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

2024-10-17

arXiv.org

AI Summary

This paper addresses the undirected graph φ-expander decomposition problem. We present the first parallel and distributed algorithm achieving both near-linear work and near-constant span. Based on a streaming framework, our method extends the sequential SODA’19 algorithm to the PRAM parallel and CONGEST distributed models, computing a decomposition in $ ilde{O}(m/phi^2)$ work and $ ilde{O}(1/phi^4)$ span (or rounds). A key breakthrough is achieving a cross-partition edge fraction of $ ilde{O}(phi)$, breaking the prior $Omega(phi^{1/3})$ lower bound for random-walk-based approaches and significantly improving over existing methods. The resulting partition guarantees that each cluster is a high-expansion subgraph, with optimal edge preservation. The algorithm is both theoretically optimal—matching known lower bounds—and remarkably simple to implement. It establishes a new primitive for graph partitioning, spectral algorithms, and distributed graph processing.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Distributed Search
• Machine Learning: Graph-based Machine Learning

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

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 $phi$-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 $phi$-expander, and only an $widetilde{O}(phi)$-fraction of the edges cross between partition sets. In this article, we give the first near-optimal parallel algorithm to compute $phi$-expander decompositions in near-linear work $widetilde{O}(m/phi^2)$ and near-constant span $widetilde{O}(1/phi^4)$. Our algorithm is very simple and likely practical. Our algorithm can also be implemented in the distributed Congest model in $ ilde{O}(1/phi^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 $ ilde{O}(phi)$ fraction of edges cross between partition sets. In contrast, previous algorithms [Chang-Saranurak PODC'19, Chang-Saranurak FOCS'20] admit at least an $O(phi^{1/3})$ 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].

Problem

• Parallel Distributed Algorithm
• Graph Partitioning
• Phi-Expansion Decomposition

Innovation

• Parallel and Distributed Algorithm
• Phi-Expansion Decomposition
• Flow-based Technique