A Note on Preconditioning by Low-Stretch Spanning Trees

TL;DR

This paper improves the theoretical understanding of solving Laplacian linear systems by analyzing eigenvalue distributions, leading to a faster algorithm using low-stretch spanning trees for preconditioning.

Contribution

It provides a refined analysis of eigenvalues in preconditioned systems, demonstrating a faster convergence time for solving Laplacian systems with low-stretch spanning trees.

Findings

Preconditioned conjugate gradient solves systems in ((m^{4/3} \u00a0 ext{log}(1/\u03b5))) time.

Eigenvalue distribution analysis explains the improved convergence.

Theoretical bounds on solving Laplacian systems are tightened.

Abstract

Boman and Hendrickson observed that one can solve linear systems in Laplacian matrices in time $\bigO{m^{3/2 + o (1)} \ln (1/\epsilon)}$ 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 $\softO{m^{4/3} \ln (1/\epsilon)}$.