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Optimization Problems in Optical Networks

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

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

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Lightpaths ADM Data in electronic form

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

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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 )

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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:

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

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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:

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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).

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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:

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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.

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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 :

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

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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

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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:

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

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