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 other applications. 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 distortion $O(1)$ always exist, and moreover, we give new efficient distributed algorithms to construct them. This clustering is based on Voronoi cells centered at the vertices of a maximal independent set in a suitable power graph. Applications include low-energy simulation of distributed algorithms in the LOCAL, CONGEST, and RADIO-CONGEST models and efficient approximate solutions to distributed combinatorial optimization problems. We also investigate related lower bounds.

翻译：Miller、Peng和Xu（SPAA 2013）提出的著名聚类算法在许多应用中非常有用，包括低直径分解和低能耗分布式算法。Chang、Dani、Hayes和Pettie（PODC 2020）的先前工作表明，该聚类的一个优良性质是，聚类图中的距离是原始图中距离的缩放版本，缩放因子至多为$O(\log n)$，并存在舍入问题。最小化这一失真因子对于聚类计算效率及其他应用至关重要。我们证明存在一些图，对于任何聚类，$\Omega((\log n)^{1/3})$的失真因子是不可避免的。我们还考虑一类称为一致有界独立图的好图，例如路径、网格图和“稠密”单位圆盘图。对于这些图，我们证明始终存在失真为$O(1)$的聚类，并提出新的高效分布式算法来构造它们。该聚类基于以合适幂图中最大独立集顶点为中心的Voronoi细胞。应用包括在LOCAL、CONGEST和RADIO-CONGEST模型中低能耗模拟分布式算法，以及分布式组合优化问题的高效近似解。我们还研究了相关的下界。