1

Electronic ADM

2

3

ADM (add-drop multiplexer)

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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: , | ( , ) , | ( , )

| ( ) |,

v v

v v v vv V

v V S p P p v x E p P p x v

ADM w S E ADM ADM

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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

W

vADM

ADM

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number of wavelengthsSwitching cost

ADM

7

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

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ring (Eilam, Moran, Zaks, 2002) reduction from coloring of circular arc

graphs.

NP-complete

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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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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)

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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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|ADMs|=7=7+0

|ADMs|=9=6+3

|ADMs| = N + |chains|

Basic observationN lightpaths

cycles

chains

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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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|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