Dynamic Effective Resistances and Approximate Schur Complement on Separable Graphs

Authors Gramoz Goranci, Monika Henzinger, Pan Peng



PDF
Thumbnail PDF

File

LIPIcs.ESA.2018.40.pdf
  • Filesize: 0.52 MB
  • 15 pages

Document Identifiers

Author Details

Gramoz Goranci
  • University of Vienna, Faculty of Computer Science, Vienna, Austria
Monika Henzinger
  • University of Vienna, Faculty of Computer Science, Vienna, Austria
Pan Peng
  • Department of Computer Science, University of Sheffield, Sheffield, UK

Cite AsGet BibTex

Gramoz Goranci, Monika Henzinger, and Pan Peng. Dynamic Effective Resistances and Approximate Schur Complement on Separable Graphs. In 26th Annual European Symposium on Algorithms (ESA 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 112, pp. 40:1-40:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)
https://doi.org/10.4230/LIPIcs.ESA.2018.40

Abstract

We consider the problem of dynamically maintaining (approximate) all-pairs effective resistances in separable graphs, which are those that admit an n^{c}-separator theorem for some c<1. We give a fully dynamic algorithm that maintains (1+epsilon)-approximations of the all-pairs effective resistances of an n-vertex graph G undergoing edge insertions and deletions with O~(sqrt{n}/epsilon^2) worst-case update time and O~(sqrt{n}/epsilon^2) worst-case query time, if G is guaranteed to be sqrt{n}-separable (i.e., it is taken from a class satisfying a sqrt{n}-separator theorem) and its separator can be computed in O~(n) time. Our algorithm is built upon a dynamic algorithm for maintaining approximate Schur complement that approximately preserves pairwise effective resistances among a set of terminals for separable graphs, which might be of independent interest. We complement our result by proving that for any two fixed vertices s and t, no incremental or decremental algorithm can maintain the s-t effective resistance for sqrt{n}-separable graphs with worst-case update time O(n^{1/2-delta}) and query time O(n^{1-delta}) for any delta>0, unless the Online Matrix Vector Multiplication (OMv) conjecture is false. We further show that for general graphs, no incremental or decremental algorithm can maintain the s-t effective resistance problem with worst-case update time O(n^{1-delta}) and query-time O(n^{2-delta}) for any delta >0, unless the OMv conjecture is false.

Subject Classification

ACM Subject Classification
  • Theory of computation → Dynamic graph algorithms
Keywords
  • Dynamic graph algorithms
  • effective resistance
  • separable graphs
  • Schur complement
  • conditional lower bounds

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads

References

  1. Amir Abboud and Søren Dahlgaard. Popular conjectures as a barrier for dynamic planar graph algorithms. In Proc. of the 57th FOCS, pages 477-486, 2016. Google Scholar
  2. Ittai Abraham, Shiri Chechik, and Cyril Gavoille. Fully dynamic approximate distance oracles for planar graphs via forbidden-set distance labels. In Proc. of the 44th STOC, pages 1199-1218, 2012. Google Scholar
  3. Ittai Abraham, David Durfee, Ioannis Koutis, Sebastian Krinninger, and Richard Peng. On fully dynamic graph sparsifiers. In Proc. of the 57th FOCS, pages 335-344, 2016. Google Scholar
  4. Nima Anari and Shayan Oveis Gharan. Effective-resistance-reducing flows, spectrally thin trees, and asymmetric tsp. In Proc. of the 56th FOCS, pages 20-39, 2015. Google Scholar
  5. Daniel K Blandford, Guy E Blelloch, and Ian A Kash. Compact representations of separable graphs. In Proceedings of the fourteenth annual ACM-SIAM symposium on Discrete algorithms, pages 679-688. Society for Industrial and Applied Mathematics, 2003. Google Scholar
  6. James R Bunch and John E Hopcroft. Triangular factorization and inversion by fast matrix multiplication. Mathematics of Computation, 28(125):231-236, 1974. Google Scholar
  7. Dehua Cheng, Yu Cheng, Yan Liu, Richard Peng, and Shang-Hua Teng. Efficient sampling for gaussian graphical models via spectral sparsification. In Conference on Learning Theory, pages 364-390, 2015. Google Scholar
  8. Paul Christiano, Jonathan A. Kelner, Aleksander Madry, Daniel A. Spielman, and Shang-Hua Teng. Electrical flows, laplacian systems, and faster approximation of maximum flow in undirected graphs. In Proc. of the 43rd STOC, pages 273-282, 2011. Google Scholar
  9. Michael B. Cohen, Rasmus Kyng, Gary L. Miller, Jakub W. Pachocki, Richard Peng, Anup B. Rao, and Shen Chen Xu. Solving SDD linear systems in nearly mlog^1/2n time. In Proc. of the 46th STOC, pages 343-352, 2014. Google Scholar
  10. Krzysztof Diks and Piotr Sankowski. Dynamic plane transitive closure. In European Symposium on Algorithms, pages 594-604. Springer, 2007. Google Scholar
  11. Michael Dinitz, Robert Krauthgamer, and Tal Wagner. Towards resistance sparsifiers. In Proc. of the 18th APPROX, pages 738-755, 2015. Google Scholar
  12. Peter G Doyle and J Laurie Snell. Random Walks and Electric Networks. Carus Mathematical Monographs. Mathematical Association of America, 1984. Google Scholar
  13. David Durfee, Yu Gao, Gramoz Goranci, and Richard Peng. Fully Dynamic Effective Resistances. ArXiv e-prints, apr 2018. URL: http://arxiv.org/abs/1804.04038.
  14. David Durfee, Rasmus Kyng, John Peebles, Anup B Rao, and Sushant Sachdeva. Sampling random spanning trees faster than matrix multiplication. In Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing, pages 730-742. ACM, 2017. Google Scholar
  15. David Eppstein, Zvi Galil, Giuseppe F. Italiano, and Thomas H. Spencer. Separator based sparsification. i. planary testing and minimum spanning trees. J. Comput. Syst. Sci., 52(1):3-27, 1996. Google Scholar
  16. Gramoz Goranci, Monika Henzinger, and Pan Peng. The power of vertex sparsifiers in dynamic graph algorithms. In Proc. of the 25th ESA, volume 87, pages 45:1-45:14, 2017. Google Scholar
  17. Gramoz Goranci, Monika Henzinger, and Pan Peng. Dynamic effective resistances and approximate schur complement on separable graphs. CoRR, abs/1802.09111, 2018. URL: http://arxiv.org/abs/1802.09111.
  18. Prahladh Harsha, Thomas P Hayes, Hariharan Narayanan, Harald Räcke, and Jaikumar Radhakrishnan. Minimizing average latency in oblivious routing. In Proceedings of the nineteenth annual ACM-SIAM symposium on Discrete algorithms, pages 200-207. Society for Industrial and Applied Mathematics, 2008. Google Scholar
  19. Monika Henzinger, Sebastian Krinninger, Danupon Nanongkai, and Thatchaphol Saranurak. Unifying and strengthening hardness for dynamic problems via the online matrix-vector multiplication conjecture. In Proc. of the 47th STOC, pages 21-30, 2015. Google Scholar
  20. Jacob Holm, Giuseppe F Italiano, Adam Karczmarz, Jakub Łacki, Eva Rotenberg, and Piotr Sankowski. Contracting a planar graph efficiently. In 25th European Symposium on Algorithms, ESA 2017. Schloss Dagstuhl-Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing, 2017. Google Scholar
  21. Oscar H Ibarra, Shlomo Moran, and Roger Hui. A generalization of the fast lup matrix decomposition algorithm and applications. Journal of Algorithms, 3(1):45-56, 1982. Google Scholar
  22. Giuseppe F Italiano, Adam Karczmarz, Jakub Łącki, and Piotr Sankowski. Decremental single-source reachability in planar digraphs. In Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing, pages 1108-1121. ACM, 2017. Google Scholar
  23. Giuseppe F. Italiano, Yahav Nussbaum, Piotr Sankowski, and Christian Wulff-Nilsen. Improved algorithms for min cut and max flow in undirected planar graphs. In Proc. of the 43rd STOC, pages 313-322, 2011. Google Scholar
  24. Arun Jambulapati and Aaron Sidford. Efficient Õ(n/ε) spectral sketches for the laplacian and its pseudoinverse. In Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 2487-2503. SIAM, 2018. Google Scholar
  25. Adam Karczmarz. Decrementai transitive closure and shortest paths for planar digraphs and beyond. In Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 73-92. SIAM, 2018. Google Scholar
  26. Ken-ichi Kawarabayashi and Bruce Reed. A separator theorem in minor-closed classes. In Proc. of the 51st FOCS, pages 153-162. IEEE, 2010. Google Scholar
  27. Jonathan A Kelner, Gary L Miller, and Richard Peng. Faster approximate multicommodity flow using quadratically coupled flows. In Proceedings of the forty-fourth annual ACM symposium on Theory of computing, pages 1-18. ACM, 2012. Google Scholar
  28. Douglas J Klein and Milan Randić. Resistance distance. Journal of mathematical chemistry, 12(1):81-95, 1993. Google Scholar
  29. Philip N. Klein and Sairam Subramanian. A fully dynamic approximation scheme for shortest paths in planar graphs. Algorithmica, 22(3):235-249, 1998. Google Scholar
  30. Rasmus Kyng, Yin Tat Lee, Richard Peng, Sushant Sachdeva, and Daniel A. Spielman. Sparsified cholesky and multigrid solvers for connection laplacians. In Proc. of the 48th STOC, 2016. Google Scholar
  31. Rasmus Kyng and Sushant Sachdeva. Approximate gaussian elimination for laplacians - fast, sparse, and simple. In Proc. of the 57th FOCS, pages 573-582, 2016. Google Scholar
  32. Huan Li, Stacy Patterson, Yuhao Yi, and Zhongzhi Zhang. Maximizing the Number of Spanning Trees in a Connected Graph. ArXiv e-prints, 2018. URL: http://arxiv.org/abs/1804.02785.
  33. Huan Li and Zhongzhi Zhang. Kirchhoff index as a measure of edge centrality in weighted networks: Nearly linear time algorithms. In Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 2377-2396. SIAM, 2018. Google Scholar
  34. Richard J. Lipton, Donald J. Rose, and Robert Endre Tarjan. Generalized nested dissection. SIAM Journal on Numerical Analysis, 16(2):346-358, 1979. Google Scholar
  35. Richard J Lipton and Robert Endre Tarjan. A separator theorem for planar graphs. SIAM Journal on Applied Mathematics, 36(2):177-189, 1979. Google Scholar
  36. Aleksander Madry. Navigating central path with electrical flows: From flows to matchings, and back. In Proc. of the 54th FOCS, pages 253-262, 2013. Google Scholar
  37. Aleksander Madry. Computing maximum flow with augmenting electrical flows. In Proc. of the 57th FOCS, pages 593-602, 2016. Google Scholar
  38. Aleksander Mądry, Damian Straszak, and Jakub Tarnawski. Fast generation of random spanning trees and the effective resistance metric. In Proceedings of the twenty-sixth annual ACM-SIAM symposium on Discrete algorithms, pages 2019-2036. Society for Industrial and Applied Mathematics, 2015. Google Scholar
  39. Gary L. Miller and Richard Peng. Approximate maximum flow on separable undirected graphs. In Proc. of the 24th SODA, pages 1151-1170, 2013. Google Scholar
  40. Gary L Miller, Shang-Hua Teng, William Thurston, and Stephen A Vavasis. Separators for sphere-packings and nearest neighbor graphs. Journal of the ACM (JACM), 44(1):1-29, 1997. Google Scholar
  41. Cameron Musco, Praneeth Netrapalli, Aaron Sidford, Shashanka Ubaru, and David P. Woodruff. Spectrum Approximation Beyond Fast Matrix Multiplication: Algorithms and Hardness. LIPIcs, 94:8:1-8:21, 2018. Google Scholar
  42. Piotr Sankowski. Dynamic transitive closure via dynamic matrix inverse. In Proceedings of the 45th Annual IEEE Symposium on Foundations of Computer Science, pages 509-517. IEEE Computer Society, 2004. Google Scholar
  43. Aaron Schild. An almost-linear time algorithm for uniform random spanning tree generation. arXiv preprint arXiv:1711.06455, 2017. Google Scholar
  44. Daniel A. Spielman and Nikhil Srivastava. Graph sparsification by effective resistances. SIAM J. Comput., 40(6):1913-1926, 2011. Google Scholar
  45. Sairam Subramanian. A fully dynamic data structure for reachability in planar digraphs. In Proc. of the 1st ESA, pages 372-383, 1993. Google Scholar
  46. Virginia Vassilevska Williams. Multiplying matrices faster than coppersmith-winograd. In Proceedings of the forty-fourth annual ACM symposium on Theory of computing, pages 887-898. ACM, 2012. Google Scholar
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail