## 4 Computational Efficiency
### Evaluating Clustering Result Methods  
The performance of algorithms, comparisons with other methods, and evaluations of stability are all crucial. Through performance evaluation, we can effectively measure the accuracy and efficiency of a method, gaining insight into its practical application. Numeric evaluations not only facilitate comparisons between different methods but also highlight the advantages and limitations of each. This evaluation process provides objective indicators of the robustness and stability of the method, which is essential for real-world applications. To further assess our experimental results, we will introduce the Silhouette Score and Davies-Bouldin Index as standard metrics for evaluating the quality of clustering outcomes.  
The Silhouette Coefficient is a metric used to evaluate the quality of clustering results. [PERSON] et al. (2017) [32] utilized the Silhouette Coefficient to assess the quality of clustering. The Silhouette Coefficient is used to measure the compactness and separation of clustering results. This content appears in Section \"Evaluating Clustering Result Methods\" on page 9 of the paper, where Equation (4) details the calculation method of the Silhouette Coefficient. At the same time, the same section also introduces the Davies-Bouldin Index (DBI) as a metric for evaluating clustering results, including the DBI formula (Equation (5)) and its related description. The theoretical concept is to calculate the average distance of the same clustering data, representing the compactness of that cluster, as well as to compute the core distance between different clusters, representing the separation between different clusters. The values range from \(-1\) to \(1\), where a value close to \(1\) indicates that the data are correctly assigned to their respective cluster, \(-1\) indicates that they have been assigned to the wrong cluster, and \(0\) indicates that they are on the boundary between two clusters. The formula for the Silhouette Coefficient is as follows in Equation (4):  
\[SC=\frac{b(i)-a(i)}{max(a(i),b(i))}\quad-1\leq S\leq 1 \tag{4}\]  
Here, \(SC\) is the Silhouette Coefficient for data point \(i\), \(a(i)\) is the average distance between data point i and other data points in the same cluster. \(b(i)\) is the average core distance between different clusters. The SC allows us to objectively evaluate the rationality of the HDBSCAN clustering results and the clustering performance under different parameter settings.  
The Davies-Bouldin Index (DBI) is a metric used to evaluate clustering results, as described by [PERSON] et al. (2023) [33]. This index measures the compactness of a cluster using the average distance \(s_{i}\) and evaluates the dissimilarity between different clusters' core distance \(d_{ij}\). When clusters are more compact or the dissimilarity between clusters is greater, the DBI value becomes smaller. A smaller DBI value indicates better clustering performance, signifying that clusters are effectively distinguishable. A DBI value close to \(0\) implies perfect separation. The calculation formula is as follows, Equation (5):  
\begin{table}
\begin{tabular}{l l l} \hline
**Algorithm** & **Time Complexity** & **Execution Time** \\ \hline HDBSCAN & \(O(n\cdot logn)\) & 38.37 s \\ DBSCAN & \(O(n^{2})\) to \(O(n\cdot logn)\) & 213.16 s \\ K-means & \(O(n\cdot k\cdot i)\) & 336.74 s \\ OPTICS & \(O(n^{2})\) to \(O(n\cdot logn)\) & 3091.87 s \\ Douglas-Peucker & \(O(n\cdot logn)\) & - \\ SQUISH & \(O(n)\) to \(O(n^{2})\) & - \\ MD-Distance & \(O(n^{2}\cdot p^{2})\) & - \\ \hline \end{tabular} where \(n\) is the number of Data, \(k\) is the clusters, \(i\) is the iterations.  
\end{table}
Table 3: Time complexity and execution time of clustering algorithms.  
\[DBI=\frac{1}{k}\sum_{i=1}^{k}max(\frac{s_{i}+s_{j}}{d_{ij}})\quad i\
eq j \tag{5}\]  
where \(k\) is the number of clusters, \(S_{i}\) is the average distance from data points within cluster \(i\) to its centroid, and \(d_{ij}\) is the distance between the centroids of clusters \(i\) and \(j\).