Spectral graph sparsification in nearly-linear time leveraging efficient spectral perturbation analysis

Document Type

Conference Proceeding

Publication Date



Department of Electrical and Computer Engineering, Center for Scalable Architectures and Systems


Spectral graph sparsification aims to find an ultra-sparse subgraph whose Laplacian matrix can well approximate the original Laplacian matrix in terms of its eigenvalues and eigenvectors. The resultant sparsified subgraph can be efficiently leveraged as a proxy in a variety of numerical computation applications and graph-based algorithms. This paper introduces a practically efficient, nearly-linear time spectral graph sparsification algorithm that can immediately lead to the development of nearly-linear time symmetric diagonally-dominant (SDD) matrix solvers. Our spectral graph sparsification algorithm can efficiently build an ultra-sparse subgraph from a spanning tree subgraph by adding a few "spectrally-critical" off-tree edges back to the spanning tree, which is enabled by a novel spectral perturbation approach and allows to approximately preserve key spectral properties of the original graph Laplacian. Extensive experimental results confirm the nearly-linear runtime scalability of an SDD matrix solver for large-scale, real-world problems, such as VLSI, thermal and finite-element analysis problems, etc. For instance, a sparse SDD matrix with 40 million unknowns and 180 million nonzeros can be solved (1E-3 accuracy level) within two minutes using a single CPU core and about 6GB memory.

Publication Title

Proceedings of the 53rd Annual Design Automation Conference