1 Distributed Computing Optical networks: switching cost and traffic grooming Shmuel Zaks ©

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

3 Optical switch lightpath

4 A virtual topology

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 lightpaths p1 p2 Valid coloring

7 number of wavelengths

8 Switching cost ADM OADM

9 Electronic ADM

10 p1 p2 Valid coloring Switching cost: number of ADMs

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

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

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

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

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

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

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

24 for every set cover SC.

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

26 Conclusion: For k = g ln g :

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

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

30 w/o the assumption we have:

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