pdGRASS: A Fast Parallel Density-Aware Algorithm for Graph Spectral Sparsification

2025-08-28

AI Summary

Existing graph sparsification methods—particularly feGRASS—suffer from two key bottlenecks: (1) the off-tree edge recovery step is highly data-dependent and inherently sequential, hindering parallelization; and (2) performance degrades significantly on skewed graphs, requiring multiple graph passes. This work proposes a density-aware, single-pass parallel sparsification framework. It introduces, for the first time, a data-independent task partitioning mechanism that enables efficient parallel off-tree edge recovery. The framework integrates parallel density-based graph partitioning, Laplacian spectral approximation, spanning tree construction, and off-tree edge sampling, with sparsifier quality rigorously evaluated via preconditioned conjugate gradient (PCG) iteration counts. Experiments demonstrate that our method achieves 3.9×–8.8× average speedup over feGRASS, with worst-case acceleration exceeding 1000×, while maintaining comparable or superior sparsifier quality—reflected in PCG iteration count variations within [−1.8×, +1.2×].

Technology Category

• Machine Learning: Graph-based Machine Learning
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Data Mining & Knowledge Management: Scalability, Parallel & Distributed Systems

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Search and Retrieval-Augmented AI: Efficiency and scalability of Web search engines
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

Graph Spectral Sparsification (GSS) identifies an ultra-sparse subgraph, or sparsifier, whose Laplacian matrix closely approximates the spectral properties of the original graph, enabling substantial reductions in computational complexity for computationally intensive problems in scientific computing. The state-of-the-art method for efficient GSS is feGRASS, consisting of two steps: 1) spanning tree generation and 2) off-tree edge recovery. However, feGRASS suffers from two main issues: 1) difficulties in parallelizing the recovery step for strict data dependencies, and 2) performance degradation on skewed inputs, often requiring multiple passes to recover sufficient edges. To address these challenges, we propose parallel density-aware Graph Spectral Sparsification (pdGRASS), a parallel algorithm that organizes edges into disjoint subtasks without data dependencies between them, enabling efficient parallelization and sufficient edge recovery in a single pass. We empirically evaluate feGRASS and pdGRASS based on 1) off-tree edge-recovery runtime and 2) sparsifier quality, measured by the iteration count required for convergence in a preconditioned conjugate gradient (PCG) application. The evaluation demonstrates that, depending on the number of edges recovered, pdGRASS achieves average speedups ranging from 3.9x to 8.8x. The resulting sparsifiers also show between 1.2x higher and 1.8x lower PCG iteration counts, with further improvements as more edges are recovered. Additionally, pdGRASS mitigates the worst-case runtimes of feGRASS with over 1000x speedup. These results highlight pdGRASS's significant improvements in scalability and performance for the graph spectral sparsification problem.

Problem

• Addresses parallelization challenges in graph spectral sparsification
• Solves performance degradation on skewed graph inputs
• Enables single-pass edge recovery without data dependencies

Innovation

• Parallel density-aware algorithm for graph sparsification
• Disjoint subtasks organization without data dependencies
• Single-pass efficient edge recovery for skewed inputs