We consider the distributed and parallel construction of low-diameter decompositions with strong diameter for (weighted) graphs and (weighted) graphs that can be separated through $k \in \tilde{O}(1)$ shortest paths. This class of graphs includes planar graphs, graphs of bounded treewidth, and graphs that exclude a fixed minor $K_r$. We present algorithms in the PRAM, CONGEST, and the novel HYBRID communication model that are competitive in all relevant parameters. Given $\mathcal{D} > 0$, our low-diameter decomposition algorithm divides the graph into connected clusters of strong diameter $\mathcal{D}$. For a arbitrary graph, an edge $e \in E$ of length $\ell_e$ is cut between two clusters with probability $O(\frac{\ell_e\cdot\log(n)}{\mathcal{D} })$. If the graph can be separated by $k \in \tilde{O}(1)$ paths, the probability improves to $O(\frac{\ell_e\cdot\log \log n}{\mathcal{D} })$. In either case, the decompositions can be computed in $\tilde{O}(1)$ depth and $\tilde{O}(kn)$ work in the PRAM and $\tilde{O}(1)$ time in the HYBRID model. In CONGEST, the runtimes are $\tilde{O}(HD + \sqrt{n})$ and $\tilde{O}(HD)$ respectively. All these results hold w.h.p. Broadly speaking, we present distributed and parallel implementations of sequential divide-and-conquer algorithms where we replace exact shortest paths with approximate shortest paths. In contrast to exact paths, these can be efficiently computed in the distributed and parallel setting [STOC '22]. Further, and perhaps more importantly, we show that instead of explicitly computing vertex-separators to enable efficient parallelization of these algorithms, it suffices to sample a few random paths of bounded length and the nodes close to them. Thereby, we do not require complex embeddings whose implementation is unknown in the distributed and parallel setting.

翻译：本文研究了在分布式与并行计算模型中，为（加权）图以及可通过$k \in \tilde{O}(1)$条最短路径分离的（加权）图构建具有强直径的低直径分解。此类图包括平面图、有界树宽图以及排除固定小图$K_r$的图。我们在PRAM、CONGEST及新型HYBRID通信模型中提出了算法，这些算法在所有相关参数上均具有竞争力。给定$\mathcal{D} > 0$，我们的低直径分解算法将图划分为强直径为$\mathcal{D}$的连通簇。对于任意图，长度为$\ell_e$的边$e \in E$被划分到不同簇的概率为$O(\frac{\ell_e\cdot\log(n)}{\mathcal{D} })$。若图可通过$k \in \tilde{O}(1)$条路径分离，该概率改进为$O(\frac{\ell_e\cdot\log \log n}{\mathcal{D} })$。在两种情况下，分解均可在PRAM模型中以$\tilde{O}(1)$深度与$\tilde{O}(kn)$工作量，以及在HYBRID模型中以$\tilde{O}(1)$时间完成计算。在CONGEST模型中，运行时间分别为$\tilde{O}(HD + \sqrt{n})$与$\tilde{O}(HD)$。所有结果均以高概率成立。概括而言，我们实现了顺序分治算法的分布式与并行版本，其中将精确最短路径替换为近似最短路径。与精确路径相比，后者可在分布式与并行环境中高效计算[STOC '22]。更重要的是，我们证明无需显式计算顶点分割器来实现这些算法的并行化，只需采样若干条有界长度的随机路径及其邻近节点即可。因此，我们避免了需要复杂嵌入的方法——这类嵌入在分布式与并行环境中的实现尚不明确。