Faster Spectral Sparsification and Numerical Algorithms for SDD Matrices

We study algorithms for spectral graph sparsification. The input is a graph <i>G</i> with <i>n</i> vertices and <i>m</i> edges, and the output is a sparse graph <i>&Gtilde;</i> that approximates <i>G</i> in an algebraic sense. Concretely, for all vectors <i>x</i> and any ε > 0, the graph <i>&Gtilde;</i> satisfies (1-ε )<i>x<sup><i>T</i></sup> <i>L</i><sub><i>G</i></sub><sup><i>x</i></sup></i> ≤ <i>x</i><sup><i>T</i></sup> <i>L</i><i>&Gtilde;</i><sup><i>x</i></sup> ≤ (1+ε)<i>x</i><sup><i>T</i></sup> <i>L</i><sub><i>G</i></sub><sup><i>x</i></sup>, where <i>L<sub>G</sub></i> and <i>&Gtilde;</i> are the Laplacians of <i>G</i> and <i>&Gtilde;</i> respectively. The first contribution of this article applies to all existing sparsification algorithms that rely on solving solving linear systems on graph Laplacians. These algorithms are the fastest known to date. Specifically, we show that less precision is required in the solution of the linear systems, leading to speedups by an <i>O</i>(log <i>n</i>) factor. We also present faster sparsification algorithms for slightly dense graphs: — An <i>O</i>(<i>m</i>log <i>n</i>) time algorithm that generates a sparsifier with <i>O</i>(<i>n</i>log <sup>3</sup><i>n</i>/ε<sup>2</sup>) edges. — An <i>O</i>(<i>m</i>log log <i>n</i>) time algorithm for graphs with more than <i>n</i>log <sup>5</sup><i>n</i>log log <i>n</i> edges. — An <i>O</i>(<i>m</i>) algorithm for graphs with more than <i>n</i>log <sup>10</sup><i>n</i> edges. — An <i>O</i>(<i>m</i>) algorithm for unweighted graphs with more than <i>n</i>log <sup>8</sup><i>n</i> edges. These bounds hold up to factors that are in <i>O</i>(<i>poly</i>(log log <i>n</i>)) and are conjectured to be removable.

[1]  Jakub W. Pachocki,et al.  Solving SDD linear systems in nearly mlog1/2n time , 2014, STOC.

[2]  Robert E. Tarjan,et al.  A linear-time algorithm for a special case of disjoint set union , 1983, J. Comput. Syst. Sci..

[3]  W. B. Johnson,et al.  Extensions of Lipschitz mappings into Hilbert space , 1984 .

[4]  Richard Peng,et al.  Improved Spectral Sparsification and Numerical Algorithms for SDD Matrices , 2012, STACS.

[5]  Robert E. Tarjan,et al.  Applications of Path Compression on Balanced Trees , 1979, JACM.

[6]  Daniel A. Spielman,et al.  A Note on Preconditioning by Low-Stretch Spanning Trees , 2009, ArXiv.

[7]  Gary L. Miller,et al.  Iterative Row Sampling , 2012, 2013 IEEE 54th Annual Symposium on Foundations of Computer Science.

[8]  Jonathan A. Kelner,et al.  Spectral Sparsification in the Semi-streaming Setting , 2013, Theory of Computing Systems.

[9]  F. Chung.  Random walks and local cuts in graphs , 2007 .

[10]  Nikhil Srivastava,et al.  Graph sparsification by effective resistances , 2008, SIAM J. Comput..

[11]  Richard Peng,et al.  Uniform Sampling for Matrix Approximation , 2014, ITCS.

[12]  G. Miller,et al.  Combinatorial and algebraic tools for optimal multilevel algorithms , 2007 .

[13]  Gary L. Miller,et al.  Approaching Optimality for Solving SDD Linear Systems , 2010, 2010 IEEE 51st Annual Symposium on Foundations of Computer Science.

[14]  Gary L. Miller,et al.  A Nearly-m log n Time Solver for SDD Linear Systems , 2011, 2011 IEEE 52nd Annual Symposium on Foundations of Computer Science.

[15]  Piotr Indyk,et al.  Approximate nearest neighbors: towards removing the curse of dimensionality , 1998, STOC '98.

[16]  Dimitris Achlioptas,et al.  Database-friendly random projections: Johnson-Lindenstrauss with binary coins , 2003, J. Comput. Syst. Sci..

[17]  Debmalya Panigrahi,et al.  A general framework for graph sparsification , 2010, STOC '11.

[18]  Shang-Hua Teng,et al.  Spectral Sparsification of Graphs , 2008, SIAM J. Comput..

[19]  Ittai Abraham,et al.  Nearly Tight Low Stretch Spanning Trees , 2008, 2008 49th Annual IEEE Symposium on Foundations of Computer Science.

[20]  D. Spielman,et al.  Spectral sparsification and restricted invertibility , 2010 .

[21]  Shang-Hua Teng,et al.  Nearly-linear time algorithms for graph partitioning, graph sparsification, and solving linear systems , 2003, STOC '04.

[22]  Ioannis Koutis,et al.  Simple Parallel and Distributed Algorithms for Spectral Graph Sparsification , 2016, TOPC.

[23]  Peter G. Doyle,et al.  Random Walks and Electric Networks: REFERENCES , 1987 .

[24]  Piotr Indyk,et al.  Approximate Nearest Neighbor: Towards Removing the Curse of Dimensionality , 2012, Theory Comput..

[25]  Gary L. Miller,et al.  Approaching optimality for solving SDD systems , 2010, ArXiv.

[26]  David R. Karger,et al.  Approximating s – t Minimum Cuts in ~ O(n 2 ) Time , 2007 .

[27]  Dimitris Achlioptas,et al.  Database-friendly random projections , 2001, PODS.

[28]  Ittai Abraham,et al.  Using petal-decompositions to build a low stretch spanning tree , 2012, STOC '12.

[29]  Jakub W. Pachocki,et al.  Stretching Stretch , 2014, ArXiv.

[30]  Shang-Hua Teng,et al.  Nearly-Linear Time Algorithms for Preconditioning and Solving Symmetric, Diagonally Dominant Linear Systems , 2006, SIAM J. Matrix Anal. Appl..

[31]  David R. Karger.  Better random sampling algorithms for flows in undirected graphs , 1998, SODA '98.