# DELER: Distributed Clustering with Local Direction Centrality and Density Measure
## 3 The DEALER Algorithm
### Data Partitioning
#### 3.3.2 Data Partition Adjustment
Through the proposed density-enhanced local direction centrality measure, DEALER mitigates the weak connectivity problem while maintaining the ability to recognize sparse clusters. Further, the \(z\)-value index is designed to reduce the number of distance computations for KNN and enable the communication-free computation of the Direction Centrality Metric \(DCM\) and density \(\rho\) for the distributed large-scale clustering. We also theoretically prove that DEALER is able to reduce the time complexity of the CDC algorithm from \(O\big{(}n^{2}\big{)}\) to \(O(n\log n)\). A comprehensive experimental evaluation shows that DEALER outperforms the competitors in terms of effectiveness and efficiency and is scalable to large-scale datasets. Facing large-scale data clustering challenges such as those in social networks [26] and genomics [27], the distributed clustering algorithm based on directional centrality discussed in this paper holds significant potential for practical applications. Recent advances have introduced scalable and interpretable clustering frameworks for high-dimensional data [28], including density-based spatial clustering approaches specifically designed for high-dimensional environments [29]. As part of future work, we plan to investigate extensions to the DEALER algorithm that could potentially enhance the performance of these high-dimensional clustering techniques. In the future, we will explore how to extend DEALER to high-dimensional clustering, where the \(z\)-value index does not perform very well. Conceptualization, Y.Z. ; methodology, X.L. and Y.Z. ; software, X.L. and Y.Z. ; validation, X.L., Z.Z. and Y.Z. ; formal analysis, X.L., Z.Z. and Y.Z. ; data curation, X.L., Z.Z. and Y.Z. ; writing--original draft preparation, X.L. ; writing--review and editing, X.L., Z.Z. and Y.Z. ; visualization, Y.Z. and X.L. All authors have read and agreed to the published version of the manuscript. This work was supported by the National Natural Science Foundation of China (No. 62432003 and 62032013). Not applicable. The data presented in this study are available on request from the corresponding author. We extend our sincere gratitude to the anonymous reviewers for their valuable insights and feedback. The authors declare no conflicts of interest. ## References

* (1) [PERSON]; [PERSON]; [PERSON] Efficient clustering in data mining applications based on harmony search and k-medoids. _Soft Comput._**2024**, _28_, 13245-13268. [CrossRef]
* (2) [PERSON]; [PERSON]; [PERSON]; [PERSON] Category-Level Contrastive Learning for Unsupervised Hashing in Cross-Modal Retrieval. _Data Sci. Eng._**2024**, \(9\), 251-263. * (3) [PERSON]; [PERSON]; [PERSON]; [PERSON] Graph-Enhanced Prompt Learning for Personalized Review Generation. _Data Sci. Eng._**2024**, \(9\), 309-324. * (4) [PERSON] A clustering-based adaptive Neighborhood Retrieval Visualizer. _Neural Netw._**2021**, _140_, 247-260. [PubMed]
* (5) [PERSON]; [PERSON]; [PERSON]; [PERSON] Cluster-based information retrieval using pattern mining. _Appl. Intell._**2021**, _51_, 1888-1903. [CrossRef]
* (6) [PERSON]; [PERSON] K-Means Clustering-Based Data Compression Scheme for Wireless Imaging Sensor Networks. _IEEE Syst. J._**2017**, _11_, 2652-2662. * (7) [PERSON]; [PERSON] Experimental Comparisons of Clustering Approaches for Data Representation. _ACM Comput. Surv._**2023**, _55_, 45:1-45:33. * _[PERSON] et al. (2023)_ [PERSON]; [PERSON] [PERSON]; [PERSON] [PERSON]; [PERSON] [PERSON]; [PERSON] [PERSON] Stages of cluster analysis in the diagnosis of Lyme disease in children. In Proceedings of the ITAP'2023: 3 rd International Workshop on Information Technologies: Theoretical and Applied Problems Ternopil, Ukraine, Opole, Poland, 22-24 November 2023; pp. 79-92. * _[PERSON] et al. (2024)_ [PERSON]; [PERSON]; [PERSON] [PERSON]; [PERSON] [PERSON]; [PERSON] [PERSON]; [PERSON] [PERSON] Explainable Gait Analysis for Early Detection of Neurodegenerative Diseases Using Unsupervised Clustering Techniques.