Advertisement

Algorithms, Graph Theory, and the Solution of Laplacian Linear Equations

  • Daniel A. Spielman
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7392)

Abstract

In this talk, we survey major developments in the design of algorithms for solving Laplacian linear equations, by which we mean systems of linear equations in the Laplacian matrices of graphs and their submatrices. We begin with a few examples of where such equations arise, including the analysis of networks of resistors, the analysis of networks of springs, and the solution of maximum flow problems by interior point methods.

Keywords

Span Tree Interior Point Method Annual IEEE Symposium 49th Annual IEEE Symposium Laplacian Matrice 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Vaidya, P.M.: Solving linear equations with symmetric diagonally dominant matrices by constructing good preconditioners. Unpublished manuscript UIUC 1990. A talk based on the manuscript was presented at the IMA Workshop on Graph Theory and Sparse Matrix Computation, October 1991, Minneapolis (1990)Google Scholar
  2. 2.
    Joshi, A.: Topics in Optimization and Sparse Linear Systems. PhD thesis, UIUC (1997)Google Scholar
  3. 3.
    Chen, D., Toledo, S.: Vaidya’s preconditioners: implementation and experimental study. Electronic Transactions on Numerical Analysis 16, 30–49 (2003)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Bern, M., Gilbert, J., Hendrickson, B., Nguyen, N., Toledo, S.: Support-graph preconditioners. SIAM J. Matrix Anal. & Appl. 27(4), 930–951 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Gremban, K.: Combinatorial Preconditioners for Sparse, Symmetric, Diagonally Dominant Linear Systems. PhD thesis, Carnegie Mellon University, CMU-CS-96-123 (1996)Google Scholar
  6. 6.
    Boman, E.G., Hendrickson, B.: Support theory for preconditioning. SIAM Journal on Matrix Analysis and Applications 25(3), 694–717 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Boman, E., Hendrickson, B.: On spanning tree preconditioners. Manuscript, Sandia National Lab. (2001)Google Scholar
  8. 8.
    Alon, N., Karp, R.M., Peleg, D., West, D.: A graph-theoretic game and its application to the k-server problem. SIAM Journal on Computing 24(1), 78–100 (1995)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Spielman, D.A., Teng, S.H.: Nearly-linear time algorithms for graph partitioning, graph sparsification, and solving linear systems. In: Proceedings of the Thirty-Sixth Annual ACM Symposium on Theory of Computing, pp. 81–90 (2004), Full version available at http://arxiv.org/abs/cs.DS/0310051
  10. 10.
    Spielman, D.A., Teng, S.H.: Nearly-linear time algorithms for preconditioning and solving symmetric, diagonally dominant linear systems. CoRR abs/cs/0607105 (2008), http://www.arxiv.org/abs/cs.NA/0607105 (submitted to SIMAX)
  11. 11.
    Elkin, M., Emek, Y., Spielman, D.A., Teng, S.H.: Lower-stretch spanning trees. SIAM Journal on Computing 32(2), 608–628 (2008)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Abraham, I., Bartal, Y., Neiman, O.: Nearly tight low stretch spanning trees. In: Proceedings of the 49th Annual IEEE Symposium on Foundations of Computer Science, pp. 781–790 (October 2008)Google Scholar
  13. 13.
    Abraham, I., Neiman, O.: Using petal-decompositions to build a low stretch spanning tree. In: Proceedings of The Fourty-Fourth Annual ACM Symposium on The Theory of Computing (STOC 2012) (to appear, 2012)Google Scholar
  14. 14.
    Andersen, R., Chung, F., Lang, K.: Local graph partitioning using pagerank vectors. In: Proceedings of the 47th Annual Symposium on Foundations of Computer Science, pp. 475–486 (2006)Google Scholar
  15. 15.
    Andersen, R., Peres, Y.: Finding sparse cuts locally using evolving sets. In: STOC 2009: Proceedings of the 41st Annual ACM Symposium on Theory of Computing, pp. 235–244. ACM, New York (2009)CrossRefGoogle Scholar
  16. 16.
    Orecchia, L., Sachdeva, S., Vishnoi, N.K.: Approximating the exponential, the lanczos method and an \(\tilde{O}(m)\)-time spectral algorithm for balanced separator. In: Proceedings of The Fourty-Fourth Annual ACM Symposium on The Theory of Computing (STOC 2012) (to appear, 2012)Google Scholar
  17. 17.
    Spielman, D.A., Srivastava, N.: Graph sparsification by effective resistances. In: Proceedings of the 40th Annual ACM Symposium on Theory of Computing, pp. 563–568 (2008)Google Scholar
  18. 18.
    Batson, J.D., Spielman, D.A., Srivastava, N.: Twice-Ramanujan sparsifiers. In: Proceedings of the 41st Annual ACM Symposium on Theory of Computing, pp. 255–262 (2009)Google Scholar
  19. 19.
    Levin, A., Koutis, I., Peng, R.: Improved spectral sparsification and numerical algorithms for sdd matrices. In: Proceedings of the 29th Symposium on Theoretical Aspects of Computer Science (STACS) (to appear, 2012)Google Scholar
  20. 20.
    Koutis, I., Miller, G., Peng, R.: Approaching optimality for solving sdd linear systems. In: 2010 51st Annual IEEE Symposium on Foundations of Computer Science (FOCS), pp. 235–244 (2010)Google Scholar
  21. 21.
    Koutis, I., Miller, G., Peng, R.: A nearly-mlogn time solver for sdd linear systems. In: 2011 52nd Annual IEEE Symposium on Foundations of Computer Science, FOCS (2011)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Daniel A. Spielman
    • 1
  1. 1.Departments of Computer Science and Mathematics, Program in Applied MathematicsYale UniversityUSA

Personalised recommendations