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Near-Optimal Directed Low-Diameter Decompositions

Authors Karl Bringmann , Nick Fischer, Bernhard Haeupler , Rustam Latypov

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ACM Subject Classification
  • Theory of computation → Graph algorithms analysis
Keywords
  • Low Diameter Decompositions
  • Expander Decompositions
  • Directed Graphs

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Abstract

Low Diameter Decompositions (LDDs) are invaluable tools in the design of combinatorial graph algorithms. While historically they have been applied mainly to undirected graphs, in the recent breakthrough for the negative-length Single Source Shortest Path problem, Bernstein, Nanongkai, and Wulff-Nilsen [FOCS '22] extended the use of LDDs to directed graphs for the first time. Specifically, their LDD deletes each edge with probability at most O(1/D ⋅ log²n), while ensuring that each strongly connected component in the remaining graph has a (weak) diameter of at most D.
In this work, we make further advancements in the study of directed LDDs. We reveal a natural and intuitive (in hindsight) connection to Expander Decompositions, and leveraging this connection along with additional techniques, we establish the existence of an LDD with an edge-cutting probability of O(1/D ⋅ log n log log n). This improves the previous bound by nearly a logarithmic factor and closely approaches the lower bound of Ω(1/D ⋅ log n). With significantly more technical effort, we also develop two efficient algorithms for computing our LDDs: a deterministic algorithm that runs in time Õ(m poly(D)) and a randomized algorithm that runs in near-linear time Õ(m).
We believe that our work provides a solid conceptual and technical foundation for future research relying on directed LDDs, which will undoubtedly follow soon.

Cite As Get BibTex

Karl Bringmann, Nick Fischer, Bernhard Haeupler, and Rustam Latypov. Near-Optimal Directed Low-Diameter Decompositions. In 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 334, pp. 35:1-35:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.ICALP.2025.35

Author Details

Karl Bringmann
  • Saarland University and Max Planck Institute for Informatics, Saarland Informatics Campus, Saarbücken, Germany
Nick Fischer
  • INSAIT, Sofia University "St. Kliment Ohridski", Bulgaria
Bernhard Haeupler
  • INSAIT, Sofia University "St. Kliment Ohridski", Bulgaria
  • ETH Zürich, Switzerland
Rustam Latypov
  • Aalto University, Finland

Funding

  • Bringmann, Karl: This work is part of the project TIPEA that has received funding from the European Research Council (ERC) under the European Unions Horizon 2020 research and innovation programme (grant agreement No. 850979).
  • Fischer, Nick: Partially funded by the Ministry of Education and Science of Bulgaria’s support for INSAIT as part of the Bulgarian National Roadmap for Research Infrastructure.
  • Haeupler, Bernhard: Partially funded by the Ministry of Education and Science of Bulgaria’s support for INSAIT as part of the Bulgarian National Roadmap for Research Infrastructure and through the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program (ERC grant agreement 949272).
  • Latypov, Rustam: Supported by the Research Council of Finland (grant 334238), The Finnish Foundation for Technology Promotion and The Nokia Foundation

References

  1. Ittai Abraham, Yair Bartal, and Ofer Neiman. Nearly tight low stretch spanning trees. In 49th Annual Symposium on Foundations of Computer Science (FOCS 2008), pages 781-790. IEEE Computer Society, 2008. URL: https://doi.org/10.1109/FOCS.2008.62.
  2. Ittai Abraham, Shiri Chechik, Michael Elkin, Arnold Filtser, and Ofer Neiman. Ramsey spanning trees and their applications. ACM Trans. Algorithms, 16(2):19:1-19:21, 2020. URL: https://doi.org/10.1145/3371039.
  3. Ittai Abraham and Ofer Neiman. Using petal-decompositions to build a low stretch spanning tree. SIAM J. Comput., 48(2):227-248, 2019. URL: https://doi.org/10.1137/17M1115575.
  4. Noga Alon, Richard M. Karp, David Peleg, and Douglas B. West. A graph-theoretic game and its application to the k-server problem. SIAM J. Comput., 24(1):78-100, 1995. URL: https://doi.org/10.1137/S0097539792224474.
  5. Sanjeev Arora, Elad Hazan, and Satyen Kale. The multiplicative weights update method: a meta-algorithm and applications. Theory of Computing, 8(6):121-164, 2012. URL: https://doi.org/10.4086/toc.2012.v008a006.
  6. Vikrant Ashvinkumar, Aaron Bernstein, and Adam Karczmarz. Faster approximation algorithms for restricted shortest paths in directed graphs. In Yossi Azar and Debmalya Panigrahi, editors, 36th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2025), pages 5263-5277. SIAM, 2025. URL: https://doi.org/10.1137/1.9781611978322.180.
  7. Baruch Awerbuch. Complexity of network synchronization. J. ACM, 32(4):804-823, 1985. URL: https://doi.org/10.1145/4221.4227.
  8. Baruch Awerbuch, Bonnie Berger, Lenore Cowen, and David Peleg. Fast network decomposition (extended abstract). In Norman C. Hutchinson, editor, 11th Annual ACM Symposium on Principles of Distributed Computing (PODC 1992), pages 169-177. ACM, 1992. URL: https://doi.org/10.1145/135419.135456.
  9. Baruch Awerbuch, Andrew V. Goldberg, Michael Luby, and Serge A. Plotkin. Network decomposition and locality in distributed computation. In 30th Annual Symposium on Foundations of Computer Science (FOCS 1989), pages 364-369. IEEE Computer Society, 1989. URL: https://doi.org/10.1109/SFCS.1989.63504.
  10. Baruch Awerbuch and David Peleg. Routing with polynomial communication-space trade-off. SIAM J. Discret. Math., 5(2):151-162, 1992. URL: https://doi.org/10.1137/0405013.
  11. Yair Bartal. Probabilistic approximations of metric spaces and its algorithmic applications. In 37th Annual Symposium on Foundations of Computer Science (FOCS 1996), pages 184-193. IEEE Computer Society, 1996. URL: https://doi.org/10.1109/SFCS.1996.548477.
  12. Yair Bartal. On approximating arbitrary metrices by tree metrics. In Jeffrey Scott Vitter, editor, 30th Annual ACM Symposium on Theory of Computing (STOC 1998), pages 161-168. ACM, 1998. URL: https://doi.org/10.1145/276698.276725.
  13. Aaron Bernstein, Joakim Blikstad, Thatchaphol Saranurak, and Ta-Wei Tu. Maximum flow by augmenting paths in n^2+o(1) time. In 65th Annual Symposium on Foundations of Computer Science (FOCS 2024). IEEE, 2024. To appear, see https://arxiv.org/abs/2406.03648. URL: https://doi.org/10.48550/arXiv.2406.03648.
  14. Aaron Bernstein, Maximilian Probst Gutenberg, and Thatchaphol Saranurak. Deterministic decremental reachability, scc, and shortest paths via directed expanders and congestion balancing. In Sandy Irani, editor, 61st Annual Symposium on Foundations of Computer Science (FOCS 2020), pages 1123-1134. IEEE, 2020. URL: https://doi.org/10.1109/FOCS46700.2020.00108.
  15. Aaron Bernstein, Maximilian Probst Gutenberg, and Christian Wulff-Nilsen. Near-optimal decremental SSSP in dense weighted digraphs. In Sandy Irani, editor, 61st Annual Symposium on Foundations of Computer Science (FOCS 2020), pages 1112-1122. IEEE, 2020. URL: https://doi.org/10.1109/FOCS46700.2020.00107.
  16. Aaron Bernstein, Danupon Nanongkai, and Christian Wulff-Nilsen. Negative-weight single-source shortest paths in near-linear time. In 63rd Annual Symposium on Foundations of Computer Science (FOCS 2022), pages 600-611. IEEE, 2022. URL: https://doi.org/10.1109/FOCS54457.2022.00063.
  17. Guy E. Blelloch, Anupam Gupta, Ioannis Koutis, Gary L. Miller, Richard Peng, and Kanat Tangwongsan. Nearly-linear work parallel SDD solvers, low-diameter decomposition, and low-stretch subgraphs. Theory Comput. Syst., 55(3):521-554, 2014. URL: https://doi.org/10.1007/s00224-013-9444-5.
  18. Allan Borodin and Ran El-Yaniv. Online computation and competitive analysis. Cambridge University Press, 1998. Google Scholar
  19. Karl Bringmann, Alejandro Cassis, and Nick Fischer. Negative-weight single-source shortest paths in near-linear time: Now faster! In 64th Annual Symposium on Foundations of Computer Science (FOCS 2023), pages 515-538. IEEE, 2023. URL: https://doi.org/10.1109/FOCS57990.2023.00038.
  20. Shiri Chechik, Yang P. Liu, Omer Rotem, and Aaron Sidford. Constant girth approximation for directed graphs in subquadratic time. In Konstantin Makarychev, Yury Makarychev, Madhur Tulsiani, Gautam Kamath, and Julia Chuzhoy, editors, 52nd Annual ACM Symposium on Theory of Computing (STOC 2020), pages 1010-1023. ACM, 2020. URL: https://doi.org/10.1145/3357713.3384330.
  21. Shiri Chechik and Tianyi Zhang. Dynamic low-stretch spanning trees in subpolynomial time. In Shuchi Chawla, editor, 31st Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2020), pages 463-475. SIAM, 2020. URL: https://doi.org/10.1137/1.9781611975994.28.
  22. Julia Chuzhoy and Sanjeev Khanna. Maximum bipartite matching in n^2+o(1) time via a combinatorial algorithm. In Bojan Mohar, Igor Shinkar, and Ryan O'Donnell, editors, 56th Annual ACM Symposium on Theory of Computing (STOC 2024), pages 83-94. ACM, 2024. URL: https://doi.org/10.1145/3618260.3649725.
  23. Edith Cohen. Size-estimation framework with applications to transitive closure and reachability. J. Comput. Syst. Sci., 55(3):441-453, 1997. URL: https://doi.org/10.1006/JCSS.1997.1534.
  24. Michael Elkin, Yuval Emek, Daniel A. Spielman, and Shang-Hua Teng. Lower-stretch spanning trees. SIAM J. Comput., 38(2):608-628, 2008. URL: https://doi.org/10.1137/050641661.
  25. Jittat Fakcharoenphol, Satish Rao, and Kunal Talwar. A tight bound on approximating arbitrary metrics by tree metrics. J. Comput. Syst. Sci., 69(3):485-497, 2004. URL: https://doi.org/10.1016/J.JCSS.2004.04.011.
  26. Sebastian Forster and Gramoz Goranci. Dynamic low-stretch trees via dynamic low-diameter decompositions. In Moses Charikar and Edith Cohen, editors, 51st Annual ACM Symposium on Theory of Computing (STOC 2019), pages 377-388. ACM, 2019. URL: https://doi.org/10.1145/3313276.3316381.
  27. Sebastian Forster, Martin Grösbacher, and Tijn de Vos. An improved random shift algorithm for spanners and low diameter decompositions. In Quentin Bramas, Vincent Gramoli, and Alessia Milani, editors, 25th International Conference on Principles of Distributed Systems (OPODIS 2021), volume 217 of LIPIcs, pages 16:1-16:17. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2021. URL: https://doi.org/10.4230/LIPIcs.OPODIS.2021.16.
  28. Bernhard Haeupler, D. Ellis Hershkowitz, and Goran Zuzic. Tree embeddings for hop-constrained network design. In Samir Khuller and Virginia Vassilevska Williams, editors, 53rd Annual ACM Symposium on Theory of Computing (STOC 2021), pages 356-369. ACM, 2021. URL: https://doi.org/10.1145/3406325.3451053.
  29. Yiding Hua, Rasmus Kyng, Maximilian Probst Gutenberg, and Zihang Wu. Maintaining expander decompositions via sparse cuts. In Nikhil Bansal and Viswanath Nagarajan, editors, 34th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2023), pages 48-69. SIAM, 2023. URL: https://doi.org/10.1137/1.9781611977554.CH2.
  30. Rohit Khandekar, Satish Rao, and Umesh Vazirani. Graph partitioning using single commodity flows. J. ACM, 56(4), July 2009. URL: https://doi.org/10.1145/1538902.1538903.
  31. Ioannis Koutis, Gary L. Miller, and Richard Peng. A nearly-m log n time solver for SDD linear systems. In Rafail Ostrovsky, editor, 52th Annual Symposium on Foundations of Computer Science (FOCS 2011), pages 590-598. IEEE Computer Society, 2011. URL: https://doi.org/10.1109/FOCS.2011.85.
  32. Nathan Linial and Michael E. Saks. Low diameter graph decompositions. Comb., 13(4):441-454, 1993. URL: https://doi.org/10.1007/BF01303516.
  33. Gary L. Miller, Richard Peng, and Shen Chen Xu. Parallel graph decompositions using random shifts. In Guy E. Blelloch and Berthold Vöcking, editors, 25th ACM Symposium on Parallelism in Algorithms and Architectures (SPAA 2013), pages 196-203. ACM, 2013. URL: https://doi.org/10.1145/2486159.2486180.
  34. Jakub Pachocki, Liam Roditty, Aaron Sidford, Roei Tov, and Virginia Vassilevska Williams. Approximating cycles in directed graphs: Fast algorithms for girth and roundtrip spanners. In Artur Czumaj, editor, 29th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2018), pages 1374-1392. SIAM, 2018. URL: https://doi.org/10.1137/1.9781611975031.91.
  35. Harald Räcke, Chintan Shah, and Hanjo Täubig. Computing cut-based hierarchical decompositions in almost linear time. In Chandra Chekuri, editor, 25th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2014), pages 227-238. SIAM, 2014. URL: https://doi.org/10.1137/1.9781611973402.17.
  36. Paul D. Seymour. Packing directed circuits fractionally. Comb., 15(2):281-288, 1995. URL: https://doi.org/10.1007/BF01200760.
  37. Goran Zuzic, Gramoz Goranci, Mingquan Ye, Bernhard Haeupler, and Xiaorui Sun. Universally-optimal distributed shortest paths and transshipment via graph-based 𝓁₁-oblivious routing. In Joseph (Seffi) Naor and Niv Buchbinder, editors, 33rd Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2022), pages 2549-2579. SIAM, 2022. URL: https://doi.org/10.1137/1.9781611977073.100.
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