# Evolutionary Approach to the Euclidean Steiner Tree Problem in n-Space
## 3 Results
### Results for the Algorithm with Population Initialization Heuristics  
For problems with 17 points, Table 3 presents the values of the best solutions obtained by different versions of the genetic algorithm, while Table 4 presents the average values from ten runs. The columns labeled '1 point' and 'uniform' represent GA versions without preliminary sorting (best results from Tables 1 and 2). The column 'nn2-1p' denotes population initialization using the first heuristic, where two nearest neighbors were sought and a single-point crossover operator was applied. The column 'nn2-u' represents the same for uniform crossover. The meanings of the names of the other columns are analogous for subsequent values of the number of nearest neighbors searched in the applied heuristic of initial population initialization. From the presented results, the general positive impact of the applied heuristic is evident. Considering the minimum values, the basic versions of GA provided the best results only in 11 cases, of which only 6 are unique. For average values, there were only two cases. However, there is no clear winner in terms of the combination of the number of nearest neighbors and the crossover operator used. \begin{table}
\begin{tabular}{l l l l l l l l l} \hline
**Problem** & \multicolumn{4}{c}{**1 point**} & \multicolumn{4}{c}{**uniform**} \\ \hline  & **min** & **Mean (std)** & **max** & **min** & **Mean (std)** & **max** \\ \hline
3d100-38 & 35.973 & 36.743 (0.622) & 37.942 & **34.461** & 35.920 (0.649) & 37.169 \\ \hline
3d100-39 & 33.648 & 33.923 (0.171) & 34.231 & **31.600** & 32.404 (0.523) & **33.211** \\ \hline
3d100-40 & 35.333 & 36.134 (0.421) & 36.759 & **34.191** & 34.829 (0.474) & 35.574 \\ \hline
3d100-41 & 34.658 & 35.773 (0.494) & 36.418 & 33.960 & 34.509 (0.304) & **34.944** \\ \hline
3d100-42 & 33.927 & 34.812 (0.410) & 35.394 & **33.164** & 33.695 (0.291) & 34.218 \\ \hline
3d100-43 & 35.062 & 36.208 (0.524) & 37.178 & **33.914** & 34.551 (0.504) & 35.664 \\ \hline
3d100-44 & 36.050 & 36.548 (0.380) & 37.379 & **33.875** & 34.686 (0.528) & 35.312 \\ \hline
3d100-45 & 35.893 & 36.235 (0.342) & 36.944 & **34.190** & 34.881 (0.444) & 35.642 \\ \hline
3d100-46 & 35.284 & 36.006 (0.446) & 36.775 & **33.162** & 34.878 (0.804) & 35.858 \\ \hline
3d100-47 & 33.986 & 34.665 (0.458) & 35.326 & **32.761** & 33.584 (0.435) & 34.075 \\ \hline
3d100-48 & 35.787 & 36.367 (0.273) & 36.784 & **34.452** & 35.053 (0.374) & 35.596 \\ \hline
3d100-49 & 35.566 & 36.724 (0.637) & 37.565 & **33.847** & **35.492 (0.587)** & 36.042 \\ \hline
3d100-50 & 34.048 & 35.201 (0.502) & 35.748 & **32.914** & **33.867 (0.627)** & **34.773** \\ \hline \end{tabular}
\end{table} 
Table 2: _Cont._ \begin{table}
\begin{tabular}{l l l l l l l l l l l} \hline
**Problem** & **1 point** & **uniform** & **nn2-1p** & **nn2-u** & **nn3-1p** & **nn3-u** & **nn4-1p** & **nn4-u** & **nn5-1p** & **nn5-u** \\ \hline
3d17-01 & 4.333 & 4.396 & 4.390 & 4.348 & 4.339 & **4.333** & 4.421 & 4.339 & 4.339 & 4.391 \\ \hline
3d17-02 & 4.545 & 4.535 & **4.496** & 4.504 & 4.535 & 4.531 & 4.535 & 4.525 & 4.543 & 4.535 \\ \hline
3d17-03 & 4.466 & 4.495 & 4.460 & 4.514 & 4.452 & **4.430** & 4.483 & 4.465 & 4.508 & 4.519 \\ \hline
3d17-04 & 4.355 & 4.399 & 4.344 & 4.374 & 4.366 & 4.387 & 4.344 & **4.320** & 4.325 & 4.336 \\ \hline
3d17-05 & 3.768 & **3.750** & 3.765 & **3.750** & 3.768 & **3.750** & 3.771 & 3.766 & 3.780 & 3.766 \\ \hline
3d17-06 & 4.921 & 4.900 & **4.852** & 4.876 & 4.901 & 4.867 & 4.865 & 4.885 & 4.930 & 4.854 \\ \hline
3d17-07 & 4.644 & 4.624 & **4.562** & 4.576 & 4.607 & 4.605 & 4.579 & 4.585 & 4.603 & 4.611 \\ \hline
3d17-08 & 4.599 & **4.593** & 4.620 & 4.609 & 4.595 & 4.609 & 4.613 & 4.607 & 4.613 & 4.621 \\ \hline
3d17-09 & 4.185 & 4.129 & 4.153 & **4.124** & 4.159 & 4.159 & 4.164 & 4.129 & 4.164 & 4.129 \\ \hline \end{tabular}
\end{table} 
Table 3: Results (minimum values) of a genetic algorithm (initialization with random sorting and nearest neighbor selection) for 3d17 problems. The best result for a given problem is indicated in bold font.