

6. 6 lightpaths p1 p2 Valid coloring

7. 7 number of wavelengths

8. 8 Switching cost ADM OADM

9. 9 Electronic ADM

10. 10 p1 p2 Valid coloring Switching cost: number of ADMs

11. 11 W=2, ADM=8 W=3, ADM=7

12. 12 ring (Eilam, Moran, Zaks, 2002) reduction from coloring of circular arc graphs. NP-complete

13. 13 |ADMs|=7=7+0 |ADMs|=9=6+3 |ADMs| = N + |chains| Basic observation N lightpaths cycles chains

14. 14 In the approximation algorithms there are two common techniques for saving ADMs: Eliminate cycles of lightpaths Find matchings of lightpaths |ADMs| = N + |chains|

15. 15 w/out grooming: 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

16. 16 3/2 - Calinescu, Wan, 2002 10/7+  - Shalom, Z., 2004 10/7 - Epstein, Levin, 2004 ALG  2 x OPT Previous Work - ring

17. 17 low capacity requests can be groomed into high capacity wavelengths (colors). colors can be assigned such that at most g lightpaths with the same color can share an edge g is the grooming factor Traffic grooming

18. 18 W=2, ADM=8 W=1, ADM=7 g=2

19. 19 R: # of lightpaths ALG: # of ADMs used by the algorithm OPT: # of ADMs used by optimal solution w/ grooming: R/g  ALG  2R R/g  OPT  2R ALG  2g x OPT

20. 20 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);

21. 21 Algorithm (cont’d) Step 3: COVER  an approximation to the Minimum Weight Set Cover of S, using [Chvatal 79] Step 4: Convert COVER to a PARTITION Output: the coloring induced by PARTITION

22. 22 Legal coloring For any fixed g, the number of subsets constructed in the first phase is

23. 23 Analysis Legal coloring, B is 1-colorable  A is 1-colorable (  correctness). (and cost(A)  cost(B).)

24. 24 for every set cover SC.

25. 25 Lemma: There is a set cover SC, s.t.: for every set cover SC.

26. 26 Conclusion: For k = g ln g :

27. 27 Proof of Lemma Lemma: There is a set cover SC, s.t.:

28. 28  Consider OPT  x - a color of OPT.  P x - the set of paths colored x.  endpoints(P x ) - the set of ADMs operating at wavelength x.  (assume |endpoints(P x )|= ) Partition endpoints(P x ) into sets of k consecutive nodes.

29. 29 kk k k S 1 S 2 S m m=4 k=6

30. 30 w/o the assumption we have:

31. 31 Minimizing # of ADMs – Gerstel, Lin, Sasaki, 1998 … Traffic grooming – Gerstel, Ramaswamy, Sasaki, 1998 Zhu, Mukherjee, 2003 … References