A simple, combinatorial algorithm for solving SDD systems in nearly-linear time

Original manuscript January 28, 2013In this paper, we present a simple combinatorial algorithm that solves symmetric diagonally dominant (SDD) linear systems in nearly-linear time. It uses 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 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 the simplicity of the algorithm and the insights yielded by its analysis will be useful in both theory and practice.National Science Foundation (U.S.) (Award 0843915)National Science Foundation (U.S.) (Award 1111109)Alfred P. Sloan Foundation (Research Fellowship)National Science Foundation (U.S.). Graduate Research Fellowship Program (Grant 1122374