#### Date of Original Version

4-7-2011

#### Type

Conference Proceeding

#### Rights Management

© 2011 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other uses, in any current or future media, including reprinting/republishing this material for advertising or promotional purposes, creating new collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in other works

#### Abstract or Description

We present an improved algorithm for solving symmetrically diagonally dominant linear systems. On input of an n×n symmetric diagonally dominant matrix A with m non-zero entries and a vector b such that Ax̅ = b for some (unknown) vector x̅, our algorithm computes a vector x such that ∥x-x̅∥A≤ϵ∥x̅∥A1in time Õ (m log n log (1/ϵ))2. The solver utilizes in a standard way a 'preconditioning' chain of progressively sparser graphs. To claim the faster running time we make a two-fold improvement in the algorithm for constructing the chain. The new chain exploits previously unknown properties of the graph sparsification algorithm given in [Koutis,Miller,Peng, FOCS 2010], allowing for stronger preconditioning properties.We also present an algorithm of independent interest that constructs nearly-tight low-stretch spanning trees in time Õ (m log n), a factor of O (log n) faster than the algorithm in [Abraham,Bartal,Neiman, FOCS 2008]. This speedup directly reflects on the construction time of the preconditioning chain.

#### DOI

10.1109/FOCS.2011.85

#### Published In

Proceedings of the IEEE Symposium on Foundations of Computer Science (FOCS), 2011, 590-598.

## Comments

Presented as "A Nearly-m log n Time Solver for SDD Linear Systems"