# Construction of Nonblocking Wavelength/Space Switches with AWGs and WSSes
## 2 A New Decomposition Technique for Constructing AWG/WC-Based Nonblocking Switches
### A New Decomposition Scheme  
Due to the limited size of AWGs, if \(r>N\) or \(n>N\), the network in Figure 1 cannot be implemented. Neither can the case of \(r>n\) be implemented, because the number of wavelengths required to implement the AWGs and the middle WC module stage is not obtained. The decomposition described below can solve this problem. To simplify the discussion, we assume \(n=N\) in the following example. We will explain later why the method applies directly to the case \(n>N\). Let \(r\) be a multiple of \(k\), where \(k\leq n\). Suppose \(r=k\times r_{1}\). Let us decompose an \(r\times r\) switch of the middle stage in Figure 2 into a three-stage C(\(k\), \(k\), \(r_{1}\)) network, as shown in Figure 3, where an example with \(k=4\) and \(r=12\) is given. Based on the Clos theory, the new wavelength dilated topology of Figure 2 is RNB, which means that its corresponding switch in the original space and wavelength domains _is still an RNB network_. Properties 1 and 2 given below show that the number of wavelengths and the size of AWGs required to implement the corresponding switch in the original wavelength/space domains will be reduced. In the paper, starting from node 0, all of the \(k\) consecutive switches in node stage 1 of Figure 3 are considered a group. Let N\({}_{i}\) be the \(i\)th node group of stage 1, where \(0\leq i\leq r_{1}-1\). That is:  
\[\text{N}_{i}=\{\text{node }x_{1}\ \mid\ i\text{-}k\leq x_{1}\leq(i+1)\text{-}k-1\}. \tag{3}\]  
We also consider all of the outputs \((x_{1},x_{2})\) of switches in the same group as one output group. Let O\({}_{i}\) be the \(i\)th output group of stage 1. That is:  
\[\text{O}_{i}=\{(x_{1},x_{2})\ \mid\ i\text{-}k\leq x_{1}\leq(i+1)\text{-}k-1,0 \leq x_{2}\leq n-1\} \tag{4}\]  
Figure 2: A three-stage C (4, 4, 12) network. Property 1 shows that the mapping describing the interconnection pattern between the first and the second node stages can be broken into _non-overlapping mappings_, meaning that they have no commonly shared links among these mappings. **Property 2. **_Assume \(r=k\times r_{1}\). After decomposing a middle stage switch into a three-stage C(\(k\), \(k\), \(r_{1}\)) network (see Figure 3), the number of wavelengths required to implement the corresponding switch in the original space and wavelength domains is reduced to max\(\{r_{1}\), \(n\}\) from max\(\{r\), \(n\}\)._  
**Proof. ** Property 1 shows that after decomposition, the mapping describing the interconnection pattern between node stage 1 and node stage 2 (see Figure 3) can be broken into \(r_{1}\) non-overlapping mappings. Thus, the implementation corresponding to the new dilated topology can be derived as shown in Figure 4, wherein each \(r\)-WC module in the middle stage is now broken into \(2r_{1}\)\(k\)-WC modules and \(k\)\(r_{1}\)-WC modules, the first (or second) \(r\times n\) (or \(n\times r\)) AWG is broken into \(r_{1}\) independent AWGs, and the size of each AWG is \(k\times n\) (or \(n\times k\)), where \(k\leq n\). In addition, \(2n\) additional AWGs of size \(r_{1}\times k\) and \(k\times r_{1}\) are generated. As a result, the number of wavelengths required to implement the corresponding topology in the original space and wavelength domains is reduced to max\(\{r_{1}\), \(n\}\) from max\(\{r\), \(n\}\). \(\Box\)  
From the proof of Property 2, we can also see that the size of the AWGs is reduced. If \(r_{1}>n\), we can then apply the same principle to further decompose \(r_{1}\), as described below. For a given \(n\), let us factor \(r\) as \(n_{1}\times n_{2}\times\dots\times n_{s}\), where \(n_{i}\) is the largest factor of \(r/(n_{1}\times n_{2}\times\dots\times n_{i-1})\) with \(1<n_{i}\leq n\) for \(1\leq i\leq s\). Such factorization is called a compact factorization of \(r\) in the paper. For example, given \(r=24\) and \(n=4\), the compact factorization of \(r\) is \(4\times 3\times 2\). Using the method described previously, there will be \(2s+1\) node stages in the final topology in the space domain.