## 2 Materials and Methods
### Developing the Improved Federated RHLP Algorithm for Efficient Client Selection
Similarly, in the context of low Non-IID issues in the FMNIST dataset, the improved Fed-RHLP attains a maximum global model accuracy of 90.97% with a selection proportion of 8%. This represents incremental performance improvements of: 0.38% higher than FedAvg (highest accuracy: 90.59% at an 8% selection proportion), 0.20% above PoC (highest accuracy: 90.77% at an 8% selection proportion), and 0.34% greater than FedChoice (highest accuracy: 90.63% at an 8% selection proportion). Additionally, the improved Fed-RHLP underwent assessment for its convergence speed and execution time, benchmarked against alternative algorithms across both high and low levels of Non-IID challenges. The evaluation focused on achieving the highest global model accuracy under the following parameters: High Non-IID Conditions: MNIST: 10% client selection proportion; FMNIST: 10% client selection proportion; and in Low Non-IID Conditions: MNIST: 4% client selection proportion; FMNIST: 8% client selection proportion, respectively. The results are presented as follows. Figure 7a-d show that the improved Fed-RHLP achieves higher global model accuracy, faster convergence speed, and reduced execution time compared to FedAvg, PoC, and FedChoice when tested on MNIST and FMNIST under both high and low levels of Non-IID issues. The insights from the results are presented in Tables 2 and 3. Figure 6: Accuracy of the global model of the algorithms under varying proportions of client selection. (**a**) Proportion of client selection in high-level Non-IID issues in MNIST. (**b**) Proportion of client selection in low-level Non-IID issues in MNIST. (**c**) Proportion of client selection in high-level Non-IID issues in FMNIST. (**d**) Proportion of client selection in low-level Non-IID issues in FMNIST. \begin{table}
\begin{tabular}{c c c c c c c c c c c c} \hline \hline \multirow{3}{*}{**Algorithm**} & \multirow{3}{*}{**Issue Level**} & \multirow{3}{*}{
\begin{tabular}{c} **Accuracy** \\ **of Global** \\ **Model** \\ \end{tabular} } & \multicolumn{6}{c}{**Convergence Rounds and Time (Seconds) with Different Accuracy**} \\ \cline{3-10}  & & & \multicolumn{2}{c}{**60\%**} & \multicolumn{2}{c}{**70\%**} & \multicolumn{2}{c}{**80\%**} & \multicolumn{2}{c}{**90\%**} \\ \cline{3-10}  & & & **Round** & **Time** & **Round** & **Time** & **Round** & **Time** & **Round** & **Time** \\ \hline improved & High & 96.49\% & 6 & 633.6 & 11 & 1161.6 & 19 & 2006.4 & 41 & 4329.6 \\ \cline{2-10} Fed-RILP & Low & 98.91\% & 3 & 486 & 3 & 486 & 6 & 972 & 10 & 1620 \\ \hline FedAvg & High & 86.18\% & 32 & 1747.2 & 74 & 4040.4 & 126 & 6879.6 & - & - \\ \cline{2-10}  & Low & 98.03\% & 9 & 680.4 & 12 & 907.2 & 12 & 907.2 & 52 & 3931.2 \\ \hline \multirow{2}{*}{PoC} & High & 90.31\% & 43 & 3947.4 & 57 & 5232.6 & 85 & 7803 & 185 & 16,983 \\ \cline{2-10}  & Low & 98.76\% & 3 & 451.8 & 5 & 753 & 8 & 1204.8 & 15 & 2259 \\ \hline \multirow{2}{*}{FedChoice} & High & 87.80\% & 40 & 3816 & 60 & 5724 & 92 & 8776.8 & - & - \\ \cline{2-10}  & Low & 98.78\% & 3 & 464.4 & 5 & 774 & 8 & 1238.4 & 14 & 2167.2 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Performance metrics for the global model of the algorithms on the Non-IID MNIST dataset.