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

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.