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Published bySteven Merritt Modified over 9 years ago
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1 Electronic ADM
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3 ADM (add-drop multiplexer)
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4 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:
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5 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
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6 number of wavelengths Switching cost ADM
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7 W=2, ADM=8 W=3, ADM=7
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8 ring (Eilam, Moran, Zaks, 2002) reduction from coloring of circular arc graphs. NP-complete
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9 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 ’ ]
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10 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=L min. 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 L min (P ’ )=L(P ’ )=L(P). (A full instance)
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11 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.
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12 |ADMs|=7=7+0 |ADMs|=9=6+3 |ADMs| = N + |chains| Basic observation N lightpaths cycles chains
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13 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 ’ |.
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14 |P| ALG 2x|P| |P| OPT 2x|P| ALG 2 x OPT |P|: # of lightpaths ALG: # of ADMs used by the algorithm OPT: # of ADMs used by optimal solution Approximation algorithms
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