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Approximating the Traffic Grooming Problem

Approximating the Traffic Grooming Problem. Mordo Shalom Tel Hai Academic College & Technion. Joint work with Michele Flammini – L ’ Aquila Luca Moscardelli – L ’ Aquila Shmuel Zaks - Technion. Outline. Optical networks The Min ADM Problem The Traffic Grooming Problem

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Approximating the Traffic Grooming Problem

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

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

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

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

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

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

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

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

  9. 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. Approximating the Traffic Grooming Problem

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

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

  12. 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| rSOL = SOL/OPT  2g Approximating the Traffic Grooming Problem

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

  14. 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. Approximating the Traffic Grooming Problem

  15. 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); Approximating the Traffic Grooming Problem

  16. 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 Approximating the Traffic Grooming Problem

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

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

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

  20. Conclusion: Fork = g ln g : Approximating the Traffic Grooming Problem

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

  22. Proof of Lemma • Consider a color l of OPT. • Consider the set Pl of paths colored l. • Consider the set of ADMs operating at wavelength l. (i.e. endpoints(Pl) ) • Divide endpoints(Pl) into sets of k consecutive nodes. • For simplicity assume |endpoints(Pl)|=m.k Approximating the Traffic Grooming Problem

  23. M=4 k=6 k k k k S1 S2 Sm Approximating the Traffic Grooming Problem

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

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

  26. Thank You ! Approximating the Traffic Grooming Problem

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