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A Semi Brute-Force Search Approach for (Balanced) Clustering

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Abstract

Clustering is one of the most long-standing fundamental problems in the fields of computational geometry and algorithm design. In this paper, we focus on the variance-based clustering problems, included in which is the widely known k-means clustering. As the main contribution, a so-called semi brute-force search approach is proposed and analyzed from both theoretical and experimental aspects. The proposed approach samples a small percentage from the input dataset and search in a brute-force way for a k sized seed whose resulting Voronoi Diagram gives a good clustering of the original dataset. With high probability, the clustering is provably good to estimate the optimum under certain assumptions. Extensive experiments on both synthetic datasets and real-world datasets show that to obtain competitive results compared with k-means method (Llyod in IEEE Trans Inf Theory 28(2):129–137, 1982) and k-means++ method (Arthur and Vassilvitskii, (in: 18th ACM-SIAM symposium on discrete algorithms (SODA), 2007)), we only need a subset of 7% size comparing with the input dataset. If we are allowed to sample 15% from the dataset, our algorithm outperforms both the k-means method and k-means++ method in at least 80% of the clustering tasks. Also, an extended algorithm based on the same idea guarantees a balanced k-clustering result.

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Notes

  1. Note that \(V_1\), \(V_2\) are not necessary to be independent for Fact 1 to hold, but it does not hurt to say so.

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Funding

Yicheng Xu is supported by Fundamental Research Project of Shenzhen City (No. JCYJ20210324102012033) and Shenzhen Science and Technology Program (No. ZDSYS20220422103800001 and CJGJZD20210408092806017). Vincent Chau is supported by NSFC (No. 62202100), and by the Fundamental Research Funds for the Central Universities (No. 2242022R10024). Chenchen Wu is supported by National Natural Science Foundation of China (No.11971349). Yong Zhang is supported by National Key R &D Program of China (No. 2022YFE0196100) and National Natural Science Foundation of China (No. 12071460). Vassilis Zissimopoulos is supported by the CAS President’s International Fellowship Initiative (No. 2019VTA0005). Yifei Zou is supported by Shandong Science Fund for Excellent Young Scholars (No. 2023HWYQ-007).

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Authors and Affiliations

  1. Shenzhen Insititute of Advanced Technology, Chinese Academy of Sciences, Beijing, China

    Yicheng Xu & Yong Zhang

  2. Southeast University, Dhaka, China

    Vincent Chau

  3. Tianjin University of Technology, Tianjin, China

    Chenchen Wu

  4. National and Kapodistrian University of Athens, Zografou, Greece

    Vassilis Zissimopoulos

  5. Shandong University, Jinan, China

    Yifei Zou

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  1. Yicheng Xu
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  2. Vincent Chau
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  3. Chenchen Wu
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  4. Yong Zhang
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  5. Vassilis Zissimopoulos
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  6. Yifei Zou
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YX and VC wrote the main manuscript text. All other authors participated on the problem discussion and helped to review the manuscript.

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Correspondence to Vincent Chau.

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Xu, Y., Chau, V., Wu, C. et al. A Semi Brute-Force Search Approach for (Balanced) Clustering. Algorithmica 86, 130–146 (2024). https://doi.org/10.1007/s00453-023-01158-4

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