# LightR: A Fault-Tolerant Wavelength-Routed Optical Networks-on-Chip Topology +
## 3 LightR: An \(N\times N\) Scalable and Fault-Tolerant WRONoC Topology
### Wavelength Configuration  
After connecting the waveguides, we configured the wavelengths of all the MRRs in the LightR. As introduced in Section 3.1, each HashR uses eight MRRs on four different wavelengths, which can be regarded as a wavelength set denoted as \(\Lambda\). For example, in the HashR shown in Figure 7, the wavelength set \(\Lambda\) contains four wavelengths: \(\lambda_{1}\), \(\lambda_{2}\), \(\lambda_{3}\), and \(\lambda_{4}\). In this case, the task to assign wavelengths to each MRR is converted into the task to assign a wavelength set to each HashR. To avoid data conflict, we assign the wavelengths of the signals from the same masters or to the same slaves with different wavelengths and propose a simple wavelength configuration approach. For an \(N\times N\)_LightR_, we first construct a \((\lceil\frac{N}{2}\rceil-1)\times(\lceil\frac{N}{2}\rceil-1)\) wavelength-set matrix, where each entry represents a wavelength set (\(\Lambda\)). After that, we fill the matrix column by column by repeatedly iterating over an array from \(1\) to \(\lceil\frac{N}{2}\rceil\). In this way, we ensure that entries in the same row or the same column of the matrix must be different, such that every master will send signals to different slaves on different wavelengths, and every slave will receive signals from different masters on different wavelengths. Hence, data conflict can be avoided. At last, we configure the HashRs according to the matrix. Taking a network with eight nodes as an example, we need \(\lceil\frac{8}{2}\rceil(\lceil\frac{8}{2}\rceil-1)/2=6\) HashRs to support the communications among the nodes. To configure the resonant wavelengths of the MRRs in the HashRs, we first construct a \(3\times 3\) wavelength-set matrix and four wavelength sets [\(\Lambda_{1}\),\(\Lambda_{2}\),\(\Lambda_{3}\),\(\Lambda_{4}\)]. We fill the first column with \(\Lambda_{1}\), \(\Lambda_{2}\), and \(\Lambda_{3}\) as shown in Figure 10a and fill \(\Lambda_{4}\) to the first entry in the second column. Then, we begin the second iteration from \(\Lambda_{1}\) again and fill \(\Lambda_{1}\), \(\Lambda_{2}\) to the remaining two entries in the second  
Figure 9: The \(N\times N\) LightR structure. column shown in Figure 9(b). After repeating this step for the third column, we have a filled \(3\times 3\) wavelength-set matrix, as shown in Figure 9(c). Since only six HashRs are required for this topology, the entries below the counter-diagonal are replaced by 0, as shown in Figure 9(d). To construct an \(8\times 8\) LightR, we place and connect the six HashRs according to the steps stated in Section 3.2. Assuming that \(\Lambda_{1}=(\lambda_{1}\),\(\lambda_{2}\),\(\lambda_{3}\),\(\lambda_{4}\)), \(\Lambda_{2}=(\lambda_{5}\),\(\lambda_{6}\),\(\lambda_{7}\),\(\lambda_{8}\)), \(\Lambda_{3}=(\lambda_{9}\),\(\lambda_{10}\), \(\lambda_{11}\),\(\lambda_{12}\)), and \(\Lambda_{4}=(\lambda_{13}\), \(\lambda_{14}\), \(\lambda_{15}\), \(\lambda_{16})\), we configure the MRRs of the \(8\times 8\) LightR with the 3\(\times\)3 wavelength-set matrix, shown in Figure 9(d). The \(8\times 8\) LightR topology is presented in Figure 10.  
Specifically, if \(m_{i}\) is directly connected to \(s_{j}\) by a waveguide, the signals should be off-resonance with the MRRs along the waveguide. For example, \(m_{1}\) can communicate with \(s_{5}\) using the signals on the wavelengths \(\lambda_{13}\), \(\lambda_{14}\), \(\lambda_{15}\), and \(\lambda_{16}\), which are off-resonance with the MRRs along the waveguide connecting \(m_{1}\) and \(s_{5}\). A PSE in LightR is formed by two waveguides: one connects \(m_{i}\) and \(s_{j}\), and the other one connects \(m_{p}\) and \(s_{q}\). Then, the wavelengths of the signals from \(m_{i}\) to \(s_{q}\) and from \(m_{p}\) to \(s_{j}\) are equal to the resonant wavelength of the MRR in the PSE. For example, the top-left PSEs are formed by the waveguide that connects \(m_{1}\) to \(s_{5}\) and the waveguide that connects \(m_{4}\) to \(s_{8}\).