Boosting Graph Spectral Sparsification via Parallel Sparse Approximate Inverse of Cholesky Factor

Baiyu Chen, Zhiqiang Liu, Yibin Zhang, Wenjian Yu

Abstract: With the advance of very-large-scale-integrated (VLSI) systems, fast and efficient algorithms for solving equations of Laplacian matrices are increasingly significant. Graph spectral sparsification, which aims to produce an ultra-sparse subgraph while preserving properties of original graph, has aroused extensive attention thanks to its distinguished performance. For preconditioning, the effectiveness of sparsifiers produced by graph spectral sparsification algorithms may directly influence the speed of PCG iterations, while the recently proposed algorithm that pursues effective sparsifiers may result in huge time expenditure of sparsifier construction as calculating the sparse approximate inverse of Cholesky factor may be rather time-consuming. In this paper, based on domain decomposition, a parallel algorithm for calculating sparse approximate inverse of Cholesky factor is proposed, where a skill for calculating Schur complement matrix based on partial Cholesky factorization is applied. Based on the proposed parallel algorithm for calculating sparse approximate inverse of Cholesky factor, a fast and effective parallel graph spectral sparsification algorithm is proposed. Extensive experiments reveal that the proposed parallel graph spectral sparsification algorithm shows eminent speedup compared with serial approach. Moreover, for transient analysis of power grids, the proposed algorithm shows significant speedup compared with the state-of-the-art parallel iterative solver based on graph sparsification.