## 5 The Random Grouping Selection Algorithm for the JSCC-IA Scheme in the Asymmetric Networks
### The Complexity Analysis
 \tag{77}\]  
The BFS will have to take \(D\) iterations, and the corresponding overall complexity for the cases of identical and different propagation delays can be respectively represented as:  
\[C_{BFS1}=D\xi_{1} \tag{78}\]  
\[C_{BFS2}=D\xi_{2} \tag{79}\]  
On the contrary, the RGS algorithm only requires \(\Theta\) iterations at most, and the associated overall computational complexity for the cases of identical and different propagation delays can be respectively represented as:  
\[C_{RGS1}=\Theta\xi_{1} \tag{80}\]  
\[C_{RGS2}=\Theta\xi_{2} \tag{81}\]  
Table 1 lists the overall complexity of the RGS and BFS algorithms under different number of antennas, users, data streams and groups. The maximal number of iterations for RGS algorithm is chosen as \(\Theta=20\). From the results, we can see that the number of complex multiplications of the BFS algorithm increases dramatically with the network scale, and the complexity is so huge that it is infeasible in the practical systems. On the contrary, the complexity of the RGS algorithm is several orders lower than that of the BFS algorithm, and does not increase rapidly when the network scale becomes larger. From the rate of the complexity between the two algorithms, it can be seen that the proposed RGS algorithm can reduce the computational cost significantly. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \((M,K,d,K_{0},G)\) & \(\mathbf{C_{BFS1}}\) & \(\mathbf{C_{BFS2}}\) & \(\mathbf{C_{RCS1}}\) & \(\mathbf{C_{RGS2}}\) & \(\mathbf{C_{RGS1}}/\mathbf{C_{BFS1}}\) & \(\mathbf{C_{RGS2}}/\mathbf{C_{BFS2}}\) \\ \hline (2,6,1,2,3) & 5.51\(\times 10^{4}\) & 7.94 \(\times\) 10\({}^{4}\) & 1.22 \(\times\) 10\({}^{4}\) & 1.76 \(\times\) 10\({}^{4}\) & 2.22 \(\times\) 10\({}^{-1}\) & 2.22 \(\times\) 10\({}^{-1}\) \\ (2,8,1,2,4) & 2.42 \(\times\) 10\({}^{6}\) & 3.69 \(\times\) 10\({}^{6}\) & 1.92 \(\times\) 10\({}^{4}\) & 2.93 \(\times\) 10\({}^{4}\) & 7.94 \(\times\) 10\({}^{-3}\) & 7.94 \(\times\) 10\({}^{-3}\) \\ (2,10,1,2,5) & 1.56 \(\times\) 10\({}^{8}\) & 2.48 \(\times\) 10\({}^{8}\) & 2.76 \(\times\) 10\({}^{4}\) & 4.38 \(\times\) 10\({}^{4}\) & 1.76 \(\times\) 10\({}^{-4}\) & 1.76 \(\times\) 10\({}^{-4}\) \\ (4,10,1,5,2) & 2.98 \(\times\) 10\({}^{6}\) & 3.50 \(\times\) 10\({}^{6}\) & 2.36 \(\times\) 10\({}^{5}\) & 2.78 \(\times\) 10\({}^{5}\) & 7.94 \(\times\) 10\({}^{-2}\) & 7.94 \(\times\) 10\({}^{-2}\) \\ (4,15,1,5,3) & 1.47 \(\times\) 10\({}^{10}\) & 1.84 \(\times\) 10\({}^{10}\) & 3.89 \(\times\) 10\({}^{5}\) & 4.86 \(\times\) 10\({}^{5}\) & 2.64 \(\times\) 10\({}^{-5}\) & 2.64 \(\times\) 10\({}^{-5}\) \\ (4,20,1,5,4) & 3.31 \(\times\) 10\({}^{14}\) & 4.34 \(\times\) 10\({}^{14}\) & 5.65 \(\times\) 10\({}^{5}\) & 7.40 \(\times\) 10\({}^{5}\) & 1.70 \(\times\) 10\({}^{-9}\) & 1.70 \(\times\) 10\({}^{-9}\) \\ (4,25,1,5,5) & 2.38 \(\times\) 10\({}^{19}\) & 3.24 \(\times\) 10\({}^{19}\) & 7.63 \(\times\) 10\({}^{5}\) & 1.04 \(\times\) 10\({}^{6}\) & 3.21 \(\times\) 10\({}^{-14}\) & 3.21 \(\times\) 10\({}^{-14}\) \\ (4,8,2,2,4) & 5.06 \(\times\) 10\({}^{7}\) & 6.36 \(\times\) 10\({}^{7}\) & 4.02 \(\times\) 10\({}^{5}\) & 5.05 \(\times\) 10\({}^{5}\) & 7.94 \(\times\) 10\({}^{-3}\) & 7.94 \(\times\) 10\({}^{-3}\) \\ (4,10,2,2,5) & 3.06 \(\times\) 10\({}^{9}\) & 4.00 \(\times\) 10\({}^{9}\) & 5.39 \(\times\) 10\({}^{5}\) & 7.05 \(\times\) 10\({}^{5}\) & 1.76 \(\times\) 10\({}^{-4}\) & 1.76 \(\times\) 10\({}^{-4}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The number of overall complex multiplications required by the RGS and BFS algorithms.