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

  • Authors:

  • Jonathan A. Kelner;Lorenzo Orecchia;Aaron Sidford;Zeyuan Allen Zhu

  • Affiliations:

  • Massachusetts Institute of Technology, Cambridge, MA, USA;Massachusetts Institute of Technology, Cambridge, MA, USA;Massachusetts Institute of Technology, Cambridge, MA, USA;Massachusetts Institute of Technology, Cambridge, MA, USA

  • Venue:
  • Proceedings of the forty-fifth annual ACM symposium on Theory of computing
  • Year:
  • 2013

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Abstract

In 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.