# DELER: Distributed Clustering with Local Direction Centrality and Density Measure
## 3 The DEALER Algorithm
### Data Partitioning
#### 3.3.2 Data Partition Adjustment
Accuracy Comparison: Table 3 presents the performance of the five clustering algorithms across the four evaluation metrics. In each table, the best value for each metric is highlighted in bold. The quality of clustering results requires evaluation through specific metrics, which can be divided into internal and external metrics. Internal metrics assess the clustering results without referring to the true data labels, while external metrics compare the predicted labels from the clustering algorithm with the true labels to evaluate clustering quality. Common external metrics include Clustering Accuracy (ACC), Adjusted Mutual Information (AMI), \begin{table}
\begin{tabular}{l c c c c c} \hline \hline  & \multicolumn{5}{c}{DS1} \\ \cline{2-6} algorithm & Clusters number k & knn-k & \(\epsilon\) & _MinPts_ & \(T_{DCM}\) \\ \hline k-means & 10 & — & — & — & — \\ DBSCAN & — & — & 0.3 & — & — \\ DPC & — & — & 0.3 & — & — \\ CDC & — & 30 & — & — & 0.2 \\ DEALER & — & 30 & — & — & 0.2 \\ \hline \hline \end{tabular} \begin{tabular}{l c c c c c} \hline \hline  & \multicolumn{5}{c}{DS2} \\ \cline{2-6} algorithm & Clusters number k & knn-k & \(\epsilon\) & _MinPts_ & \(T_{DCM}\) \\ \hline k-means & 6 & — & — & — & — \\ DBSCAN & — & — & 1.5 & 9 & — \\ DPC & — & — & 2 & — & — \\ CDC & — & 4 & — & — & 0.1 \\ DEALER & — & 4 & — & — & 0.2 \\ \hline \hline \end{tabular} \
\begin{tabular}{l c c c c c} \hline \hline  & \multicolumn{5}{c}{Flame} \\ \cline{2-6} algorithm & Clusters number k & knn-k & \(\epsilon\) & _MinPts_ & \(T_{DCM}\) \\ \hline k-means & 2 & — & — & — & — \\ DBSCAN & — & — & 1.5 & 14 & — \\ DPC & — & — & 2 & — & — \\ CDC & — & 10 & — & — & 0.04 \\ DEALER & — & 10 & — & — & 0.04 \\ \hline \hline \end{tabular}
\begin{tabular}{l c c c c c} \hline \hline  & \multicolumn{5}{c}{Spiral} \\ \cline{2-6} algorithm & Clusters number k & knn-k & \(\epsilon\) & _MinPts_ & \(T_{DCM}\) \\ \hline k-means & 3 & — & — & — & — \\ DBSCAN & — & — & 2.5 & 5 & — \\ DPC & — & — & 2.5 & — & — \\ CDC & — & 6 & — & — & — \\ DEALER & — & 6 & — & — & 0.5 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The clustering-related parameters of various algorithms on the datasets. Adjusted Rand Index (ARI), and F-score. This paper primarily uses external metrics to evaluate the clustering algorithms. The ACC metric measures the proportion of data points that are correctly clustered. The ARI metric focuses on pairs of correctly classified data points, rather than individual points, and is an improvement of the Rand Index [23]. NMI, based on information theory, is another clustering evaluation metric. The F-score combines precision and recall into a single measurement. These four metrics are positively correlated with the quality of the experimental results, with higher values indicating better performance. Table 3 presents the evaluation metrics for clustering quality across four datasets using five different algorithms, with the best values highlighted in bold. It is evident that, regardless of the clustering metric used, the DEALER algorithm consistently achieves the highest number of optimal results. Even in cases where DEALER is not the top performer, its performance is only marginally below the best. In contrast, other algorithms achieve fewer optimal results and occasionally exhibit poor performance on specific datasets due to their underlying clustering principles. Based on the results and analysis, it is clear that DEALER demonstrates excellent clustering performance across various complex datasets, consistently ranking among the top two. Its accuracy is generally above 95%, indicating strong robustness. ### Davies-Bouldin Index and Silhouette Coefficient

This section presents comparative experiments evaluating the DEALER clustering algorithm against four other clustering algorithms on two synthetic datasets (DS1 and DS2) and two real-world datasets (Flame and Spiral). The clustering accuracy is assessed using metrics such as the Davies-Bouldin Index (DB Index) [24] and the Silhouette Coefficient [25].