# Evolutionary Approach to the Euclidean Steiner Tree Problem in n-Space
## 3 Results
### Results for the Algorithm with Population Initialization Heuristics
\begin{table}
\begin{tabular}{c c} \hline
**Algorithm** & **Ranking** \\ \hline mst2-uniform & 2.28 \\ \hline mst3-uniform & 2.56 \\ \hline mst4-uniform & 3.64 \\ \hline mst5-uniform & 4 \\ \hline mst2-1 point & 4.84 \\ \hline mst3-1 point & 6.36 \\ \hline mst4-1 point & 7.84 \\ \hline mst5-1 point & 8.88 \\ \hline nn3-uniform & 9.76 \\ \hline nn2-uniform & 10.02 \\ \hline nn4-uniform & 10.38 \\ \hline nn5-uniform & 11.22 \\ \hline nn2-1 point & 12.54 \\ \hline default-uniform & 13.62 \\ \hline nn4-1 point & 15.1 \\ \hline nn3-1 point & 15.12 \\ \hline nn5-1 point & 15.56 \\ \hline default-1 point & 17.28 \\ \hline \end{tabular}
\end{table} 
Table 13: Friedman test ranking for 3d17 data. \(p\)-value computed by the Friedman test: \(2.023\times 10^{-10}\). \begin{table}
\begin{tabular}{c c} \hline
**Algorithm** & \(p\)**-Value (Post Hoc)** \\ \hline mst4-uniform & 0.311771 \\ \hline mst5-uniform & 0.177439 \\ \hline mst2-1 point & 0.032727 \\ \hline mst3-1 point & 0.000372 \\ \hline mst4-1 point & **0.000001** \\ \hline mst5-1 point & **0** \\ \hline nn3-uniform & **0** \\ \hline nn4-uniform & **0** \\ \hline nn5-uniform & **0** \\ \hline nn2-1 point & **0** \\ \hline default-uniform & **0** \\ \hline nn4-1 point & **0** \\ \hline nn3-1 point & **0** \\ \hline default-1 point & **0** \\ \hline \end{tabular}
\end{table} 
Table 15: [PERSON]’s post hoc \(p\)-values for 3d17 data for pair-wise comparisons of mst3-uniform vs. other algorithms. [PERSON]’s procedure rejects those hypotheses that have an unadjusted \(p\)-value \(\leq\) 0.000327 (highlighted in bold font). \begin{table}
\begin{tabular}{c c} \hline
**Algorithm** & \(p\)**-Value (Post Hoc)** \\ \hline mst4-uniform & 0.311771 \\ \hline mst5-uniform & 0.177439 \\ \hline mst2-1 point & 0.032727 \\ \hline mst3-1 point & 0.000372 \\ \hline mst4-1 point & **0.000001** \\ \hline mst5-1 point & **0** \\ \hline nn3-uniform & **0** \\ \hline nn2-uniform & **0** \\ \hline nn4-uniform & **0** \\ \hline nn5-uniform & **0** \\ \hline default-uniform & **0** \\ \hline nn4-1 point & **0** \\ \hline nn3-1 point & **0** \\ \hline nn5-1 point & **0** \\ \hline default-1 point & **0** \\ \hline \end{tabular}
\end{table} 
Table 14: [PERSON]’s post hoc \(p\)-values for 3d17 data for pair-wise comparisons of mst2-uniform vs. other algorithms. [PERSON]’s procedure rejects those hypotheses that have an unadjusted \(p\)-value \(\leq\) 0.000327 (highlighted in bold font). Table 16 presents the results of the [PERSON] statistical test. A \(p\)-value to the tenth decimal place clearly indicates differences in the performance of various versions. It is evident that configurations utilizing MST-based sorting and the smallest number of nearest neighbors occupy the top spots on the list. The choice of crossover operator is of less significance. Table 17 shows the results of post hoc tests where the top-ranked configuration,'mst2-1 point', is compared with others. The findings suggest that based on the conducted studies, there is no basis to judge its performance as statistically significantly better than the configurations'mst2-uniform','mst3-uniform','mst3-1 point', 'nn2-1 point', and 'nn2-uniform'. Thus, even without MST sorting, the greedy choice of nearest neighbors is significant. On the other hand, there are clear reasons to assert that the'mst2-1 point' configuration is distinctly better than the other configurations, including those that do not consider point order or make less greedy choices of nearest neighbors. Again, the choice of crossover operator is not as crucial. Similarly, Table 18 presents the results of post hoc tests for the second-ranked configuration,'mst2-uniform', compared pairwise with the others (except'mst2-1 point'). There is no basis to claim that it performs statistically better than'mst3-uniform','mst3-1 point', 'nn2-1 point', and 'nn2-uniform'. However, there are grounds to assert that it performs statistically better than the remaining configurations.