1. Dynamic Wavelength Allocation in All-optical Ring Networks Ori Gerstel and Shay Kutten Proceedings of ICC'97

2. Static WLA in Rings Definitions: For any we define: and :

3. Static WLA in Rings Algorithm: Choose a node v such that. Duplicate node v and cut the cycle to form a line graph. Color the paths in using the greedy algorithm for line graphs with at most L colors. Color the paths in using colors: Number of colors used by the algorithm:

4. Static WLA in Rings vv

5. L=4 W=7 The above algorithm is optimal for some instances:

6. Dynamic Routing in Rings Input : A sequence of node pairs (s i,t i ). Output : for each (s i,t i ) decide online, “CLOCKWISE” or “COUNTER CLOCKWISE.” Goal : minimize L. An online algorithm: –Does not have any knowledge of subsequent inputs (j > i). –Can not change its decision on previous input elements (j < i).

7. Shortest Path Routing Algorithm SHORT: Given a pair (s i,t i ) route it on the shortest path on the ring. Claim: SHORT is 2 competitive ( )

8. Shortest Path Routing Proof: Consider an edge e such that : Let In OPT, there are at least x paths that use the edge e’ opposite to e. Then:

9. Dynamic WLA in Rings Algorithm WLA-1(L alg ). –It depends on an additional parameter which is the maximum anticipated load (L<=L alg ). – Pools of 2 L alg wavelengths each. 1.Given a path p, let l(p) its length. Choose i such that: 2.If the request is insert 3.If the request is delete

10. Dynamic WLA in Rings Claim: as long as L<=L alg, upon entering to step 2. Corollary: the algorithm colors all the paths using at most wavelengths.

11. Dynamic WLA in Rings Proof: Assume A0A0 B0B0

12. Dynamic WLA in Rings A0A0 B0B0 There are at most L alg such paths traversing A 0. They can be colored using at most L alg colors The paths not traversing A 0,do traverse B 0. They can be colored using at most L alg colors. We use at most 2 L alg colors for paths from class 0.

13. Dynamic WLA in Rings AiAi There are two types of paths: Paths traversing (exactly) one A i edge, (A) Paths traversing no A i edge, (B). These edges traverse exactly one B edge. AiAi BiBi BiBi We have two sets of L colors for each of the A and B paths.

14. Dynamic WLA in Rings Algorithm WLA-2. –Same as WLA-1, except… –The pools are not static. We have a global pool of wavelengths. –As long as L<=L alg, Algorithm WLA-3 –No a priori allocation. –

15. Dynamic WLA in Rings A lower bound: –Assume L=2 –We describe an adversary, which works in phases. –Phase i ends when the algorithm uses i wavelengths. –At each phase the adversary issues requests of length at most 2 i. –There are phases. Therefore:

16. Dynamic WLA in Rings

17. For any even L, the above adversary will issue L/2 requests instead one. We get again the same lower bound: