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

TL;DR

The paper advances distributed and parallel construction of low-diameter decompositions by replacing expensive exact shortest-path computations with efficient approximate SetSSP routines across PRAM, CONGEST, and HYBRID models. It delivers two main results: (i) for general weighted graphs, an LDD with strong diameter and quality O(log n) using only ~O(1) SetSSP calls and minor-aggregation steps, achieving near-optimal depth in PRAM and favorable CONGEST runtimes; (ii) for universally k-path separable graphs (including planar, bounded-treewidth, and minor-free families), an LDD with strong diameter and quality O(log log n) via a backbone-and-refinement scheme built on weak path separators and approximate shortest paths. A core methodological theme is the pseudo-padded decomposition and blurry ball growing, which enables tight edge-cut guarantees without requiring complex graph embeddings. The results unify divide-and-conquer strategies across several computation models, enabling efficient routing, embeddings-inspired constructions, and fast distributed solvers for a broad class of graphs. The techniques have potential impact on light spanners, metric embeddings, and distributed implementations of tree-based reductions, with practical relevance to large-scale networks and parallel processing frameworks.

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.

Figures (3)

• Figure A1: We choose a two-dimensional mesh for illustration. While the operations in Figures B and C are straightforward to prove, the core of our analysis we will be the claim we make in Figure D.
• Figure A2: An illustration of the boundary $\mathcal{B}(\mathcal{Z}(\mathcal{D}'))$. The inner circle denotes the core set $\mathcal{Z}(\mathcal{D}')$ and the outer circle denotes the ball $B_{G^{(i-1)}}(\mathcal{Z}(\mathcal{D}'), \mathcal{D}')$. $\mathcal{S}(\mathcal{D}')$ is the critical separator, consisting of several shortest paths. The red parts of $\mathcal{S}(\mathcal{D}')$ make up the boundary $\mathcal{B}(\mathcal{Z}(\mathcal{D}'))$. It suffices to remove the boundary s.t. a constant fraction of nodes have a distance greater than $\mathcal{D}'$ from the core set.
• Figure A3: An example for the chunks $C^{(\epsilon)}(P)$ of distance $\epsilon$ of a path $P := v_1, \ldots, v_8$. The solid lines denote edges between two nodes of the same chunk, the dashed lines denote edges between chunks.

Theorems & Definitions (88)

• definition A1: Probabilistic Low-diameter Decomposition (LDD)
• definition A2: $k$-path separatorDBLP:conf/podc/AbrahamG06
• definition A3: Approximate SetSSP with Virtual Nodes
• theorem A1
• theorem A2: LDDs for General Graphs
• theorem A3: LDDs for $k$-Path Seperable Graphs
• definition A4: Truncated Exponential Distribution
• theorem A4: Pseudo-Padded Decomposition for General Graphs
• lemma A1
• theorem A5: A Generic Clustering Theorem