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Electronic ADM

Electronic ADM. ADM (add-drop multiplexer). Cost Measure: # of ADMs. Each lightpath requires 2 ADM ’ s, one at each endpoint, as described before. A total of 2| P | ADM ’ s.

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Electronic ADM

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  1. Electronic ADM

  2. ADM (add-drop multiplexer)

  3. 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:

  4. 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

  5. Switching cost number of wavelengths ADM

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

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

  8. 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’]

  9. 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=Lmin. • 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 Lmin(P’)=L(P’)=L(P). (A full instance)

  10. 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.

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

  12. 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’|.

  13. Approximation algorithms • |P|: # of lightpaths • ALG: # of ADMs used by the algorithm • OPT: # of ADMs used byoptimalsolution |P|  ALG  2x|P| |P|  OPT  2x|P| ALG  2 x OPT

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