Dynamic Wavelength Allocation Strategies in All-Optical Ring Networks

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 v v

5. Static WLA in Rings The above algorithm is optimal for some instances: • L=4 • W=7

6. Dynamic Routing in Rings • Input : A sequence of node pairs (si,ti). • Output : for each (si,ti) 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 (si,ti) 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 : Suppose In OPT, there are at least x paths that use the edge e’ opposite to e. Therefore:

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

10. B0 Dynamic WLA in Rings • Claim: as long as L<=Lalg,upon entering to step 2 • Corollary: the algorithm colors all the paths using at most wavelengths. • Assume A0

11. A0 B0 Dynamic WLA in Rings There are at most Lalg such paths traversing A0. They can be colored using at most Lalg colors The paths not traversing A0 ,do traverse B0. They can be colored using at most Lalg colors. We use at most 2 Lalg colors for paths from class 0.

12. Bi Bi Ai Dynamic WLA in Rings • There are three types of paths: • Paths traversing one Ai edge, but no Bi edge. (A) • Paths traversing one Bi edge, but no Ai edge. (B) • Paths traversing one Ai edge, and one Bi edge. (AB-BA) Ai We have two sets of L colors for each of the A and B paths. Given an AB (BA) path, we use an unused color from the A (B) set .

13. 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<=Lalg, • Algorithm WLA-3 • No a priori allocation.

14. 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 2i. • There are phases. Therefore:

15. Dynamic WLA in Rings

16. Dynamic WLA in Rings • For any even L, the above adversary will issue L/2 requests instead one. • We get again the same lower bound: