Distributed And Parallel Low-Diameter Decompositions for Arbitrary and Restricted Graphs

TL;DR

This paper develops efficient distributed and parallel algorithms for constructing low-diameter graph decompositions, applicable to various graph classes, with improved probabilities and runtime guarantees, using approximate shortest paths and random sampling.

Contribution

It introduces novel algorithms for low-diameter decompositions in distributed and parallel models, leveraging approximate shortest paths and random path sampling, applicable to graphs with special structural properties.

Findings

Algorithms achieve low-diameter decompositions with high probability.

Runtime is polylogarithmic in the number of nodes for various models.

Probabilities of cutting edges depend on edge length and graph structure.

Abstract

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.