Center-based clustering techniques are fundamental in some areas of machine learning such as data summarization. Generic $k$-center algorithms can produce biased cluster representatives so there has been a recent interest in fair $k$-center clustering. Our main theoretical contributions are two new $(3+\epsilon)$-approximation algorithms for solving the fair $k$-center problem in (1) the dynamic incremental, i.e., one-pass streaming, model and (2) the MapReduce model. Our dynamic incremental algorithm is the first such algorithm for this problem (previous streaming algorithms required two passes) and our MapReduce one improves upon the previous approximation factor of $(17+\epsilon).$ Both algorithms work by maintaining a small coreset to represent the full point set and their analysis requires that the underlying metric has finite-doubling dimension. We also provide related heuristics for higher dimensional data and experimental results that compare the performance of our algorithms to existing ones.

翻译：基于中心的聚类技术是机器学习领域（如数据摘要）的基本方法。通用 $k$-中心算法可能产生有偏的聚类代表，因此最近出现了对公平 $k$-中心聚类的关注。我们的主要理论贡献是两种新的 $(3+\epsilon)$-近似算法，分别用于解决（1）动态增量（即单遍流）模型和（2）MapReduce 模型中的公平 $k$-中心问题。我们的动态增量算法是该问题的首个此类算法（以往的流算法需要两遍遍历），而 MapReduce 算法则改进了先前 $(17+\epsilon)$ 的近似因子。这两种算法均通过维护一个小型核心集来代表全部点集，其分析要求底层度量具有有限倍增维数。我们还针对高维数据提供了相关启发式方法，并通过实验结果将我们的算法与现有算法的性能进行了比较。