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Optimization Problems in Optical Networks

Optimization Problems in Optical Networks. Wavelength Division Multiplexing (WDM). Directed:. Optic Fiber. Symmetric:. Optic Fiber. Undirected:. Optic Fiber. Optic Switches. No to inputs with the same wavelength should be routed on the same edge. Lightpaths. ADM. ADM.

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Optimization Problems in Optical Networks

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  1. Optimization Problems in Optical Networks

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

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

  4. Lightpaths ADM ADM Data in electronic form Data in electronic form

  5. The Routing Problem • Input : • A graph G=(V,E) • A set or sequence of node pairs (ai,bi) • Output: • A set or sequence of paths pi =(ai, v1, …, bi)

  6. 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)=|Pe| • The (maximum, minimum, average) load on the network:

  7. 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:

  8. 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. • Formally we define:

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

  10. Routing and WLA (RLA) • Input : • A graph G=(V,E) • A set or sequence of node pairs (ai,bi) • Output: • A set or sequence of paths pi =(ai, v1, …, bi) • A coloring w of the paths: • Constraint:

  11. 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.

  12. 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 p1, …, p|L| using e, because the paths are simple. • Therefore :

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

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

  15. 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:

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