A nearly-mlogn time solver for SDD linear systems

TL;DR

This paper introduces a nearly logarithmic time algorithm for solving symmetric diagonally dominant linear systems, improving the construction of preconditioning chains and nearly-tight low-stretch spanning trees.

Contribution

It presents a faster method for constructing preconditioning chains and low-stretch spanning trees, leading to more efficient solvers for SDD linear systems.

Findings

Achieves ${ ilde O}(m ext{log} n ext{log} (1/ ext{epsilon}))$ solve time.

Develops a faster algorithm for constructing low-stretch spanning trees.

Improves preconditioning chain construction using new graph sparsification properties.

Abstract

We present an improved algorithm for solving symmetrically diagonally dominant linear systems. On input of an $n\times n$ symmetric diagonally dominant matrix $A$ with $m$ non-zero entries and a vector $b$ such that $A\bar{x} = b$ for some (unknown) vector $\bar{x}$, our algorithm computes a vector $x$ such that $||{x}-\bar{x}||_A < \epsilon ||\bar{x}||_A $ {$||\cdot||_A$ denotes the A-norm} in time $${\tilde O}(m\log n \log (1/\epsilon)).$$ 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 $\tilde{O}(m\log{n})$, 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.