A simple, combinatorial algorithm for solving SDD systems in nearly-linear time
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Kelner_A simple combinatorial.pdf
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1.88 MB
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Author(s)
Orecchia, Lorenzo
Zhu, Zeyuan Allen
Kelner, Jonathan Adam
Sidford, Aaron D.
Date Issued
June 2013
Journal
Proceedings of the 45th annual ACM symposium on Symposium on theory of computing (STOC '13)
Publisher
Association for Computing Machinery (ACM)
Citation
Jonathan A. Kelner, Lorenzo Orecchia, Aaron Sidford, and Zeyuan Allen Zhu. 2013. A simple, combinatorial algorithm for solving SDD systems in nearly-linear time. In Proceedings of the 45th annual ACM symposium on Symposium on theory of computing (STOC '13). ACM, New York, NY, USA, 911-920.
Version
Original manuscript
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.
Description
Original manuscript January 28, 2013
MIT Department
Massachusetts Institute of Technology. Computer Science and Artificial Intelligence Laboratory
Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science
Massachusetts Institute of Technology. Department of Mathematics
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Creative Commons Attribution-Noncommercial-Share Alike 3.0
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DOI of Published Version
https://doi.org/10.1145/2488608.2488724
