# Evolutionary Approach to the Euclidean Steiner Tree Problem in n-Space
## 3 Results
### Results for the Algorithm with Population Initialization Heuristics
It has thus been shown that considering the distances of the sequentially connected points during the initialization of FST is crucial. The second heuristic, which additionally pre-sorts the points according to their usage in the construction of the minimum spanning tree using [PERSON]'s algorithm, was described earlier. Intuition suggests that such sorting should further positively impact the performance of the algorithm. Indeed, the obtained results confirm this. For problems with 17 points, Table 7 shows improvement in the best results, while Table 8 shows improvement in averaged results. The column name'mst2-1p' indicates that the sorting heuristic using [PERSON]'s algorithm starting from a random point was applied during the construction of the FST, a random point was chosen from the two nearest neighbors, and the crossover operator used was single-point crossover. Similarly, in'mst2-u', uniform crossover was used. The names of the other columns have analogous meanings for other values of the number of nearest neighbors. \begin{table}
\begin{tabular}{l l l l l l l l l l} \hline \hline
**Problem** & **1 point** & **uniform** & **nn2-1p** & **nn2-u** & **nn3-1p** & **nn3-u** & **nn4-1p** & **nn4-u** & **nn5-1p** & **nn5-u** \\ \hline
3d100-39 & 33.923 & 32.404 & 16.983 & **16.940** & 18.350 & 18.454 & 1  
\begin{table}
\begin{tabular}{l c c c c c c c c} \hline \hline
**Problem** & **mst2-1p** & **mst2-u** & **mst3-1p** & **mst3-u** & **mst4-1p** & **mst4-u** & **mst5-1p** & **mst5-u** \\ \hline
3d17-13 & 4.789 & 4.783 & 4.799 & **4.783** & 4.788 & 4.786 & 4  
\begin{table}
\begin{tabular}{l l l l l l l l l} \hline \hline
**Problem** & **mst2-1p** & **mst2-u** & **mst3-1p** & **mst3-u** & **mst4-1p** & **mst4-u** & **mst5-1p** & **mst5-u** \\ \hline
3d17-01 & 4.359 & **4.342** & 4.385 & 4.367 & 4.388 & 4.356 & 4.417 & 4.374 \\ \hline
3d17-02 &  
From Tables 7 and 8, no clearly best configuration is evident. However, for problems with 100 points (Table 9 with the values of the best solutions found and Table 10 for averaged values), it is evident that the greedy choice of a random point among the two nearest neighbors prevails over other configurations and is more significant than the choice of crossover operator. The minimal and maximal values of standard deviation for problems with 17 points and 100 points are presented in Tables 11 and 12. \begin{table}
\begin{tabular}{l c c c c c c c c} \hline \hline
**Problem** & **mst2-1p** & **mst2-u** & **mst3-1p** & **mst3-u** & **mst4-1p** & **mst4-u** & **mst5-1p** & **mst5-u** \\ \hline
3d17-38 & 5.012 & 4.984 & 4.992 & 4.997 & 5.010 & **4.975** & 5.008 & 5.005 \\ \hline
3d17-39 & 4.362 & **4.348** & 4.364 & 4.354 &  
\begin{table}
\begin{tabular}{l c c c c c c c c} \hline
**Problem** & **mst2-1p** & **mst2-u** & **mst3-1p** & **mst3-u** & **mst4-1p** & **mst4-u** & **mst5-1p** & **mst5-u** \\ \hline
3d100-16 & 16.831 & **16.822** & 18.463 & 18.472 & 19.868 & 19.473  
\begin{table}
\begin{tabular}{l l l l l l l l l} \hline \hline
**Problem** & **mst2-1p** & **mst2-u** & **mst3-1p** & **mst3-u** & **mst4-1p** & **mst4-u** & **mst5-1p** & **mst5-u** \\ \hline
3d100-01 & **17.123** & 17.164 & 19.052 & 19.076 & 20.413 & 20.500 & 21.8  
The Friedman test evaluates the mean behavior across different algorithms. It considers only the average results for each problem and each algorithm. The test assesses all problems and algorithms simultaneously to identify potential discrepancies in the algorithms' average performance across all problems. The null hypothesis of the Friedman test is that there are no significant differences among the methods based on their mean performances in the given problems. The \(p\)-value obtained from the Friedman test indicates the likelihood that the null hypothesis holds true. Consequently, low \(p\)-values (typically below 0.05 or 0.01 are enough) provide grounds for rejecting the null hypothesis. At this stage, it was determined that significant differences exist among the algorithms.