1. Optimization Problems in Optical Networks

2. Wavelength Division Multiplexing (WDM) Directed: Symmetric: Optic Fiber

3. Optic Switches No two inputs with the same wavelength should be routed on the same edge.

4. Lightpaths ADM Data in electronic form

5. Wavelength Division Multiplexing (WDM) Directed: Symmetric: Undirected: Optic Fiber

6. Routing Input : –A graph G=(V,E) –A sequence of node pairs (a i,b i ) Output: –A sequence of paths p i =(a i, v 1, …, b i )

7. The Load Given a graph G=(V,E) and a set P of paths on the graph, we define: for any edge e of the graph: – –the load on this edge l(e)=|P e | The (maximum, minimum, average) load on the network:

8. Wavelength Assignment (WLA) Input: –A graph G=(V,E). –A set or sequence of paths P. Output: –A coloring w of the paths: Constraint:

9. Wavelength Assignment (WLA) For any legal coloring w of the paths we define: Each lightpath requires 2 ADM’s, one at each endpoint, as described before. A total of 2.|P| ADM’s. 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 define:

10. Wavelength Assignment Problem (WLA) Possible goals: –MINW: Minimize W. or –MAXPC: Maximize |Domain(w)| under the constraint W<=W max. or –MINADM: Minimize ADM(w).

11. Routing and WLA (RLA) Input : –A graph G=(V,E) –A sequence of node pairs (a i,b i ) Output: –A sequence of paths p i =(a i, v 1, …, b i ) –A coloring w of the paths: –Constraint:

12. Static vs. Dynamic vs. Incremental Static: The input is a set (of pairs or paths), the algorithm calculates its output based on the input. Incremental (Online): –The input is a sequence of input elements (i.e. node pairs or paths). –It is supplied to the algorithm one element at a time. –The output corresponding to the input element is calculated w/o knowledge of the subsequent input elements. Dynamic: –Similar to incremental –The sequence may contain deletion requests for previous elements.

13. WLA (A trivial lower bound) For any instance of the WLA problem: W>=L. Proof: –Consider an edge e, such that L=l(e). –There are L paths p 1, …, p |L| using e, because the paths are simple. – –Therefore :

14. WLA (A trivial lower bound) For some instances W > L. L=2 W=3

15. Static WLA in Line Graphs The GREEDY algorithm: // The set of integers for i = 1 to |V| do –for each path p=(…,i) do –for each path p=(i,…) do

16. Static WLA in Path Graphs Correctness: We prove by induction on i that, after node i is processed, the following holds: This implies that at any time: Therefore:

17. Static WLA in Path Graphs GREEDY minimizes ADM(w) Proof: –For any v, GREEDY minimizes ADM v (w).