Derandomization Beyond Connectivity: Undirected Laplacian Systems in Nearly Logarithmic Space

We give a deterministic \tilde{O}(\log n)-space algorithm for approximately solving linear systems given by Laplacians of undirected graphs, and consequently also approximating hitting times, commute times, and escape probabilities for undirected graphs. Previously, such systems were known to be solvable by randomized algorithms using O(\log n) space (Doron, Le Gall, and Ta-Shma, 2017) and hence by deterministic algorithms using O(\log^{3/2} n) space (Saks and Zhou, FOCS 1995 and JCSS 1999).Our algorithm combines ideas from time-efficient Laplacian solvers (Spielman and Teng, STOC 04; Peng and Spielman, STOC 14) with ideas used to show that Undirected S-T Connectivity is in deterministic logspace (Reingold, STOC 05 and JACM 08; Rozenman and Vadhan, RANDOM 05).

[1]  Yin Tat Lee,et al.  Efficient Accelerated Coordinate Descent Methods and Faster Algorithms for Solving Linear Systems , 2013, 2013 IEEE 54th Annual Symposium on Foundations of Computer Science.

[2]  Dieter van Melkebeek,et al.  Graph Nonisomorphism Has Subexponential Size Proofs Unless the Polynomial-Time Hierarchy Collapses , 2002, SIAM J. Comput..

[3]  Luca Trevisan,et al.  Better Pseudorandom Generators from Milder Pseudorandom Restrictions , 2012, 2012 IEEE 53rd Annual Symposium on Foundations of Computer Science.

[4]  Shang-Hua Teng,et al.  Electrical flows, laplacian systems, and faster approximation of maximum flow in undirected graphs , 2010, STOC '11.

[5]  Yin Tat Lee,et al.  Efficient Inverse Maintenance and Faster Algorithms for Linear Programming , 2015, 2015 IEEE 56th Annual Symposium on Foundations of Computer Science.

[6]  Gary L. Miller,et al.  Combinatorial preconditioners and multilevel solvers for problems in computer vision and image processing , 2009, Comput. Vis. Image Underst..

[7]  Omer Reingold,et al.  S-T Connectivity on Digraphs with a Known Stationary Distribution , 2007, Twenty-Second Annual IEEE Conference on Computational Complexity (CCC'07).

[8]  Salil P. Vadhan,et al.  Derandomized Squaring of Graphs , 2005, APPROX-RANDOM.

[9]  Salil P. Vadhan,et al.  Pseudorandomness , 2012, Found. Trends Theor. Comput. Sci..

[10]  Amnon Ta-Shma,et al.  Inverting well conditioned matrices in quantum logspace , 2013, STOC '13.

[11]  Aleksander Madry,et al.  Faster Generation of Random Spanning Trees , 2009, 2009 50th Annual IEEE Symposium on Foundations of Computer Science.

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

[13]  Luca Trevisan,et al.  Pseudorandom walks on regular digraphs and the RL vs. L problem , 2006, STOC '06.

[14]  Thomas Steinke,et al.  Pseudorandomness and Fourier-Growth Bounds for Width-3 Branching Programs , 2014, Theory Comput..

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

[16]  Richard Peng,et al.  Almost-linear-time algorithms for Markov chains and new spectral primitives for directed graphs , 2016, STOC.

[17]  Thomas Steinke.  Pseudorandomness for Permutation Branching Programs Without the Group Theory , 2012, Electron. Colloquium Comput. Complex..

[18]  Eric Allender,et al.  Uniform constant-depth threshold circuits for division and iterated multiplication , 2002, J. Comput. Syst. Sci..

[19]  Pavel Pudlák,et al.  Pseudorandom generators for group products: extended abstract , 2011, STOC '11.

[20]  Neil D. Jones,et al.  New problems complete for nondeterministic log space , 1976, Mathematical systems theory.

[21]  Michael Saks,et al.  BP H SPACE(S)⊆DSPACE(S 3/2 ) , 1999, FOCS 1999.

[22]  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.

[23]  Shang-Hua Teng,et al.  A Local Clustering Algorithm for Massive Graphs and Its Application to Nearly Linear Time Graph Partitioning , 2008, SIAM J. Comput..

[24]  Moni Naor,et al.  Small-bias probability spaces: efficient constructions and applications , 1990, STOC '90.

[25]  Richard Peng,et al.  An efficient parallel solver for SDD linear systems , 2013, STOC.

[26]  Avi Wigderson,et al.  Entropy waves, the zig-zag graph product, and new constant-degree expanders and extractors , 2000, Proceedings 41st Annual Symposium on Foundations of Computer Science.

[27]  Joshua Brody,et al.  The Coin Problem and Pseudorandomness for Branching Programs , 2010, 2010 IEEE 51st Annual Symposium on Foundations of Computer Science.

[28]  Amnon Ta-Shma,et al.  Probabilistic logarithmic-space algorithms for Laplacian solvers , 2017, Electron. Colloquium Comput. Complex..

[29]  Yin Tat Lee,et al.  An Almost-Linear-Time Algorithm for Approximate Max Flow in Undirected Graphs, and its Multicommodity Generalizations , 2013, SODA.

[30]  Noam Nisan,et al.  Pseudorandom generators for space-bounded computation , 1992, Comb..

[31]  Jonah Sherman,et al.  Breaking the Multicommodity Flow Barrier for O(vlog n)-Approximations to Sparsest Cut , 2009, 2009 50th Annual IEEE Symposium on Foundations of Computer Science.

[32]  Christos H. Papadimitriou,et al.  Symmetric Space-Bounded Computation , 1982, Theor. Comput. Sci..

[33]  Carme Àlvarez,et al.  A compendium of problems complete for symmetric logarithmic space , 2000, computational complexity.

[34]  Russell Impagliazzo,et al.  Pseudorandomness from Shrinkage , 2012, 2012 IEEE 53rd Annual Symposium on Foundations of Computer Science.

[35]  Dieter van Melkebeek,et al.  Graph nonisomorphism has subexponential size proofs unless the polynomial-time hierarchy collapses , 1999, STOC '99.

[36]  Avi Wigderson,et al.  P = BPP if E requires exponential circuits: derandomizing the XOR lemma , 1997, STOC '97.

[37]  Omer Reingold,et al.  Undirected connectivity in log-space , 2008, JACM.

[38]  Richard Peng,et al.  Faster Algorithms for Computing the Stationary Distribution, Simulating Random Walks, and More , 2016, 2016 IEEE 57th Annual Symposium on Foundations of Computer Science (FOCS).

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

[40]  Thomas Steinke,et al.  Pseudorandomness for Regular Branching Programs via Fourier Analysis , 2013, APPROX-RANDOM.

[41]  Ran Raz,et al.  Pseudorandom Generators for Regular Branching Programs , 2010, 2010 IEEE 51st Annual Symposium on Foundations of Computer Science.

[42]  Nisheeth K. Vishnoi,et al.  Towards an SDP-based approach to spectral methods: a nearly-linear-time algorithm for graph partitioning and decomposition , 2010, SODA '11.

[43]  Joan Feigenbaum,et al.  Proceedings of the forty-fifth annual ACM symposium on Theory of computing , 2013, STOC 2013.

[44]  Vladimir Trifonov,et al.  An O(log n log log n) space algorithm for undirected st-connectivity , 2005, STOC '05.

[45]  Avi Wigderson,et al.  In search of an easy witness: exponential time vs. probabilistic polynomial time , 2001, Proceedings 16th Annual IEEE Conference on Computational Complexity.

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

[47]  Stephen A. Cook,et al.  Log Depth Circuits for Division and Related Problems , 1986, SIAM J. Comput..

[48]  Russell Impagliazzo,et al.  Derandomizing Polynomial Identity Tests Means Proving Circuit Lower Bounds , 2003, STOC '03.

[49]  Stanislav Zák,et al.  Almost k-Wise Independent Sets Establish Hitting Sets for Width-3 1-Branching Programs , 2011, CSR.

[50]  Zeyuan Allen Zhu,et al.  A simple, combinatorial algorithm for solving SDD systems in nearly-linear time , 2013, STOC '13.

[51]  Richard J. Lipton,et al.  Random walks, universal traversal sequences, and the complexity of maze problems , 1979, 20th Annual Symposium on Foundations of Computer Science (sfcs 1979).