Theory of Computing Systems

, Volume 55, Issue 3, pp 521–554

Nearly-Linear Work Parallel SDD Solvers, Low-Diameter Decomposition, and Low-Stretch Subgraphs

  • Guy E. Blelloch
  • Anupam Gupta
  • Ioannis Koutis
  • Gary L. Miller
  • Richard Peng
  • Kanat Tangwongsan
Article

DOI: 10.1007/s00224-013-9444-5

Cite this article as:
Blelloch, G.E., Gupta, A., Koutis, I. et al. Theory Comput Syst (2014) 55: 521. doi:10.1007/s00224-013-9444-5

Abstract

We present the design and analysis of a nearly-linear work parallel algorithm for solving symmetric diagonally dominant (SDD) linear systems. On input an SDD n-by-n matrix A with m nonzero entries and a vector b, our algorithm computes a vector \(\tilde{x}\) such that \(\|\tilde{x} - A^{+}b\|_{A} \leq\varepsilon\cdot\|{A^{+}b}\|_{A}\) in \(O(m\log^{O(1)}{n}\log {\frac{1}{\varepsilon}})\) work and \(O(m^{1/3+\theta}\log\frac{1}{\varepsilon})\) depth for any θ>0, where A+ denotes the Moore-Penrose pseudoinverse of A.

The algorithm relies on a parallel algorithm for generating low-stretch spanning trees or spanning subgraphs. To this end, we first develop a parallel decomposition algorithm that in O(mlogO(1)n) work and polylogarithmic depth, partitions a graph with n nodes and m edges into components with polylogarithmic diameter such that only a small fraction of the original edges are between the components. This can be used to generate low-stretch spanning trees with average stretch O(nα) in O(mlogO(1)n) work and O(nα) depth for any α>0. Alternatively, it can be used to generate spanning subgraphs with polylogarithmic average stretch in O(mlogO(1)n) work and polylogarithmic depth. We apply this subgraph construction to derive a parallel linear solver.

By using this solver in known applications, our results imply improved parallel randomized algorithms for several problems, including single-source shortest paths, maximum flow, minimum-cost flow, and approximate maximum flow.

Keywords

Parallel algorithms Linear systems Low-stretch spanning trees Low-stretch subgraphs Low-diameter decomposition 

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • Guy E. Blelloch
    • 1
  • Anupam Gupta
    • 1
  • Ioannis Koutis
    • 2
  • Gary L. Miller
    • 1
  • Richard Peng
    • 1
  • Kanat Tangwongsan
    • 3
  1. 1.Carnegie Mellon UniversityPittsburghUSA
  2. 2.University of Puerto RicoRio PiedrasPuerto Rico
  3. 3.Yorktown HeightsUSA

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