Low-Distortion Clustering in Bounded Growth Graphs

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.