1. 1 Electronic ADM

2. 2

3. 3 ADM (add-drop multiplexer)

4. 4 Cost Measure: # of ADMs Each lightpath requires 2 ADM ’ s, one at each endpoint, as described before. A total of 2|P | ADM ’ s. But two paths p=(a, …,b) and p ’ =(b, …,c), such that w(p)=w(p ’ ) can share the ADM in their common endpoint b. This saves one ADM. For graphs with max degree at most 2 we fix an arbitrary orientation and define:

5. 5 Static WLA in Line Graphs Note: After a slight modification, the algorithm solves optimally the MINADM problem too: At each node, first use the colors added to at this step. It ’ s straightforward to show that this: Does not harm the optimality w.r.t. to the MINW prb. Minimizes for every node v. Therefore minimizes

6. 6 number of wavelengths Switching cost ADM

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

8. 8 ring (Eilam, Moran, Zaks, 2002) reduction from coloring of circular arc graphs. NP-complete

9. 9 Coloring of Circular arc Graphs Consider: a ring H (the host graph) and A set of paths P in H. The graph G=(P,E) constructed as follows is a circular arc graph: There is an edge (p1,p2) in e if and only if p1 and p2 have a common edge in H. The problem of finding the chromatic number of a circular arc graph is NP-Hard [Tuc 75 ’ ]

10. 10 The reduction The min W problem is exactly the circular arc coloring problem. But we will show NP- hardness even of the special case L=L min. Given an instance C,P where C is the ring and P is the set of paths, we construct an instance C, P ’ (by adding paths of length 1 to P) such that L min (P ’ )=L(P ’ )=L(P). (A full instance)

11. 11 The reduction (cont ’ d) Claim: P is L-colorable iff P ’ is L-colorable. Therefore: Circular Arc Graph Coloring is NP-Hard even for full instances.

12. 12 |ADMs|=7=7+0 |ADMs|=9=6+3 |ADMs| = N + |chains| Basic observation N lightpaths cycles chains

13. 13 The reduction (cont ’ d) Let P’ a full instance of Circular Arcs P ’ is L-colorable iff P ’ can be partitioned into L cycles iff ADM(P ’ )=|P ’ |.

14. 14 |P|  ALG  2x|P| |P|  OPT  2x|P| ALG  2 x OPT |P|: # of lightpaths ALG: # of ADMs used by the algorithm OPT: # of ADMs used by optimal solution Approximation algorithms