# Evolutionary Approach to the Euclidean Steiner Tree Problem in n-Space
## 3 Results
### Results for the Algorithm with Population Initialization Heuristics
To determine which algorithms perform significantly better or worse, subsequent post hoc pairwise comparisons of the algorithms were conducted using an appropriate test (the Shaffer test in this study). This dual-step testing approach is designed to minimize the occurrence of false positive results among the compared pairs of algorithms, meaning that Shaffer pairwise comparisons are conducted only if the Friedman test rejects its null hypothesis. A false positive result arises when a test erroneously rejects the null hypothesis, suggesting differences between algorithms where none actually exist. Additionally, the Friedman test assigns rankings to the algorithms, helping to organize them from the best (with lower rank values) to the worst performing (with higher rank values). Further details about the testing method are available in [26]. All calculations were performed using the Keel software package [27; 28]. For the results for problems with 17 points, the algorithm ranking is presented in Table 13. A \(p\)-value on the order of ten decimal places clearly indicates differences in the performance of the algorithms. Initialization using MST has a clear advantage over the others. For problems of small size, the uniform crossover operator seems to have an advantage. However, its advantage over single-point crossover is not fully demonstrated. Pairwise \begin{table}
\begin{tabular}{c c c} \hline
**Algorithm** & **min std** & **max std** \\ \hline default-1 point & 0.171 & 0.720 \\ \hline default-uniform & 0.261 & 0.804 \\ \hline nn2-1 point & 0.081 & 0.231 \\ \hline nn2-uniform & 0.094 & 0.233 \\ \hline nn3-1 point & 0.092 & 0.405 \\ \hline nn3-uniform & 0.071 & 0.342 \\ \hline nn4-1 point & 0.088 & 0.425 \\ \hline nn4-uniform & 0.092 & 0.418 \\ \hline nn5-1 point & 0.131 & 0.511 \\ \hline nn5-uniform & 0.136 & 0.415 \\ \hline mst2-1 point & 0.073 & 0.262 \\ \hline mst2-uniform & 0.081 & 0.285 \\ \hline mst3-1 point & 0.112 & 0.361 \\ \hline mst3-uniform & 0.105 & 0.362 \\ \hline mst4-1 point & 0.140 & 0.370 \\ \hline mst4-uniform & 0.108 & 0.430 \\ \hline mst5-1 point & 0.154 & 0.546 \\ \hline mst5-uniform & 0.101 & 0.451 \\ \hline \end{tabular}
\end{table} 
Table 12: Standard deviation minimal and maximal values for 3d100 problems. comparisons of the algorithms were also performed using [PERSON]'s post hoc tests. Due to their large number, only the most significant are presented. Table 14 shows the results of post hoc tests between the best-performing configuration'mst2-uniform' and the others. It indicates that there is no basis to demonstrate its advantage over the configurations'mst3-uniform','mst4-uniform','mst5-uniform', and'mst2-1 point'. On the other hand, the obtained results indicate a significant advantage over the remaining configurations. Similarly, Table 15 presents the results of post hoc tests for the'mst3-uniform' configuration pairwise with the others (except'mst2-uniform'). This shows that there is no basis for demonstrating significant differences between'mst3-uniform' and the configurations'mst4-uniform','mst5-uniform','mst2-1 point','mst3-1 point'. The results, however, suggest that there are strong grounds to claim that it performs significantly better than the rest. From the statistical analysis of the results for problems with 17 points, it appears that the sorting heuristic based on MST has the greatest impact. These results also indicate a certain advantage of uniform crossover. However, the statistical analysis of results for more challenging problems with 100 points shows that choosing the appropriate number of nearest neighbors becomes more important and that a greedy approach, or a random choice from just two nearest neighbors, works best. It is worth adding that we do not simply choose the closest neighbor because we want to ensure diversity in the initial population, which is also important in the genetic algorithm.