Document Open Access Logo

Practical Expander Decomposition

Authors Lars Gottesbüren, Nikos Parotsidis, Maximilian Probst Gutenberg

PDF
Thumbnail PDF

File

LIPIcs.ESA.2024.61.pdf
  • Filesize: 1.14 MB
  • 17 pages

Document Identifiers

Author Details

Lars Gottesbüren
  • Karlsruhe Institute of Technology, Germany
Nikos Parotsidis
  • Google Research, Zürich, Switzerland
Maximilian Probst Gutenberg
  • ETH Zürich, Switzerland

Acknowledgements

We thank Isaac Arvestad for open-sourcing a clean and extensible C++ implementation of the Saranurak-Wang algorithm.

Cite AsGet BibTex

Lars Gottesbüren, Nikos Parotsidis, and Maximilian Probst Gutenberg. Practical Expander Decomposition. In 32nd Annual European Symposium on Algorithms (ESA 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 308, pp. 61:1-61:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.ESA.2024.61

Abstract

The expander decomposition of a graph decomposes the set of vertices into clusters such that the induced subgraph of each cluster is a subgraph with high conductance, and there is only a small number of inter-cluster edges. Expander decompositions are at the forefront of recent theoretical developments in the area of efficient graph algorithms and act as a central component in several state-of-the-art graph algorithms for fundamental problems like maximum flow, min-cost flow, Gomory-Hu trees, global min-cut, and more. Despite this crucial role and the existence of theoretically efficient expander decomposition algorithms, little is known on their behavior in practice. In this paper we explore the engineering design space in implementations for computing expander decompositions. We base our implementation on the near-linear time algorithm of Saranurak and Wang [SODA'19], and enhance it with practical optimizations that accelerate its running time in practice and at the same time preserve the theoretical runtime and approximation guarantees. We evaluate our algorithm on real-world graphs with up to tens of millions of edges. We demonstrate significant speedups of up to two orders of magnitude over the only prior implementation. To the best of our knowledge, our implementation is the first to compute expander decompositions at this scale within reasonable time.

Subject Classification

ACM Subject Classification
  • Theory of computation → Graph algorithms analysis
Keywords
  • Expander Decomposition
  • Clustering
  • Graph Algorithms

Metrics

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

Supplementary Materials

References

  1. Amir Abboud, Robert Krauthgamer, and Ohad Trabelsi. Apmf< apsp? gomory-hu tree for unweighted graphs in almost-quadratic time. In 2021 IEEE 62nd Annual Symposium on Foundations of Computer Science (FOCS), pages 1135-1146. IEEE, 2022. Google Scholar
  2. Amir Abboud, Jason Li, Debmalya Panigrahi, and Thatchaphol Saranurak. All-pairs max-flow is no harder than single-pair max-flow: Gomory-hu trees in almost-linear time. In 2023 IEEE 64th Annual Symposium on Foundations of Computer Science (FOCS), pages 2204-2212. IEEE, 2023. Google Scholar
  3. Daniel Agassy, Dani Dorfman, and Haim Kaplan. Expander decomposition with fewer inter-cluster edges using a spectral cut player. In Kousha Etessami, Uriel Feige, and Gabriele Puppis, editors, 50th International Colloquium on Automata, Languages, and Programming, ICALP 2023, July 10-14, 2023, Paderborn, Germany, volume 261 of LIPIcs, pages 9:1-9:20. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2023. URL: https://doi.org/10.4230/LIPICS.ICALP.2023.9.
  4. Reid Andersen, Fan Chung, and Kevin Lang. Local graph partitioning using pagerank vectors. In 2006 47th Annual IEEE Symposium on Foundations of Computer Science (FOCS'06), pages 475-486. IEEE, 2006. Google Scholar
  5. Alexandr Andoni, Jiecao Chen, Robert Krauthgamer, Bo Qin, David P Woodruff, and Qin Zhang. On sketching quadratic forms. In Proceedings of the 2016 ACM Conference on Innovations in Theoretical Computer Science, pages 311-319, 2016. Google Scholar
  6. Isaac Arvestad. Near-linear Time Expander Decomposition in Practice. Master’s thesis, KTH Royal Institute of Technology, 2022. Google Scholar
  7. Deepayan Chakrabarti, Yiping Zhan, and Christos Faloutsos. R-mat: A recursive model for graph mining. In Proceedings of the 2004 SIAM International Conference on Data Mining, pages 442-446. SIAM, 2004. Google Scholar
  8. Li Chen, Rasmus Kyng, Yang P Liu, Simon Meierhans, and Maximilian Probst Gutenberg. Almost-linear time algorithms for incremental graphs: Cycle detection, sccs, s-t shortest path, and minimum-cost flow. to appear at STOC'24, 2024. Google Scholar
  9. Li Chen, Rasmus Kyng, Yang P Liu, Richard Peng, Maximilian Probst Gutenberg, and Sushant Sachdeva. Maximum flow and minimum-cost flow in almost-linear time. In 2022 IEEE 63rd Annual Symposium on Foundations of Computer Science (FOCS), pages 612-623. IEEE, 2022. Google Scholar
  10. Timothy Chu, Yu Gao, Richard Peng, Sushant Sachdeva, Saurabh Sawlani, and Junxing Wang. Graph sparsification, spectral sketches, and faster resistance computation via short cycle decompositions. SIAM Journal on Computing, pages FOCS18-85, 2020. Google Scholar
  11. Julia Chuzhoy, Yu Gao, Jason Li, Danupon Nanongkai, Richard Peng, and Thatchaphol Saranurak. A deterministic algorithm for balanced cut with applications to dynamic connectivity, flows, and beyond. In 2020 IEEE 61st Annual Symposium on Foundations of Computer Science (FOCS), pages 1158-1167. IEEE, 2020. Google Scholar
  12. Michael B Cohen, Jonathan 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 SIGACT Symposium on Theory of Computing, pages 410-419, 2017. Google Scholar
  13. Timothy A Davis and Yifan Hu. The university of florida sparse matrix collection. ACM Transactions on Mathematical Software (TOMS), 38(1):1-25, 2011. Google Scholar
  14. Yefim Dinitz. Algorithm for Solution of a Problem of Maximum Flow in a Network with Power Estimation. Soviet Mathematics-Doklady, 11(5):1277-1280, September 1970. Google Scholar
  15. Monika Henzinger, Satish Rao, and Di Wang. Local flow partitioning for faster edge connectivity. SIAM Journal on Computing, 49(1):1-36, 2020. Google Scholar
  16. Yiding Hua, Rasmus Kyng, Maximilian Probst Gutenberg, and Zihang Wu. Maintaining expander decompositions via sparse cuts. In Proceedings of the 2023 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 48-69. SIAM, 2023. Google Scholar
  17. 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
  18. Ravi Kannan, Santosh Vempala, and Adrian Vetta. On clusterings: Good, bad and spectral. Journal of the ACM (JACM), 51(3):497-515, 2004. Google Scholar
  19. G. Karypis and V. Kumar. A Fast and High Quality Multilevel Scheme for Partitioning Irregular Graphs. SIAM Journal on Scientific Computing, 20(1):359-392, 1998. Google Scholar
  20. Ken-ichi Kawarabayashi and Mikkel Thorup. Deterministic edge connectivity in near-linear time. Journal of the ACM (JACM), 66(1):1-50, 2018. Google Scholar
  21. Jonathan A Kelner, Yin Tat Lee, Lorenzo Orecchia, and Aaron Sidford. An almost-linear-time algorithm for approximate max flow in undirected graphs, and its multicommodity generalizations. In Proceedings of the twenty-fifth annual ACM-SIAM symposium on Discrete algorithms, pages 217-226. SIAM, 2014. Google Scholar
  22. Rohit Khandekar, Satish Rao, and Umesh Vazirani. Graph partitioning using single commodity flows. Journal of the ACM (JACM), 56(4):1-15, 2009. Google Scholar
  23. Scott P Kolodziej, Mohsen Aznaveh, Matthew Bullock, Jarrett David, Timothy A Davis, Matthew Henderson, Yifan Hu, and Read Sandstrom. The suitesparse matrix collection website interface. Journal of Open Source Software, 4(35):1244, 2019. Google Scholar
  24. Rasmus Kyng, Simon Meierhans, and Maximilian Probst. Derandomizing directed random walks in almost-linear time. In 2022 IEEE 63rd Annual Symposium on Foundations of Computer Science (FOCS), pages 407-418. IEEE, 2022. Google Scholar
  25. Jure Leskovec and Andrej Krevl. SNAP Datasets: Stanford large network dataset collection. http://snap.stanford.edu/data, June 2014.
  26. Jason Li. Deterministic mincut in almost-linear time. In Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing, pages 384-395, 2021. Google Scholar
  27. Jason Li, Danupon Nanongkai, Debmalya Panigrahi, Thatchaphol Saranurak, and Sorrachai Yingchareonthawornchai. Vertex connectivity in poly-logarithmic max-flows. In Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing, pages 317-329, 2021. Google Scholar
  28. Jason Li, Debmalya Panigrahi, and Thatchaphol Saranurak. A nearly optimal all-pairs min-cuts algorithm in simple graphs. In 2021 IEEE 62nd Annual Symposium on Foundations of Computer Science (FOCS), pages 1124-1134. IEEE, 2022. Google Scholar
  29. Danupon Nanongkai and Thatchaphol Saranurak. Dynamic spanning forest with worst-case update time: adaptive, las vegas, and o (n1/2-ε)-time. In Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing, pages 1122-1129, 2017. Google Scholar
  30. Danupon Nanongkai, Thatchaphol Saranurak, and Christian Wulff-Nilsen. Dynamic minimum spanning forest with subpolynomial worst-case update time. In 2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS), pages 950-961. IEEE, 2017. Google Scholar
  31. Harald Räcke, Chintan Shah, and Hanjo Täubig. Computing cut-based hierarchical decompositions in almost linear time. In Proceedings of the twenty-fifth annual ACM-SIAM symposium on Discrete algorithms, pages 227-238. SIAM, 2014. Google Scholar
  32. Thatchaphol Saranurak. A simple deterministic algorithm for edge connectivity. In Symposium on Simplicity in Algorithms (SOSA), pages 80-85. SIAM, 2021. Google Scholar
  33. Thatchaphol Saranurak and Di Wang. Expander Decomposition and Pruning: Faster, Stronger, and Simpler. In SODA 2019, pages 2616-2635. SIAM, 2019. URL: https://doi.org/10.1137/1.9781611975482.162.
  34. Daniel A Spielman and Shang-Hua Teng. 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, pages 81-90, 2004. Google Scholar
  35. Aurelio L Sulser and Maximilian Probst Gutenberg. A simple and near-optimal algorithm for directed expander decompositions. arXiv preprint arXiv:2403.04542, 2024. Google Scholar
  36. Jan Van Den Brand, Li Chen, Richard Peng, Rasmus Kyng, Yang P Liu, Maximilian Probst Gutenberg, Sushant Sachdeva, and Aaron Sidford. A deterministic almost-linear time algorithm for minimum-cost flow. In 2023 IEEE 64th Annual Symposium on Foundations of Computer Science (FOCS), pages 503-514. IEEE, 2023. Google Scholar
  37. Santosh S Vempala. The random projection method, volume 65. American Mathematical Soc., 2005. Google Scholar
  38. Christian Wulff-Nilsen. Fully-dynamic minimum spanning forest with improved worst-case update time. In Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing, pages 1130-1143, 2017. 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
© 2023-2024 Schloss Dagstuhl – LZI GmbH Imprint Privacy Contact