Distributed and Parallel Low-Diameter Decompositions for Arbitrary and Restricted Graphs

Authors Jinfeng Dou , Thorsten Götte , Henning Hillebrandt , Christian Scheideler , Julian Werthmann



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Jinfeng Dou
  • Paderborn University, Germany
Thorsten Götte
  • University of Hamburg, Germany
Henning Hillebrandt
  • Paderborn University, Germany
Christian Scheideler
  • Paderborn University, Germany
Julian Werthmann
  • Paderborn University, Germany

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Jinfeng Dou, Thorsten Götte, Henning Hillebrandt, Christian Scheideler, and Julian Werthmann. Distributed and Parallel Low-Diameter Decompositions for Arbitrary and Restricted Graphs. In 16th Innovations in Theoretical Computer Science Conference (ITCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 325, pp. 45:1-45:26, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.ITCS.2025.45

Abstract

We consider the distributed and parallel construction of low-diameter decompositions with strong diameter. We present algorithms for arbitrary undirected, weighted graphs and also for undirected, weighted graphs that can be separated through k ∈ Õ(1) shortest paths. This class of graphs includes planar graphs, graphs of bounded treewidth, and graphs that exclude a fixed minor K_r. Our algorithms work in the PRAM, CONGEST, and the novel HYBRID communication model and are competitive in all relevant parameters. Given 𝒟 > 0, our low-diameter decomposition algorithm divides the graph into connected clusters of strong diameter 𝒟. For an arbitrary graph, an edge e ∈ E of length 𝓁_e is cut between two clusters with probability O(𝓁_e⋅log(n)/𝒟). If the graph can be separated by k ∈ Õ(1) paths, the probability improves to O(𝓁_e⋅log(log n)/𝒟). In either case, the decompositions can be computed in Õ(1) depth and Õ(m) work in the PRAM and Õ(1) time in the HYBRID model. In CONGEST, the runtimes are Õ(HD + √n) and Õ(HD) respectively. All these results hold w.h.p.
Broadly speaking, we present distributed and parallel implementations of sequential divide-and-conquer algorithms where we replace exact shortest paths with approximate shortest paths. In contrast to exact paths, these can be efficiently computed in the distributed and parallel setting [STOC '22]. Further, and perhaps more importantly, we show that instead of explicitly computing vertex-separators to enable efficient parallelization of these algorithms, it suffices to sample a few random paths of bounded length and the nodes close to them. Thereby, we do not require complex embeddings whose implementation is unknown in the distributed and parallel setting.

Subject Classification

ACM Subject Classification
  • Theory of computation → Distributed algorithms
  • Theory of computation → Parallel algorithms
  • Mathematics of computing → Graph algorithms
Keywords
  • Distributed Graph Algorithms
  • Network Decomposition
  • Excluded Minor

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References

  1. Ittai Abraham, Yair Bartal, and Ofer Neiman. Nearly tight low stretch spanning trees. In 49th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2008, October 25-28, 2008, Philadelphia, PA, USA, pages 781-790. IEEE Computer Society, 2008. URL: https://doi.org/10.1109/FOCS.2008.62.
  2. Ittai Abraham and Cyril Gavoille. Object location using path separators. In Eric Ruppert and Dahlia Malkhi, editors, Proceedings of the Twenty-Fifth Annual ACM Symposium on Principles of Distributed Computing, PODC 2006, Denver, CO, USA, July 23-26, 2006, pages 188-197. ACM, 2006. URL: https://doi.org/10.1145/1146381.1146411.
  3. Ittai Abraham, Cyril Gavoille, Anupam Gupta, Ofer Neiman, and Kunal Talwar. Cops, robbers, and threatening skeletons: padded decomposition for minor-free graphs. In David B. Shmoys, editor, Symposium on Theory of Computing, STOC 2014, New York, NY, USA, May 31 - June 03, 2014, pages 79-88. ACM, 2014. URL: https://doi.org/10.1145/2591796.2591849.
  4. Ittai Abraham, Cyril Gavoille, Anupam Gupta, Ofer Neiman, and Kunal Talwar. Cops, robbers, and threatening skeletons: Padded decomposition for minor-free graphs. SIAM J. Comput., 48(3):1120-1145, 2019. URL: https://doi.org/10.1137/17M1112406.
  5. 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.
  6. Ioannis Anagnostides, Christoph Lenzen, Bernhard Haeupler, Goran Zuzic, and Themis Gouleakis. Almost universally optimal distributed laplacian solvers via low-congestion shortcuts. Distributed Comput., 36(4):475-499, 2023. URL: https://doi.org/10.1007/S00446-023-00454-0.
  7. John Augustine, Kristian Hinnenthal, Fabian Kuhn, Christian Scheideler, and Philipp Schneider. Shortest paths in a hybrid network model. In Proceedings of the 2020 ACM-SIAM Symposium on Discrete Algorithms, SODA 2020, Salt Lake City, UT, USA, January 5-8, 2020, pages 1280-1299. SIAM, 2020. URL: https://doi.org/10.1137/1.9781611975994.78.
  8. Yair Bartal. Probabilistic approximations of metric spaces and its algorithmic applications. In 37th Annual Symposium on Foundations of Computer Science, FOCS '96, Burlington, Vermont, USA, 14-16 October, 1996, pages 184-193. IEEE Computer Society, 1996. URL: https://doi.org/10.1109/SFCS.1996.548477.
  9. Yair Bartal. On approximating arbitrary metrices by tree metrics. In Jeffrey Scott Vitter, editor, Proceedings of the Thirtieth Annual ACM Symposium on the Theory of Computing, Dallas, Texas, USA, May 23-26, 1998, pages 161-168. ACM, 1998. URL: https://doi.org/10.1145/276698.276725.
  10. Yair Bartal. Graph decomposition lemmas and their role in metric embedding methods. In Susanne Albers and Tomasz Radzik, editors, Algorithms - ESA 2004, 12th Annual European Symposium, Bergen, Norway, September 14-17, 2004, Proceedings, volume 3221 of Lecture Notes in Computer Science, pages 89-97. Springer, 2004. URL: https://doi.org/10.1007/978-3-540-30140-0_10.
  11. Surender Baswana and Sandeep Sen. A simple and linear time randomized algorithm for computing sparse spanners in weighted graphs. Random Structures & Algorithms, 30(4):532-563, 2007. URL: https://doi.org/10.1002/RSA.20130.
  12. Ruben Becker, Yuval Emek, Mohsen Ghaffari, and Christoph Lenzen. Distributed algorithms for low stretch spanning trees. In Jukka Suomela, editor, 33rd International Symposium on Distributed Computing, DISC 2019, October 14-18, 2019, Budapest, Hungary, volume 146 of LIPIcs, pages 4:1-4:14. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2019. URL: https://doi.org/10.4230/LIPIcs.DISC.2019.4.
  13. Ruben Becker, Yuval Emek, and Christoph Lenzen. Low Diameter Graph Decompositions by Approximate Distance Computation. In Thomas Vidick, editor, 11th Innovations in Theoretical Computer Science Conference (ITCS 2020), volume 151 of Leibniz International Proceedings in Informatics (LIPIcs), pages 50:1-50:29, Dagstuhl, Germany, 2020. Schloss Dagstuhl - Leibniz-Zentrum für Informatik. URL: https://doi.org/10.4230/LIPIcs.ITCS.2020.50.
  14. Guy E. Blelloch, Anupam Gupta, Ioannis Koutis, Gary L. Miller, Richard Peng, and Kanat Tangwongsan. Near linear-work parallel sdd solvers, low-diameter decomposition, and low-stretch subgraphs. In SPAA 2011: Proceedings of the 23rd Annual ACM Symposium on Parallelism in Algorithms and Architectures, San Jose, CA, USA, June 4-6, 2011 (Co-located with FCRC 2011), SPAA '11, pages 13-22, New York, NY, USA, 2011. Association for Computing Machinery. URL: https://doi.org/10.1145/1989493.1989496.
  15. Jannik Castenow, Christina Kolb, and Christian Scheideler. A bounding box overlay for competitive routing in hybrid communication networks. In Structural Information and Communication Complexity - 26th International Colloquium, SIROCCO 2019, L'Aquila, Italy, July 1-4, 2019, Proceedings, pages 345-348, 2019. URL: https://doi.org/10.1007/978-3-030-24922-9_26.
  16. Yi-Jun Chang. Efficient distributed decomposition and routing algorithms in minor-free networks and their applications. In Rotem Oshman, Alexandre Nolin, Magnús M. Halldórsson, and Alkida Balliu, editors, Proceedings of the 2023 ACM Symposium on Principles of Distributed Computing, PODC 2023, Orlando, FL, USA, June 19-23, 2023, pages 55-66. ACM, 2023. URL: https://doi.org/10.1145/3583668.3594604.
  17. Yi-Jun Chang and Hsin-Hao Su. Narrowing the LOCAL-CONGEST gaps in sparse networks via expander decompositions. In Alessia Milani and Philipp Woelfel, editors, PODC '22: ACM Symposium on Principles of Distributed Computing, Salerno, Italy, July 25 - 29, 2022, pages 301-312. ACM, 2022. URL: https://doi.org/10.1145/3519270.3538423.
  18. Sam Coy, Artur Czumaj, Christian Scheideler, Philipp Schneider, and Julian Werthmann. Routing schemes for hybrid communication networks. In Sergio Rajsbaum, Alkida Balliu, Joshua J. Daymude, and Dennis Olivetti, editors, Structural Information and Communication Complexity - 30th International Colloquium, SIROCCO 2023, Alcalá de Henares, Spain, June 6-9, 2023, Proceedings, volume 13892 of Lecture Notes in Computer Science, pages 317-338. Springer, 2023. URL: https://doi.org/10.1007/978-3-031-32733-9_14.
  19. Andrzej Czygrinow, Michal Hanckowiak, and Wojciech Wawrzyniak. Fast distributed approximations in planar graphs. In Gadi Taubenfeld, editor, Distributed Computing, 22nd International Symposium, DISC 2008, Arcachon, France, September 22-24, 2008. Proceedings, volume 5218 of Lecture Notes in Computer Science, pages 78-92. Springer, 2008. URL: https://doi.org/10.1007/978-3-540-87779-0_6.
  20. Tijn de Vos. Minimum cost flow in the CONGEST model. In Sergio Rajsbaum, Alkida Balliu, Joshua J. Daymude, and Dennis Olivetti, editors, Structural Information and Communication Complexity - 30th International Colloquium, SIROCCO 2023, Alcalá de Henares, Spain, June 6-9, 2023, Proceedings, volume 13892 of Lecture Notes in Computer Science, pages 406-426. Springer, 2023. URL: https://doi.org/10.1007/978-3-031-32733-9_18.
  21. Reinhard Diestel. Graph Theory, volume 173 of Graduate Texts in Mathematics. Springer, Heidelberg; New York, fourth edition, 2010. Google Scholar
  22. Emilie Diot and Cyril Gavoille. Path separability of graphs. In Der-Tsai Lee, Danny Z. Chen, and Shi Ying, editors, Frontiers in Algorithmics, 4th International Workshop, FAW 2010, Wuhan, China, August 11-13, 2010. Proceedings, volume 6213 of Lecture Notes in Computer Science, pages 262-273. Springer, 2010. URL: https://doi.org/10.1007/978-3-642-14553-7_25.
  23. Jinfeng Dou, Thorsten Götte, Henning Hillebrandt, Christian Scheideler, and Julian Werthmann. Brief announcement: Distributed construction of near-optimal compact routing schemes for planar graphs. In Rotem Oshman, Alexandre Nolin, Magnús M. Halldórsson, and Alkida Balliu, editors, Proceedings of the 2023 ACM Symposium on Principles of Distributed Computing, PODC 2023, Orlando, FL, USA, June 19-23, 2023, pages 67-70. ACM, 2023. URL: https://doi.org/10.1145/3583668.3594561.
  24. Jinfeng Dou, Thorsten Götte, Henning Hillebrandt, Christian Scheideler, and Julian Werthmann. Distributed and parallel low-diameter decompositions for arbitrary and restricted graphs, 2024. URL: https://arxiv.org/abs/2411.19859.
  25. 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.
  26. Michael Elkin, Arnold Filtser, and Ofer Neiman. Distributed construction of light networks. In PODC '20: ACM Symposium on Principles of Distributed Computing, Virtual Event, Italy, August 3-7, 2020, pages 483-492. ACM, 2020. URL: https://doi.org/10.1145/3382734.3405701.
  27. Michael Elkin and Ofer Neiman. On efficient distributed construction of near optimal routing schemes: Extended abstract. In George Giakkoupis, editor, Proceedings of the 2016 ACM Symposium on Principles of Distributed Computing, PODC 2016, Chicago, IL, USA, July 25-28, 2016, pages 235-244. ACM, 2016. URL: https://doi.org/10.1145/2933057.2933098.
  28. Michael Elkin and Ofer Neiman. Near-optimal distributed routing with low memory. In Proceedings of the 2018 ACM Symposium on Principles of Distributed Computing, PODC '18, pages 207-216, New York, NY, USA, 2018. Association for Computing Machinery. URL: https://doi.org/10.1145/3212734.3212761.
  29. Michael Elkin and Ofer Neiman. Distributed strong diameter network decomposition. Theor. Comput. Sci., 922:150-157, 2022. URL: https://doi.org/10.1016/J.TCS.2022.04.019.
  30. Michael Elkin, Ofer Neiman, and Shay Solomon. Light spanners. SIAM J. Discret. Math., 29(3):1312-1321, 2015. URL: https://doi.org/10.1137/140979538.
  31. 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.
  32. Arnold Filtser. On Strong Diameter Padded Decompositions. In Dimitris Achlioptas and László A. Végh, editors, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019), volume 145 of Leibniz International Proceedings in Informatics (LIPIcs), pages 6:1-6:21, Dagstuhl, Germany, 2019. Schloss Dagstuhl - Leibniz-Zentrum für Informatik. URL: https://doi.org/10.4230/LIPIcs.APPROX-RANDOM.2019.6.
  33. Arnold Filtser and Ofer Neiman. Light spanners for high dimensional norms via stochastic decompositions. Algorithmica, 84(10):2987-3007, 2022. URL: https://doi.org/10.1007/S00453-022-00994-0.
  34. Mohsen Ghaffari, Christoph Grunau, Bernhard Haeupler, Saeed Ilchi, and Václav Rozhon. Improved distributed network decomposition, hitting sets, and spanners, via derandomization. In Nikhil Bansal and Viswanath Nagarajan, editors, Proceedings of the 2023 ACM-SIAM Symposium on Discrete Algorithms, SODA 2023, Florence, Italy, January 22-25, 2023, pages 2532-2566. SIAM, 2023. URL: https://doi.org/10.1137/1.9781611977554.CH97.
  35. Mohsen Ghaffari and Bernhard Haeupler. Distributed algorithms for planar networks I: planar embedding. In George Giakkoupis, editor, Proceedings of the 2016 ACM Symposium on Principles of Distributed Computing, PODC 2016, Chicago, IL, USA, July 25-28, 2016, pages 29-38. ACM, 2016. URL: https://doi.org/10.1145/2933057.2933109.
  36. Mohsen Ghaffari and Bernhard Haeupler. Distributed algorithms for planar networks II: low-congestion shortcuts, mst, and min-cut. In Robert Krauthgamer, editor, Proceedings of the Twenty-Seventh Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2016, Arlington, VA, USA, January 10-12, 2016, pages 202-219. SIAM, 2016. URL: https://doi.org/10.1137/1.9781611974331.ch16.
  37. Mohsen Ghaffari and Bernhard Haeupler. Low-congestion shortcuts for graphs excluding dense minors. In Avery Miller, Keren Censor-Hillel, and Janne H. Korhonen, editors, PODC '21: ACM Symposium on Principles of Distributed Computing, Virtual Event, Italy, July 26-30, 2021, pages 213-221. ACM, 2021. URL: https://doi.org/10.1145/3465084.3467935.
  38. Mohsen Ghaffari, Andreas Karrenbauer, Fabian Kuhn, Christoph Lenzen, and Boaz Patt-Shamir. Near-optimal distributed maximum flow. SIAM J. Comput., 47(6):2078-2117, 2018. URL: https://doi.org/10.1137/17M113277X.
  39. Mohsen Ghaffari and Merav Parter. Near-optimal distributed DFS in planar graphs. In Andréa W. Richa, editor, 31st International Symposium on Distributed Computing, DISC 2017, October 16-20, 2017, Vienna, Austria, volume 91 of LIPIcs, pages 21:1-21:16. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2017. URL: https://doi.org/10.4230/LIPIcs.DISC.2017.21.
  40. Mohsen Ghaffari and Goran Zuzic. Universally-Optimal Distributed Exact Min-Cut. In Proceedings of the 2022 ACM Symposium on Principles of Distributed Computing, pages 281-291, Salerno Italy, July 2022. ACM. URL: https://doi.org/10.1145/3519270.3538429.
  41. Bernhard Haeupler, David Wajc, and Goran Zuzic. Universally-optimal distributed algorithms for known topologies. In Samir Khuller and Virginia Vassilevska Williams, editors, STOC '21: 53rd Annual ACM SIGACT Symposium on Theory of Computing, Virtual Event, Italy, June 21-25, 2021, pages 1166-1179. ACM, 2021. URL: https://doi.org/10.1145/3406325.3451081.
  42. Taisuke Izumi, Naoki Kitamura, Takamasa Naruse, and Gregory Schwartzman. Fully polynomial-time distributed computation in low-treewidth graphs. In Kunal Agrawal and I-Ting Angelina Lee, editors, SPAA '22: 34th ACM Symposium on Parallelism in Algorithms and Architectures, Philadelphia, PA, USA, July 11 - 14, 2022, pages 11-22. ACM, 2022. URL: https://doi.org/10.1145/3490148.3538590.
  43. Fabian Kuhn and Philipp Schneider. Routing Schemes and Distance Oracles in the Hybrid Model. In Christian Scheideler, editor, 36th International Symposium on Distributed Computing (DISC 2022), volume 246 of Leibniz International Proceedings in Informatics (LIPIcs), pages 28:1-28:22, Dagstuhl, Germany, 2022. Schloss Dagstuhl - Leibniz-Zentrum für Informatik. URL: https://doi.org/10.4230/LIPIcs.DISC.2022.28.
  44. Christoph Lenzen and Boaz Patt-Shamir. Fast routing table construction using small messages: extended abstract. In Symposium on Theory of Computing Conference, STOC'13, Palo Alto, CA, USA, June 1-4, 2013, pages 381-390, 2013. URL: https://doi.org/10.1145/2488608.2488656.
  45. Christoph Lenzen and Boaz Patt-Shamir. Fast partial distance estimation and applications. In Chryssis Georgiou and Paul G. Spirakis, editors, Proceedings of the 2015 ACM Symposium on Principles of Distributed Computing, PODC 2015, Donostia-San Sebastián, Spain, July 21 - 23, 2015, pages 153-162. ACM, 2015. URL: https://doi.org/10.1145/2767386.2767398.
  46. Christoph Lenzen, Boaz Patt-Shamir, and David Peleg. Distributed distance computation and routing with small messages. Distributed Comput., 32(2):133-157, 2019. URL: https://doi.org/10.1007/s00446-018-0326-6.
  47. Reut Levi, Moti Medina, and Dana Ron. Property testing of planarity in the CONGEST model. Distributed Comput., 34(1):15-32, 2021. URL: https://doi.org/10.1007/S00446-020-00382-3.
  48. Jason Li and Merav Parter. Planar diameter via metric compression. In Moses Charikar and Edith Cohen, editors, Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing, STOC 2019, Phoenix, AZ, USA, June 23-26, 2019, pages 152-163. ACM, 2019. URL: https://doi.org/10.1145/3313276.3316358.
  49. Daniel Lokshtanov, Saket Saurabh, and Meirav Zehavi. Efficient Graph Minors Theory and Parameterized Algorithms for (Planar) Disjoint Paths, pages 112-128. Springer International Publishing, Cham, 2020. URL: https://doi.org/10.1007/978-3-030-42071-0_9.
  50. Gary L. Miller, Richard Peng, Adrian Vladu, and Shen Chen Xu. Improved parallel algorithms for spanners and hopsets. In Proc. of the 27th ACM symposium on Parallelism in Algorithms and Architectures (SPAA), pages 192-201, 2015. URL: https://doi.org/10.1145/2755573.2755574.
  51. 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 '13, Montreal, QC, Canada - July 23 - 25, 2013, pages 196-203. ACM, 2013. URL: https://doi.org/10.1145/2486159.2486180.
  52. David Peleg. Distributed Computing: A Locality-Sensitive Approach. Society for Industrial and Applied Mathematics, USA, 2000. Google Scholar
  53. Neil Robertson and Paul D. Seymour. Graph minors. II. algorithmic aspects of tree-width. J. Algorithms, 7(3):309-322, 1986. URL: https://doi.org/10.1016/0196-6774(86)90023-4.
  54. Neil Robertson and Paul D Seymour. Graph minors. xvi. excluding a non-planar graph. Journal of Combinatorial Theory, Series B, 89(1):43-76, 2003. URL: https://doi.org/10.1016/S0095-8956(03)00042-X.
  55. Václav Rozhon, Michael Elkin, Christoph Grunau, and Bernhard Haeupler. Deterministic low-diameter decompositions for weighted graphs and distributed and parallel applications. In 63rd IEEE Annual Symposium on Foundations of Computer Science, FOCS 2022, Denver, CO, USA, October 31 - November 3, 2022, pages 1114-1121. IEEE, 2022. URL: https://doi.org/10.1109/FOCS54457.2022.00107.
  56. Václav Rozhon, Christoph Grunau, Bernhard Haeupler, Goran Zuzic, and Jason Li. Undirected (1+ε)-shortest paths via minor-aggregates: near-optimal deterministic parallel and distributed algorithms. In Stefano Leonardi and Anupam Gupta, editors, STOC '22: 54th Annual ACM SIGACT Symposium on Theory of Computing, Rome, Italy, June 20 - 24, 2022, pages 478-487. ACM, 2022. URL: https://doi.org/10.1145/3519935.3520074.
  57. Philipp Schneider. Power and limitations of hybrid communication networks. PhD thesis, University of Freiburg, Freiburg im Breisgau, Germany, 2023. URL: https://freidok.uni-freiburg.de/data/232804.
  58. Mikkel Thorup. Compact oracles for reachability and approximate distances in planar digraphs. J. ACM, 51(6):993-1024, November 2004. URL: https://doi.org/10.1145/1039488.1039493.
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