Optimization Problems in Optical Networks

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

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

Lightpaths ADM Data in electronic form

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

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:

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

Wavelength Assignment Problem (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 |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. Fo graphs with max degree at most 2 we define:

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

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

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 (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.

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 :

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

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

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