A Note on Preconditioning by Low-Stretch Spanning Trees
This provides a tighter analysis for a known preconditioning technique, offering a modest improvement in theoretical complexity for Laplacian solvers.
The paper improves the theoretical runtime for solving Laplacian linear systems via preconditioned conjugate gradient with low-stretch spanning trees from O(m^{3/2+o(1)} ln(1/ε)) to O~(m^{4/3} ln(1/ε)).
Boman and Hendrickson observed that one can solve linear systems in Laplacian matrices in time $\bigO{m^{3/2 + o (1)} \ln (1/ε)}$ by preconditioning with the Laplacian of a low-stretch spanning tree. By examining the distribution of eigenvalues of the preconditioned linear system, we prove that the preconditioned conjugate gradient will actually solve the linear system in time $\softO{m^{4/3} \ln (1/ε)}$.