The well-known clustering algorithm of Miller, Peng, and Xu (SPAA 2013) is useful for many applications, including low-diameter decomposition and low-energy distributed algorithms. One nice property of their clustering, shown in previous work by Chang, Dani, Hayes, and Pettie (PODC 2020), is that distances in the cluster graph are rescaled versions of distances in the original graph, up to an $O(\log n)$ distortion factor and rounding issues. Minimizing this distortion factor is important for efficiency in computing the clustering, as well as in further applications, once the clustering has been constructed. We prove that there exist graphs for which an $\Omega((\log n)^{1/3})$ distortion factor is necessary for any clustering. We also consider a class of nice graphs which we call uniformly bounded independence graphs. These include, for example, paths, lattice graphs, and "dense" unit disk graphs. For these graphs, we prove that clusterings of constant distortion always exist, and moreover, we give an efficient distributed algorithm to construct them. Our clustering algorithm is based on Voronoi cells centered at the vertices of a maximal independent set in a suitable power graph. Applications of our new clustering include low-energy simulation of distributed algorithms in the LOCAL, CONGEST, and RADIO-CONGEST models, as well as efficient approximate solutions to distributed combinatorial optimization problems. We complement these results with matching or nearly matching lower bounds.

翻译：Miller、Peng和Xu（SPAA 2013）提出的著名聚类算法在众多应用中具有重要价值，包括低直径分解和低能耗分布式算法。Chang、Dani、Hayes和Pettie（PODC 2020）在先前研究中证明了该聚类的一个优良特性：在忽略舍入误差的情况下，聚类图中的距离是原始图距离的重新缩放版本，其失真因子为$O(\log n)$。最小化该失真因子对于提升聚类计算效率及后续应用至关重要。我们证明了存在一类图，其任何聚类都必须具有$\Omega((\log n)^{1/3})$的失真因子。同时，我们研究了一类称为均匀有界独立图的规整图，其典型实例包括路径图、网格图及"稠密"单位圆盘图。对于此类图，我们证明始终存在恒定失真的聚类方案，并进一步提出构建该聚类的高效分布式算法。我们的聚类算法基于在适当幂图中选取最大独立集顶点作为中心的Voronoi单元构造。新聚类算法的应用包括：在LOCAL、CONGEST和RADIO-CONGEST模型中对分布式算法进行低能耗模拟，以及对分布式组合优化问题提供高效近似解。我们通过匹配或近似匹配的下界结果对这些结论进行了补充论证。