## 4 Experimental Results and Analysis
### Analysis of Multi-Way Partitioning Results  
To further evaluate the performance of the proposed algorithm, multi-way partitioning experiments were conducted with different k values (4, 8, 16). Table 5 shows the cut edge size at different k values across various circuit instances.  
\begin{table}
\begin{tabular}{l c c c c} \hline
**Circuit** & **Metis** & **KL [(9)]** & **Genetic Algorithm [(13)]** & **Proposed Algorithm** \\ \hline c432 & 1387 & 2412 & 1490 & 1470 \\ c499 & 642 & 1280 & 1012 & 620 \\ c880 & 1769 & 3806 & 2679 & 1940 \\ c1335 & 2191 & 6250 & 2458 & 2306 \\ c1908 & 3206 & 9251 & 3476 & 3151 \\ s1423 & 2131 & 6200 & 3579 & 2055 \\ s5378 & 7656 & 21,801 & 8740 & 7500 \\ s9234 & 9379 & 27,401 & 14,547 & 9250 \\ s13207 & 14,419 & 34,855 & 465,708 & 15,203 \\ s15850 & 18,997 & 39,902 & 487,709 & 27,601 \\ s35932 & 164,598 & 365,003 & 695,041 & 193,200 \\ s38417 & 114,523 & 315,000 & 647,037 & 203,500 \\ \hline \end{tabular}
\end{table}
Table 4: Comparison of average execution time (μs) for different algorithms.  
Figure 8 shows the ratio of the proposed algorithm to other algorithms in terms of the minimum cut edges for different values of k.  
Figure 8: \(Cont\).  
\begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Circuit**} & \multicolumn{3}{c}{**k = 4**} & \multicolumn{3}{c}{**k = 8**} & \multicolumn{3}{c}{**k = 16**} \\ \cline{2-10}  & **Metis** & **KL [9]** & **Mine** & **Metis** & **KL** & **Mine** & **Metis** & **KL** & **Mine** \\ \hline c432 & 99 & 121 & 87 & 136 & 148 & 117 & 287 & 304 & 182 \\ c499 & 173 & 194 & 167 & 226 & 265 & 155 & 335 & 415 & 368 \\ c880 & 119 & 131 & 97 & 218 & 226 & 212 & 337 & 388 & 279 \\ c1335 & 140 & 157 & 123 & 254 & 273 & 231 & 330 & 421 & 307 \\ c1908 & 119 & 145 & 110 & 240 & 241 & 211 & 604 & 748 & 571 \\ s1423 & 431 & 476 & 417 & 518 & 648 & 405 & 2202 & 2562 & 2105 \\ s5378 & 606 & 790 & 578 & 3527 & 4107 & 3186 & 6176 & 6417 & 5379 \\ s9234 & 1025 & 1579 & 977 & 1414 & 2280 & 1396 & 4365 & 5067 & 2871 \\ s13207 & 23,023 & 28,870 & 21,039 & 46,207 & 52,742 & 41,911 & 80,373 & 88,501 & 79,434 \\ s15850 & 2608 & 3413 & 2414 & 30,642 & 35,816 & 23,361 & 40,760 & 47,213 & 37,469 \\ s35932 & 10,644 & 11,744 & 9786 & 722,184 & 774,757 & 667,166 & 1,325,767 & 1,546,769 & 1,324,767 \\ s38417 & 172,677 & 182,776 & 174,527 & 584,839 & 659,423 & 618,039 & 634,889 & 6,919,789 & 614,776 \\ \hline \hline \end{tabular}
\end{table}
Table 5: Multi-way partitioning minimum results: k = 4, k = 8, k = 16 The experimental results demonstrate a noticeable difference in the number of cut edges between the proposed algorithm and both METIS and KL algorithms. Compared to METIS, the proposed algorithm generally performs better, particularly for k = 4 and k = 8, where the number of cut edges is consistently lower, with the average number of cut edges being 0.91 times that of METIS. When compared to KL, the proposed algorithm also performs excellently, especially for k = 16, where the number of cut edges is significantly lower than KL, with the average being 0.86 times that of KL. Overall, the proposed algorithm shows stable performance across all test circuits, outperforming both METIS and KL in most cases, with particularly notable optimization in smaller-scale circuits.