1. Approximating the Traffic Grooming Problem Mordo Shalom Tel Hai Academic College & Technion

2. Approximating the Traffic Grooming Problem2 Joint work with Michele Flammini – L ’ Aquila Luca Moscardelli – L ’ Aquila Shmuel Zaks - Technion

3. Approximating the Traffic Grooming Problem3 Outline Outline  Optical networks  The Min ADM Problem  The Traffic Grooming Problem  Algorithm GROOMBYSC

4. Approximating the Traffic Grooming Problem4 Outline Outline  Optical networks  The Min ADM Problem  The Traffic Grooming Problem  Algorithm GROOMBYSC

5. Approximating the Traffic Grooming Problem5 The MIN ADM Problem W=2, ADM=4 W=1, ADM=3

6. Approximating the Traffic Grooming Problem6 W-ADM tradeoff W=2, ADM=8 W=3, ADM=7

7. Approximating the Traffic Grooming Problem7 The Goal Given a set of lightpaths, find a valid coloring with minimum number of ADMs.

8. Approximating the Traffic Grooming Problem8 Outline Outline  Optical networks  The Min ADM Problem  The Traffic Grooming Problem  Algorithm GROOMBYSC

9. Approximating the Traffic Grooming Problem9 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.

10. Approximating the Traffic Grooming Problem10 The Traffic Grooming Problem W=2, ADM=8 W=1, ADM=7 g=2

11. Approximating the Traffic Grooming Problem11 The Goal Given a set of requests and a grooming factor g, find a valid coloring with minimum number of ADMs.

12. Approximating the Traffic Grooming Problem12 Notation & Immediate Results P: The set of paths. SOL: The # of ADMs used by a solution. OPT: The # of ADMs used by an optimal solution. |P|/g  SOL  2|P| |P|/g  OPT  2|P|  SOL = SOL/OPT  2g

13. Approximating the Traffic Grooming Problem13 Outline Outline  Optical networks  The Min ADM Problem  The Traffic Grooming Problem  Algorithm GROOMBYSC

14. Approximating the Traffic Grooming Problem14 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.

15. Approximating the Traffic Grooming Problem15 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);

16. Approximating the Traffic Grooming Problem16 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

17. Approximating the Traffic Grooming Problem17 Analysis Let, then: If B is 1-colorable then A is 1-colorable (  correctness). Cost(A)  Cost(B). Therefore: …

18. Approximating the Traffic Grooming Problem18 for every set cover SC.

19. Approximating the Traffic Grooming Problem19 Lemma: There is a set cover SC, s.t.: for any set cover SC.

20. Approximating the Traffic Grooming Problem20 Conclusion: For k = g ln g :

21. Approximating the Traffic Grooming Problem21 Proof of Lemma Lemma: There is a set cover SC, s.t.:

22. Approximating the Traffic Grooming Problem22 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

23. Approximating the Traffic Grooming Problem23 kk k k S 1 S 2 S m M=4 k=6

24. Approximating the Traffic Grooming Problem24 Analysis (cont’d) w/o the assumption we have:

25. Approximating the Traffic Grooming Problem25 Analysis (cont’d) and also 1- colorable thus Moreover Therefore Is a set cover with sets from S.

26. Approximating the Traffic Grooming Problem26