# Construction of Nonblocking Wavelength/Space Switches with AWGs and WSSes
## 3 WSS-Based Approach
\(\Box\)  
Furthermore, we have the following property. If there is a wire connecting (\(t_{1}\), \(t_{2}\)) and (\(t_{1}{}^{\prime}\), \(t_{2}\)) in the expanded space topology, if \(t_{1}\
eq t_{1}{}^{\prime}\), there is a WC converting the wavelength from \(t_{1}\) to \(t_{1}{}^{\prime}\) on link \(t_{2}\) in the original topology. This is a direct result of how the expanded space topology is constructed. \(\Box\)  
The basic principle for designing a WSS-based wavelength/space nonblocking switch is to find a variant of the Clos network in the expanded space domain that can satisfy Property 4. We classify our discussion into three cases: (1) \(n=r\), (2) \(n<r\), and (3) \(n>r\). There are many variants of the Clos topology (see Figure 0(b)) that can satisfy Property 4. One is given in Figure 8(b). For the interconnection pattern between stage-1 and stage-2 space switches, let (\(t_{1}\), \(t_{2}\)) and (\(t_{1}{}^{\prime}\), \(t_{2}{}^{\prime}\)) be defined as before. We can then describe the interconnection pattern as:  
\[(t_{1},\,t_{2})\rightarrow(t_{1}{}^{\prime}=(t_{1}+t_{2})\bmod r,\,t_{2}{}^{ \prime}=t_{2}) \tag{8}\]  
The interconnection connecting stage-2 and stage-3 space switches is the mirror image of the mapping described by Equation (8). The topology of this network is given in Figure 8(b). From this topology, we can derive from the original wavelength/space switch how many converters are needed on each link and what conversions will be performed by them (Property 5). The result is shown in Figure 8(c). The WSS-based wavelength/space switch given in Figure 8(c) is rearrangeably nonblocking. The topology of the network expanded in the space domain is shown in Figure 8(b). For any permutation between the inputs and outputs to set up, we can construct a bipartite graph where each input (output) switch represents a vertex on the left (right). If there is a connection from an input switch to an output switch, an edge is added between the two corresponding vertexes in the bipartite graph. We then use graph theory to determine the chromatic number of the bipartite graph [25]. Although the interconnection pattern for the switch in Figure 8(c) is a variant of the Clos network, the constructed bipartite graph is exactly the same as the constructed bipartite graph for the latter (Figure 0(b)). Thus, the network described by the expanded space topology is RNB. This also means that the original network in Figure 8(c) is RNB. \(\Box\)  
The example given above is for the case \(r=n\). The approach, however, also places some constraints on \(r\) and \(n\). In other words, not all (\(r\), \(n\)) combinations can be implemented. Under this condition, the approach can work if \(r\) is a multiple of \(n\). To see why it is the case, we construct a variant of the Clos topology given in Figure 9(a). Similarly, the interconnection pattern between two stages of space switches in the expanded space topology must follow Property 4. Let (\(t_{1},\,t_{2}\)and \((t_{1}\)', \(t_{2}\)') be defined as before. The mapping \((t_{1},t_{2})\rightarrow(t_{1}\)', \(t_{2}\)') describing the interconnection pattern between stage-1 and stage-2 space switches in Figure 10a is given as:  
\[ \begin{array}{l}t_{1}\prime=[t_{1}+(t_{2}\bmod n)]\bmod n,\\ t_{2}\prime=t_{2}\end{array}  \tag{9}\]  
The interconnection connecting stage-2 and stage-3 space switches is the mirror image of the mapping described by Equation (9). From the expanded topology, we can derive the original wavelength/space switch. The result is shown in Figure 10b. **Property 7**. : _The WSS-based wavelength/space switch given in Figure 10b is rearrangeably nonblocking._  
**Proof**. The proof is similar to Property 6. \(\Box\)  
Furthermore, we also consider the case where the maximum size of each WSS switch is limited to \(N\). Under this condition, the approach can work if \(r\) is a multiple of \(n^{2}\).