Document Open Access Logo

On Solving Asymmetric Diagonally Dominant Linear Systems in Sublinear Time

Authors Tsz Chiu Kwok , Zhewei Wei , Mingji Yang

PDF

File

Thumbnail PDF
PDF
LIPIcs.ITCS.2026.89.pdf
  • Filesize: 0.98 MB
  • 25 pages

Document Identifiers

Related Versions

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Spectra of graphs
  • Theory of computation → Graph algorithms analysis
  • Theory of computation → Streaming, sublinear and near linear time algorithms
Keywords
  • Spectral Graph Theory
  • Linear Systems
  • Sublinear Algorithms

Metrics

Abstract

We initiate a study of solving a row/column diagonally dominant (RDD/CDD) linear system 𝐌x = b in sublinear time, with the goal of estimating t^{⊤}x^{∗} for a given vector t ∈ ℝⁿ and a specific solution x^{∗}. This setting naturally generalizes the study of sublinear-time solvers for symmetric diagonally dominant (SDD) systems [Andoni-Krauthgamer-Pogrow, ITCS 2019] to the asymmetric case, which has remained underexplored despite extensive work on nearly-linear-time solvers for RDD/CDD systems.
Our first contributions are characterizations of the problem’s mathematical structure. We express a solution x^{∗} via a Neumann series, prove its convergence, and upper bound the truncation error on this series through a novel quantity of 𝐌, termed the maximum p-norm gap. This quantity generalizes the spectral gap of symmetric matrices and captures how the structure of 𝐌 governs the problem’s computational difficulty.
For systems with bounded maximum p-norm gap, we develop a collection of algorithmic results for locally approximating t^{⊤}x^{∗} under various scenarios and error measures. We derive these results by adapting the techniques of random-walk sampling, local push, and their bidirectional combination, which have proved powerful for special cases of solving RDD/CDD systems, particularly estimating PageRank and effective resistance on graphs. Our general framework yields deeper insights, extended results, and improved complexity bounds for these problems. Notably, our perspective provides a unified understanding of Forward Push and Backward Push, two fundamental approaches for estimating random-walk probabilities on graphs.
Our framework also inherits the hardness results for sublinear-time SDD solvers and local PageRank computation, establishing lower bounds on the maximum p-norm gap or the accuracy parameter. We hope that our work opens the door for further study into sublinear solvers, local graph algorithms, and directed spectral graph theory.

Cite As Get BibTex

Tsz Chiu Kwok, Zhewei Wei, and Mingji Yang. On Solving Asymmetric Diagonally Dominant Linear Systems in Sublinear Time. In 17th Innovations in Theoretical Computer Science Conference (ITCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 362, pp. 89:1-89:25, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.ITCS.2026.89

Author Details

Tsz Chiu Kwok
  • Shanghai University of Finance and Economics, China
Zhewei Wei
  • Renmin University of China, Beijing, China
Mingji Yang
  • Renmin University of China, Beijing, China

Funding

This research was supported by National Natural Science Foundation of China (No. 92470128, No. U2241212).

Acknowledgements

We thank the anonymous reviewers for their valuable comments. Mingji Yang wishes to thank Prof. Shang-Hua Teng for inspiring discussions and encouragement, and Guanyu Cui for helpful discussions on effective resistance.

References

  1. Reid Andersen, Christian Borgs, Jennifer T. Chayes, John E. Hopcroft, Vahab S. Mirrokni, and Shang-Hua Teng. Local computation of PageRank contributions. Internet Mathematics, 5(1):23-45, 2008. URL: https://doi.org/10.1080/15427951.2008.10129302.
  2. Reid Andersen, Fan R. K. Chung, and Kevin J. Lang. Using PageRank to locally partition a graph. Internet Mathematics, 4(1):35-64, 2007. URL: https://doi.org/10.1080/15427951.2007.10129139.
  3. Alexandr Andoni, Robert Krauthgamer, and Yosef Pogrow. On solving linear systems in sublinear time. CoRR, abs/1809.02995, 2018. URL: https://doi.org/10.48550/arXiv.1809.02995.
  4. Alexandr Andoni, Robert Krauthgamer, and Yosef Pogrow. On solving linear systems in sublinear time. In Proceedings of the 10th Innovations in Theoretical Computer Science Conference, volume 124, pages 3:1-3:19, 2019. URL: https://doi.org/10.4230/LIPIcs.ITCS.2019.3.
  5. Siddhartha Banerjee and Peter Lofgren. Fast bidirectional probability estimation in Markov models. In Advances in Neural Information Processing Systems 28, pages 1423-1431, 2015. URL: https://proceedings.neurips.cc/paper/2015/hash/ede7e2b6d13a41ddf9f4bdef84fdc737-Abstract.html.
  6. Christian Bertram, Mads Vestergaard Jensen, Mikkel Thorup, Hanzhi Wang, and Shuyi Yan. Estimating random-walk probabilities in directed graphs. CoRR, abs/2504.16481, 2025. URL: https://doi.org/10.48550/arXiv.2504.16481.
  7. Enrico Bozzo. The Moore-Penrose inverse of the normalized graph Laplacian. Linear Algebra and its Applications, 439(10):3038-3043, 2013. URL: https://doi.org/10.1016/j.laa.2013.08.039.
  8. Marco Bressan, Enoch Peserico, and Luca Pretto. Sublinear algorithms for local graph-centrality estimation. SIAM Journal on Computing, 52(4):968-1008, 2023. URL: https://doi.org/10.1137/19M1266976.
  9. Sergey Brin and Lawrence Page. The anatomy of a large-scale hypertextual web search engine. Computer Networks, 30(1-7):107-117, 1998. URL: https://doi.org/10.1016/S0169-7552(98)00110-X.
  10. Dongrun Cai, Xue Chen, and Pan Peng. Effective resistances in non-expander graphs. In Proceedings of the 31st Annual European Symposium on Algorithms, volume 274, pages 29:1-29:18, 2023. URL: https://doi.org/10.4230/LIPIcs.ESA.2023.29.
  11. Michael B. Cohen, Jonathan A. Kelner, Rasmus Kyng, John Peebles, Richard Peng, Anup B. Rao, and Aaron Sidford. Solving directed Laplacian systems in nearly-linear time through sparse LU factorizations. In Proceedings of the 59th IEEE Symposium on Foundations of Computer Science, pages 898-909, 2018. URL: https://doi.org/10.1109/FOCS.2018.00089.
  12. Michael B. Cohen, Jonathan A. Kelner, John Peebles, Richard Peng, Anup B. Rao, Aaron Sidford, and Adrian Vladu. Almost-linear-time algorithms for Markov chains and new spectral primitives for directed graphs. In Proceedings of the 49th Annual ACM Symposium on Theory of Computing, pages 410-419, 2017. URL: https://doi.org/10.1145/3055399.3055463.
  13. Michael B. Cohen, Jonathan A. Kelner, John Peebles, Richard Peng, Aaron Sidford, and Adrian Vladu. Faster algorithms for computing the stationary distribution, simulating random walks, and more. In Proceedings of the 57th IEEE Symposium on Foundations of Computer Science, pages 583-592, 2016. URL: https://doi.org/10.1109/FOCS.2016.69.
  14. Guanyu Cui, Hanzhi Wang, and Zhewei Wei. Mixing time matters: Accelerating effective resistance estimation via bidirectional method. In Proceedings of the 31st ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, pages 177-188, 2025. URL: https://doi.org/10.1145/3690624.3709298.
  15. Paul Dagum, Richard M. Karp, Michael Luby, and Sheldon M. Ross. An optimal algorithm for Monte Carlo estimation. SIAM Journal on Computing, 29(5):1484-1496, 2000. URL: https://doi.org/10.1137/S0097539797315306.
  16. Dean Doron, François Le Gall, and Amnon Ta-Shma. Probabilistic logarithmic-space algorithms for Laplacian solvers. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, volume 81, pages 41:1-41:20, 2017. URL: https://doi.org/10.4230/LIPIcs.APPROX-RANDOM.2017.41.
  17. Peter G. Doyle and J. Laurie Snell. Random walks and electric networks, volume 22. The Mathematical Association of America, 1984. Google Scholar
  18. Rajat Vadiraj Dwaraknath, Ishani Karmarkar, and Aaron Sidford. Towards optimal effective resistance estimation. In Advances in Neural Information Processing Systems 36, 2023. URL: http://papers.nips.cc/paper_files/paper/2023/hash/b8e2046160a568145af6d42eeef199f4-Abstract-Conference.html.
  19. Dániel Fogaras, Balázs Rácz, Károly Csalogány, and Tamás Sarlós. Towards scaling fully Personalized PageRank: Algorithms, lower bounds, and experiments. Internet Mathematics, 2(3):333-358, 2005. URL: https://doi.org/10.1080/15427951.2005.10129104.
  20. George E. Forsythe and Richard A. Leibler. Matrix inversion by a Monte Carlo method. Mathematics of Computation, 4(31):127-129, 1950. URL: https://www.ams.org/journals/mcom/1950-04-031/S0025-5718-1950-0038138-X/home.html.
  21. David F. Gleich. Pagerank beyond the web. SIAM Review, 57(3):321-363, 2015. URL: https://doi.org/10.1137/140976649.
  22. Oded Goldreich and Dana Ron. Property testing in bounded degree graphs. Algorithmica, 32(2):302-343, 2002. URL: https://doi.org/10.1007/S00453-001-0078-7.
  23. Oded Goldreich and Dana Ron. On testing expansion in bounded-degree graphs. In Studies in Complexity and Cryptography, volume 6650, pages 68-75. Springer, 2011. URL: https://doi.org/10.1007/978-3-642-22670-0_9.
  24. Aram W. Harrow, Avinatan Hassidim, and Seth Lloyd. Quantum algorithm for linear systems of equations. Physical Review Letters, 103(15):150502, 2009. URL: https://doi.org/10.1103/PhysRevLett.103.150502.
  25. Arun Jambulapati, Sushant Sachdeva, Aaron Sidford, Kevin Tian, and Yibin Zhao. Eulerian graph sparsification by effective resistance decomposition. In Proceedings of the 2025 ACM-SIAM Symposium on Discrete Algorithms, pages 1607-1650, 2025. URL: https://doi.org/10.1137/1.9781611978322.50.
  26. Arun Jambulapati and Aaron Sidford. Ultrasparse ultrasparsifiers and faster Laplacian system solvers. In Proceedings of the 2021 ACM-SIAM Symposium on Discrete Algorithms, pages 540-559, 2021. URL: https://doi.org/10.1137/1.9781611976465.33.
  27. Satyen Kale, Yuval Peres, and C. Seshadhri. Noise tolerance of expanders and sublinear expansion reconstruction. SIAM Journal on Computing, 42(1):305-323, 2013. URL: https://doi.org/10.1137/110837863.
  28. Tsz Chiu Kwok, Zhewei Wei, and Mingji Yang. On solving asymmetric diagonally dominant linear systems in sublinear time. CoRR, abs/2509.13891, 2025. URL: https://doi.org/10.48550/arXiv.2509.13891.
  29. Amy Nicole Langville and Carl Dean Meyer. Survey: Deeper inside PageRank. Internet Mathematics, 1(3):335-380, 2003. URL: https://doi.org/10.1080/15427951.2004.10129091.
  30. Lawrence Li and Sushant Sachdeva. A new approach to estimating effective resistances and counting spanning trees in expander graphs. In Proceedings of the 2023 ACM-SIAM Symposium on Discrete Algorithms, pages 2728-2745, 2023. URL: https://doi.org/10.1137/1.9781611977554.ch102.
  31. Peter Lofgren, Siddhartha Banerjee, and Ashish Goel. Bidirectional PageRank estimation: from average-case to worst-case. In Proceedings of the 12th International Workshop on Algorithms and Models for the Web-Graph, volume 9479, pages 164-176, 2015. URL: https://doi.org/10.1007/978-3-319-26784-5_13.
  32. Peter Lofgren, Siddhartha Banerjee, and Ashish Goel. Personalized PageRank estimation and search: a bidirectional approach. In Proceedings of the 9th ACM International Conference on Web Search and Data Mining, pages 163-172, 2016. URL: https://doi.org/10.1145/2835776.2835823.
  33. Peter Lofgren, Siddhartha Banerjee, Ashish Goel, and Seshadhri Comandur. FAST-PPR: Scaling Personalized PageRank estimation for large graphs. In Proceedings of the 20th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, pages 1436-1445, 2014. URL: https://doi.org/10.1145/2623330.2623745.
  34. László Lovász. Random walks on graphs: a survey. Combinatorics, Paul Erdös is Eighty, 2(1-46):4, 1993. Google Scholar
  35. Pan Peng, Daniel Lopatta, Yuichi Yoshida, and Gramoz Goranci. Local algorithms for estimating effective resistance. In Proceedings of the 27th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, pages 1329-1338, 2021. URL: https://doi.org/10.1145/3447548.3467361.
  36. Ronitt Rubinfeld, Gil Tamir, Shai Vardi, and Ning Xie. Fast local computation algorithms. In Proceedings of Innovations in Computer Science, pages 223-238, 2011. URL: http://conference.iiis.tsinghua.edu.cn/ICS2011/content/papers/36.html.
  37. Nitin Shyamkumar, Siddhartha Banerjee, and Peter Lofgren. Sublinear estimation of a single element in sparse linear systems. In Proceedings of the 54th Annual Allerton Conference on Communication, Control, and Computing, pages 856-860, 2016. URL: https://doi.org/10.1109/ALLERTON.2016.7852323.
  38. Daniel A. Spielman and Nikhil Srivastava. Graph sparsification by effective resistances. SIAM Journal on Computing, 40(6):1913-1926, 2011. URL: https://doi.org/10.1137/080734029.
  39. Daniel A. Spielman and Shang-Hua Teng. Nearly-linear time algorithms for graph partitioning, graph sparsification, and solving linear systems. In Proceedings of the 36th Annual ACM Symposium on Theory of Computing, pages 81-90, 2004. URL: https://doi.org/10.1145/1007352.1007372.
  40. Daniel A. Spielman and Shang-Hua Teng. Nearly linear time algorithms for preconditioning and solving symmetric, diagonally dominant linear systems. SIAM Journal on Matrix Analysis and Applications, 35(3):835-885, 2014. URL: https://doi.org/10.1137/090771430.
  41. Amnon Ta-Shma. Inverting well conditioned matrices in quantum logspace. In Proceedings of the 45th Annual ACM Symposium on Theory of Computing, pages 881-890, 2013. URL: https://doi.org/10.1145/2488608.2488720.
  42. Shang-Hua Teng. The Laplacian Paradigm: Emerging algorithms for massive graphs. In Proceedings of the 7th Annual Conference on Theory and Applications of Models of Computation, volume 6108, pages 2-14, 2010. URL: https://doi.org/10.1007/978-3-642-13562-0_2.
  43. Mikkel Thorup, Hanzhi Wang, Zhewei Wei, and Mingji Yang. PageRank centrality in directed graphs with bounded in-degree. In Proceedings of the 2026 ACM-SIAM Symposium on Discrete Algorithms, 2026. To appear. arXiv preprint at https://arxiv.org/abs/2508.01257. URL: https://doi.org/10.48550/arXiv.2508.01257.
  44. Nisheeth K. Vishnoi. L𝐱 = 𝐛. Foundations and Trends in Theoretical Computer Science, 8(1-2):1-141, 2013. URL: https://doi.org/10.1561/0400000054.
  45. Hanzhi Wang. Revisiting local PageRank estimation on undirected graphs: Simple and optimal. In Proceedings of the 30th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, pages 3036-3044, 2024. URL: https://doi.org/10.1145/3637528.3671820.
  46. Hanzhi Wang and Zhewei Wei. Estimating single-node PageRank in Õ(min{d_t,√m}) time. Proceedings of the VLDB Endowment, 16(11):2949-2961, 2023. URL: https://doi.org/10.14778/3611479.3611500.
  47. Hanzhi Wang, Zhewei Wei, Ji-Rong Wen, and Mingji Yang. Revisiting local computation of PageRank: Simple and optimal. In Proceedings of the 56th Annual ACM Symposium on Theory of Computing, pages 911-922, 2024. URL: https://doi.org/10.1145/3618260.3649661.
  48. Wolfgang R. Wasow. A note on the inversion of matrices by random walks. Mathematical Tables and Other Aids to Computation, 6(38):78-81, 1952. URL: https://doi.org/10.2307/2002546.
  49. Zhewei Wei, Ji-Rong Wen, and Mingji Yang. Approximating single-source Personalized PageRank with absolute error guarantees. In Proceedings of the 27th International Conference on Database Theory, volume 290, pages 9:1-9:19, 2024. URL: https://doi.org/10.4230/LIPIcs.ICDT.2024.9.
  50. Mingji Yang, Hanzhi Wang, Zhewei Wei, Sibo Wang, and Ji-Rong Wen. Efficient algorithms for Personalized PageRank computation: a survey. IEEE Transactions on Knowledge and Data Engineering, 36(9):4582-4602, 2024. URL: https://doi.org/10.1109/TKDE.2024.3376000.
  51. Renchi Yang and Jing Tang. Efficient estimation of pairwise effective resistance. Proceedings of the ACM International Conference on Management of Data, 1(1):16:1-16:27, 2023. URL: https://doi.org/10.1145/3588696.
  52. Yichun Yang, Rong-Hua Li, Meihao Liao, and Guoren Wang. Improved algorithms for effective resistance computation on graphs. In Proceedings of the 38th Conference on Learning Theory, volume 291, pages 5892-5920, 2025. URL: https://proceedings.mlr.press/v291/yichun25a.html.
Any Issues?
X

Feedback on the Current Page

CAPTCHA

Thanks for your feedback!

Feedback submitted to Dagstuhl Publishing

Could not send message

Please try again later or send an E-mail
© 2023-2025 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact