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RCA as a Graph Coloring Problem using barcode implementation for none control to generate, create none image in none applications.generate data matrix In this section we cast none for none the RCA problem in terms of a path interference graph, G PI , incorporating all of the admissible RCA choices and the network constraints.14 Determination of routing and channel assignments then reduces to the coloring of selected vertices of G PI , subject to certain interference constraints. Section 6.

3.7 provides alternatives such as mixed integer program (MIP) or integer linear program (ILP) formulations. All of these formulations are equivalent, but depending on the particular problem, one formulation may be more convenient than another.

As before, we assume that each required logical/optical connection is supported by an optical path. Let k = a, b, . .

. , z denote a ber path using a sequence of bers a, b . .

. z, and let p = A, B j, k denote an optical path between source-destination pair ( A, B). ISO Standards Overview 13 14. The nonblocking NAS is none for none used here as a mathematical convenience. In practice, it would probably not be cost-effective to equip each station with W F transceivers. Interference graphs have been used by [Gopal82] for channel assignment in satellite systems and by [Chlamtac+89] and [Bala+91b] for proving NP-completeness of the channel assignment problem in WRNs and LLNs, respectively.

. Multiwavelength Optical Networks using a wavelength j on none none ber path k .15 Then, a feasible solution of the RCA problem consists of a choice of an optical path (OP) (i.e.

, a ber path and wavelength) for each prescribed connection so that all network constraints are satis ed. An optimal solution of the RCA problem consists of a feasible solution that minimizes some cost function; for example, the number of wavelengths used. Because we assume no wavelength interchange at this point, the channel assignments are constrained by wavelength continuity (as well as by the DCA condition).

Two optical paths in a WRN violate DCA and thus interfere if they share a common ber and are assigned the same wavelength. In formulating the RCA problem, it is essential to identify interfering optical paths. This is done by means of G PI , which exhibits potentially interfering OPs.

Wavelength continuity is included in the network model by assigning the same wavelength to all bers on a given OP. For a given physical topology and prescribed connection set {ci }, a path interference graph, G PI , is constructed by identifying each vertex of G PI with an admissible ber path. Assuming that there is at most one connection required for each source-destination pair, and that there are K i admissible paths for connection ci , there will be i K i vertices in G PI .

Admissibility of paths is arbitrary; for example, the admissible paths for a given source-destination pair might be all minimum-hop paths, all paths less than a given physical length, or all possible paths. Two vertices of G PI are joined by an edge (i.e.

, are adjacent) if their paths share a common ber in the network. (Adjacent vertices represent potentially interfering OPs.) Having constructed G PI , solving the RCA problem consists of selecting one vertex of the graph (a ber path) for each prescribed connection and choosing a color for each selected vertex (corresponding to a wavelength for the connection) so that adjacent vertices are assigned different colors.

If the assignments are made with a minimum number of wavelengths, this is known as minimal vertex coloring of the subgraph induced by the selected vertices. It is a classic and dif cult (NP-complete) problem in graph theory (see Appendix A). In cases in which several connections are required between the same source-destination pair, they may be routed on different ber paths and/or on the same ber path provided that connections on the same ber path are assigned different wavelengths.

The latter case corresponds to a vertex coloring of G PI using multiple colors per vertex. For illustration, consider the network of Figure 6.12(a).

We wish to determine a routing and channel assignment for the connection set {(A, C), ( A, F), ( A, E), (C, E)}. To simplify matters, we assume that a single nonblocking station is attached to each node so that multiple optical connections using the same wavelength can begin and end on any node without risk of interference on the access bers. We also assume that only minimum-hop paths are admissible to reduce the number of possible routing alternatives.

All admissible ber paths that support the required connections are as follows: 1 = ab 2 = ad f. When the network links carry more than one ber pair, the speci cation of the path must distinguish a particular ber on each link..
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