# DELER: Distributed Clustering with Local Direction Centrality and Density Measure
## 3 The DEALER Algorithm
### Data Partitioning
#### 3.3.2 Data Partition Adjustment
It can be accessed from the UC Irvinerepository. The datasets CoverType, Household, Stock, and Sofia do not specify the number of clusters and are primarily used for efficiency analysis. These datasets can be obtained from the UCI Machine Learning Repository and Kaggle platform. The characteristics of these datasets are summarized in Table 1.

#### 5.1.2 Parameter Settings

In the DEALER clustering algorithm, the key parameters include the \(k\) parameter in the \"\(k\)-nearest neighbors\" method and the \(DCM\) threshold \(T_{DCM}\). * Parameter \(k\) in the \"\(k\)-nearest neighbors\" method: The value of \(k\) is set manually based on the characteristics of the dataset and can be estimated according to the dataset size. An empirical function is proposed to express the relationship between \(k\) and the dataset size \(n\) as follows: \[k= \begin{cases}\big{[}\frac{n}{50}\big{]}\sim\big{[}\frac{n}{20}\big{]},&100 \leq n\leq 1000\\ \big{[}\log_{2}n+10\big{]}\sim 5\big{[}\log_{2}n\big{]},&n>1000\end{cases} \] (10) Here, \(\lceil n\rceil\) represents the smallest integer greater than or equal to \(n\) (i.e., the ceiling of \(n\)). This function illustrates the increasing trend of \(k\) as \(n\) grows. However, since the density of points in specific datasets may vary, the fine-tuning of \(k\) is performed based on the density of the data in each particular dataset
* \(DCM\) Threshold \(T_{DCM}\): The \(DCM\) threshold \(T_{DCM}\) is used to distinguish between boundary points and core points. This threshold can be estimated using the method described in the literature [14]. ### The Efficiency Analysis

To validate the effectiveness of the DEALER clustering algorithm, this section compares it with four other clustering algorithms on two synthetic datasets (DS1, DS2) and two real-world datasets (Flame, Spiral). The comparison is based on external evaluation metrics, including Clustering Accuracy (ACC) [19], Adjusted Mutual Information (AMI) [20], Adjusted Rand Index (ARI) [21], and F-score [22]. These metrics are used to measure clustering accuracy, providing an overall performance comparison of the clustering results. (1). Comparison Algorithms: The algorithms compared in this section include the classic k-means clustering algorithm, DBSCAN, DPC, and CDC algorithms. The parameter settings are based on the original configurations provided in the respective algorithmic literature. The specific settings are as follows:

K-means: A classical clustering algorithm that requires the parameter \(k\), which represents the number of clusters. \begin{table}
\begin{tabular}{l c c c} \hline \hline
**Dataset Type** & **Dataset Name** & **Dataset Size** & **Number of Clusters** \\ \hline Synthetic & DS1 & 999 & 10 \\ Synthetic & DS2 & 459 & 6 \\ Real-World & Flame & 240 & 2 \\ Real-World & Spiral & 312 & 3 \\ Real-World & Levine & 265,627 & 14 \\ Real-World & Samusik & 841,644 & 24 \\ Real-World & UrbanGB & 360,177 & 469 \\ Real-World & CoverType & 581,012 & N/A \\ Real-World & Household & 2,049,280 & N/A \\ Real-World & Stock & 17,400,000 & N/A \\ Real-World & Sofia & 35,667,207 & N/A \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Dataset characteristics. DBSCAN: A density-based clustering algorithm requiring two parameters--\(\epsilon\), which defines the radius of the neighborhood, and _MinPts_, which sets the density threshold to distinguish core points from border points. DPC: A density-peak-based clustering algorithm that requires \(\epsilon\), the neighborhood radius parameter. CDC: A clustering algorithm proposed by [PERSON] et al. in 2022 [14], which requires the k-nearest neighbor parameter (k) and an additional parameter \(T_{DCM}\) for distinguishing core points from border points. Table 2 presents the parameter settings used for the comparison algorithms on datasets DS1, DS2, Flame, and Spiral, respectively. A dash (\"--\") in the tables indicates that the corresponding algorithm does not require that specific parameter. (2).