A Simple, Combinatorial Algorithm for Solving SDD Systems in Nearly-Linear Time

TL;DR

This paper introduces a straightforward combinatorial algorithm that efficiently solves symmetric diagonally dominant (SDD) linear systems in nearly-linear time, avoiding complex techniques used in prior methods.

Contribution

It presents a simple, non-recursive algorithm that achieves nearly-linear time complexity for SDD systems without advanced spectral techniques.

Findings

Algorithm is numerically stable and easy to implement.

Achieves the fastest known running time under the standard RAM model.

Does not require recursive preconditioning or spectral sparsification.

Abstract

In this paper, we present a simple combinatorial algorithm that solves symmetric diagonally dominant (SDD) linear systems in nearly-linear time. It uses very little of the machinery that previously appeared to be necessary for a such an algorithm. It does not require recursive preconditioning, spectral sparsification, or even the Chebyshev Method or Conjugate Gradient. After constructing a "nice" spanning tree of a graph associated with the linear system, the entire algorithm consists of the repeated application of a simple (non-recursive) update rule, which it implements using a lightweight data structure. The algorithm is numerically stable and can be implemented without the increased bit-precision required by previous solvers. As such, the algorithm has the fastest known running time under the standard unit-cost RAM model. We hope that the simplicity of the algorithm and the insights yielded by its analysis will be useful in both theory and practice.