We study (Euclidean) $k$-median and $k$-means with constraints in the streaming model. There have been recent efforts to design unified algorithms to solve constrained $k$-means problems without using knowledge of the specific constraint at hand aside from mild assumptions like the polynomial computability of feasibility under the constraint (compute if a clustering satisfies the constraint) or the presence of an efficient assignment oracle (given a set of centers, produce an optimal assignment of points to the centers which satisfies the constraint). These algorithms have a running time exponential in $k$, but can be applied to a wide range of constraints. We demonstrate that a technique proposed in 2019 for solving a specific constrained streaming $k$-means problem, namely fair $k$-means clustering, actually implies streaming algorithms for all these constraints. These work for low dimensional Euclidean space. [Note that there are more algorithms for streaming fair $k$-means today, in particular they exist for high dimensional spaces now as well.]

中文翻译：

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低维欧几里得空间中约束 k 中值和 k 均值聚类的核心集

我们研究（欧几里德）$k$-median 和 $k$-means 与流模型中的约束。除了温和的假设，如约束下可行性的多项式可计算性（如果聚类满足约束，则计算）或有效分配预言机的存在（给定一组中心，产生满足约束的中心的最佳点分配）。这些算法的运行时间指数以 $k$ 为单位，但可以应用于广泛的约束。我们证明了 2019 年提出的一种用于解决特定约束流 $k$-means 问题的技术，即公平 $k$-means 聚类，实际上意味着所有这些约束的流算法。这些适用于低维欧几里得空间。[请注意，今天有更多的流公平 $k$-means 算法，特别是它们现在也存在于高维空间中。]