# Evolutionary Approach to the Euclidean Steiner Tree Problem in n-Space
## 4 Conclusions  
In this work, we propose the application of a genetic algorithm for optimizing connection networks within a multi-dimensional Euclidean space. A network with the minimal total edge length connecting a given set of points can be obtained by adding additional points, known as Steiner points. As there are no efficient algorithms in Euclidean spaces of more than two dimensions, new heuristic approaches are needed. In this work, the problem has been reduced to a discrete optimization problem. An appropriate data structure encodes the topology of connections between Steiner points, and their positions are optimized using the well-known Smith algorithm. The genetic algorithm is employed specifically for optimizing the solution in terms of the appropriate topology. This paper demonstrates, using \(\mathbb{R}^{3}\) data, that the proposed genetic algorithm yields results close in quality to optimal solutions, as shown for problems with 17 points, for which optimal solutions can be obtained using the Smith algorithm. For more challenging problems (100 points) in 3D space, it is shown that the proper initialization of the initial population of solutions, which applies the appropriate sorting of points, plays a significant role in the quality of solutions and convergence. In this regard, it is shown that the best results are provided by the proposed sorting using the Prim algorithm to build a minimal spanning tree for a given set of points. The results obtained through its use are better than those from pure random generation of the initial population at a high level of statistical significance. Results have been presented for points in \(\mathbb{R}^{3}\); however, the proposed method can be applied unrestrictedly for data in any dimensional space. Due to the publication's volume limitations, only results for \(\mathbb{R}^{3}\) are presented in detail, yet calculations conducted for data in \(\mathbb{R}^{5}\) demonstrated analogous conclusions about the operation of various versions of the tested genetic algorithm, particularly the significance of the initialization step based on the [PERSON]'s algorithm. This does not detract from the significance of the results, as in practice, data in \(\mathbb{R}^{3}\) may be most relevant. The following conclusion about this publication can be given as a list of key points:  
* Addresses a Specific Gap. This publication tackles a gap in the usage of genetic algorithms for optimizing connection networks within multi-dimensional Euclidean space, particularly for the Euclidean Steiner Tree Problem (ESTP) in \(\mathbb{R}^{d}\) where \(d\geq 3\). * Tailored Heuristic. A specialized evolutionary heuristic is introduced, adapted to ESTP. The problem is reformulated as a discrete optimization task, where Steiner points are added to minimize the total connection length between a given set of points. * Significance of Initialization. A key finding is the high impact of the initial population generation method. In particular, sorting points based on their addition order in [PERSON]'s MST algorithm significantly improves solution quality compared to purely random initialization. * Statistical Validation. Different algorithm configurations were tested on problems with 17 and 100 points. The resulting solutions underwent statistical significance analysis, confirming that appropriate parameter settings and initialization strategies lead to better convergence and higher-quality solutions. * Extension to Higher Dimensions. While the study presents detailed results for \(\mathbb{R}^{3}\), the method is applicable to any-dimensional Euclidean space. Preliminary tests in \(\mathbb{R}^{5}\) confirm analogous improvements in solution quality, thus supporting the broader utility of the proposed approach. * Practical Relevance. Given the computational complexity of exact methods for dimensions higher than two, this heuristic offers a practical compromise.