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In this paper, we present a simple combinatorial algorithm that solvessymmetric diagonally dominant (SDD) linear systems in nearly-linear time. Ituses very little of the machinery that previously appeared to be necessary fora such an algorithm. It does not require recursive preconditioning, spectralsparsification, or even the Chebyshev Method or Conjugate Gradient. Afterconstructing a "nice" spanning tree of a graph associated with the linearsystem, the entire algorithm consists of the repeated application of a simple(non-recursive) update rule, which it implements using a lightweight datastructure. The algorithm is numerically stable and can be implemented withoutthe increased bit-precision required by previous solvers. As such, thealgorithm has the fastest known running time under the standard unit-cost RAMmodel. We hope that the simplicity of the algorithm and the insights yielded byits analysis will be useful in both theory and practice.

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