Parallel and Distributed Expander Decomposition: Simple, Fast, and Near-Optimal

TL;DR

The paper tackles the problem of computing φ-expander decompositions in undirected graphs using parallel and distributed models. It introduces a simple, flow-based trimming approach and a parallelized unit-flow routine, achieving near-linear work and polylogarithmic span, and porting to the Congest model for distributed computation. The main contributions include the first near-optimal parallel algorithm with crossing-edge quality $ ilde{O}(φ m)$ and span $ ilde{O}(1/φ^4)$, and a distributed version with $ ilde{O}(1/φ^4)$ rounds, improving over prior φ-based guarantees that degraded cross-edge quality. The techniques combine a refined trimming procedure with a parallel cut-matching backbone, leveraging flow certificates rather than random walks, and are designed to integrate with practical expander decomposition pipelines for large-scale graphs.

Abstract

Expander decompositions have become one of the central frameworks in the design of fast algorithms. For an undirected graph $G=(V,E)$, a near-optimal $φ$-expander decomposition is a partition $V_1, V_2, \ldots, V_k$ of the vertex set $V$ where each subgraph $G[V_i]$ is a $φ$-expander, and only an $\widetilde{O}(φ)$-fraction of the edges cross between partition sets. In this article, we give the first near-optimal parallel algorithm to compute $φ$-expander decompositions in near-linear work $\widetilde{O}(m/φ^2)$ and near-constant span $\widetilde{O}(1/φ^4)$. Our algorithm is very simple and likely practical. Our algorithm can also be implemented in the distributed Congest model in $\tilde{O}(1/φ^4)$ rounds. Our results surpass the theoretical guarantees of the current state-of-the-art parallel algorithms [Chang-Saranurak PODC'19, Chang-Saranurak FOCS'20], while being the first to ensure that only an $\tilde{O}(φ)$ fraction of edges cross between partition sets. In contrast, previous algorithms [Chang-Saranurak PODC'19, Chang-Saranurak FOCS'20] admit at least an $O(φ^{1/3})$ fraction of crossing edges, a polynomial loss in quality inherent to their random-walk-based techniques. Our algorithm, instead, leverages flow-based techniques and extends the popular sequential algorithm presented in [Saranurak-Wang SODA'19].

Theorems & Definitions (30)

• Theorem 1.1: Parallel Expander Decomposition
• Theorem 1.2: Distributed Expander Decomposition
• Definition 3.1: Nearly Expander
• Lemma 3.1: Parallel Trimming
• Lemma 3.2: See Proposition 3.2 in saranurak2019expander
• proof
• Lemma 3.2
• Claim 3.3
• proof
• Claim 3.4