skip to main content
10.1145/3662158.3662801acmconferencesArticle/Chapter ViewAbstractPublication PagespodcConference Proceedingsconference-collections
research-article
Open access

Computing Minimum Weight Cycle in the CONGEST Model

Published: 17 June 2024 Publication History

Abstract

Minimum Weight Cycle (MWC) is the problem of finding a simple cycle of minimum weight in a graph G = (V, E). This is a fundamental graph problem with classical sequential algorithms that run in Õ(n3) and Õ(mn) time where n = |V| and m = |E|. In recent years this problem has received significant attention in the context of fine-grained sequential complexity [3, 50] as well as in the design of faster sequential approximation algorithms [13, 26, 32, 33], though not much is known in the distributed CONGEST model.
We present near-optimal Ω(n) CONGEST lower bounds on the round complexity of computing exact and (2 − ϵ)-approximate MWC in undirected weighted graphs and in directed graphs even if unweighted. We complement these lower bounds with sublinear-round algorithms for computing 2-approximation of MWC. Our algorithms use a variety of techniques in non-trivial ways, such as in our approximate directed unweighted MWC algorithm that efficiently computes BFS from all vertices restricted to certain implicitly computed neighborhoods in sublinear rounds, and in our weighted approximation algorithms that use unweighted MWC algorithms on scaled graphs combined with a fast and streamlined method for computing multiple source approximate SSSP.

References

[1]
Amir Abboud, Keren Censor-Hillel, and Seri Khoury. 2016. Near-Linear Lower Bounds for Distributed Distance Computations, Even in Sparse Networks. In Distributed Computing - 30th International Symposium, DISC 2016 (Lecture Notes in Computer Science, Vol. 9888). Springer, Paris, France, 29--42.
[2]
Amir Abboud, Virginia Vassilevska Williams, and Joshua R. Wang. 2016. Approximation and Fixed Parameter Subquadratic Algorithms for Radius and Diameter in Sparse Graphs. In Proceedings of the Twenty-Seventh Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2016. SIAM, Arlington, VA, USA, 377--391.
[3]
Udit Agarwal and Vijaya Ramachandran. 2018. Fine-grained complexity for sparse graphs. In Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2018. ACM, Los Angeles, CA, USA, 239--252.
[4]
Udit Agarwal and Vijaya Ramachandran. 2019. Distributed Weighted All Pairs Shortest Paths Through Pipelining. In 2019 IEEE International Parallel and Distributed Processing Symposium, IPDPS 2019. IEEE, Rio de Janeiro, Brazil, 23--32.
[5]
Udit Agarwal and Vijaya Ramachandran. 2020. Faster Deterministic All Pairs Shortest Paths in Congest Model. In SPAA '20: 32nd ACM Symposium on Parallelism in Algorithms and Architectures, 2020. ACM, Virtual Event, USA, 11--21.
[6]
Bertie Ancona, Keren Censor-Hillel, Mina Dalirrooyfard, Yuval Efron, and Virginia Vassilevska Williams. 2020. Distributed Distance Approximation. In 24th International Conference on Principles of Distributed Systems, OPODIS 2020 (LIPIcs, Vol. 184). Schloss Dagstuhl - Leibniz-Zentrum für Informatik, Strasbourg, France (Virtual Conference), 30:1--30:17.
[7]
Ziv Bar-Yossef, T. S. Jayram, Ravi Kumar, and D. Sivakumar. 2004. An information statistics approach to data stream and communication complexity. J. Comput. Syst. Sci. 68, 4 (2004), 702--732.
[8]
Aaron Bernstein and Danupon Nanongkai. 2019. Distributed exact weighted all-pairs shortest paths in near-linear time. In Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing, STOC 2019. ACM, Phoenix, AZ, USA, 334--342.
[9]
Nairen Cao and Jeremy T. Fineman. 2023. Parallel Exact Shortest Paths in Almost Linear Work and Square Root Depth. In Proceedings of the 2023 ACM-SIAM Symposium on Discrete Algorithms, SODA 2023. SIAM, Florence, Italy, 4354--4372.
[10]
Nairen Cao, Jeremy T. Fineman, and Katina Russell. 2021. Brief Announcement: An Improved Distributed Approximate Single Source Shortest Paths Algorithm. In Proceedings of the 2021 ACM Symposium on Principles of Distributed Computing (Virtual Event, Italy) (PODC'21). Association for Computing Machinery, New York, NY, USA, 493--496.
[11]
Keren Censor-Hillel, Orr Fischer, Tzlil Gonen, François Le Gall, Dean Leitersdorf, and Rotem Oshman. 2020. Fast Distributed Algorithms for Girth, Cycles and Small Subgraphs. In 34th International Symposium on Distributed Computing, DISC 2020 (LIPIcs, Vol. 179). Schloss Dagstuhl - Leibniz-Zentrum für Informatik, Virtual Conference, 33:1--33:17.
[12]
Yi-Jun Chang, Seth Pettie, Thatchaphol Saranurak, and Hengjie Zhang. 2021. Near-optimal Distributed Triangle Enumeration via Expander Decompositions. Journal of the ACM (JACM) 68, 3 (2021), 1--36.
[13]
Shiri Chechik and Gur Lifshitz. 2021. Optimal Girth Approximation for Dense Directed Graphs. In Proceedings of the 2021 ACM-SIAM Symposium on Discrete Algorithms, SODA 2021. SIAM, Virtual Conference, 290--300.
[14]
Shiri Chechik and Doron Mukhtar. 2022. Single-source shortest paths in the CONGEST model with improved bounds. Distributed Comput. 35, 4 (2022), 357--374.
[15]
Andrew Drucker, Fabian Kuhn, and Rotem Oshman. 2014. On the power of the congested clique model. In ACM Symposium on Principles of Distributed Computing, PODC '14. ACM, Paris, France, 367--376.
[16]
Talya Eden, Nimrod Fiat, Orr Fischer, Fabian Kuhn, and Rotem Oshman. 2022. Sublinear-time distributed algorithms for detecting small cliques and even cycles. Distributed Comput. 35, 3 (2022), 207--234.
[17]
Michael Elkin. 2006. An unconditional lower bound on the time-approximation trade-off for the distributed minimum spanning tree problem. SIAM J. Comput. 36, 2 (2006), 433--456.
[18]
Michael Elkin and Ofer Neiman. 2019. Hopsets with Constant Hopbound, and Applications to Approximate Shortest Paths. SIAM J. Comput. 48, 4 (2019), 1436--1480.
[19]
Michael Elkin and Chhaya Trehan. 2022. (1+ϵ)-Approximate Shortest Paths in Dynamic Streams. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, APPROX/RANDOM 2022 (LIPIcs, Vol. 245). Schloss Dagstuhl - Leibniz-Zentrum für Informatik, USA (Virtual Conference), 51:1--51:23.
[20]
Sebastian Forster and Danupon Nanongkai. 2018. A Faster Distributed Single-Source Shortest Paths Algorithm. In 59th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2018. IEEE Computer Society, Paris, France, 686--697.
[21]
Pierre Fraigniaud, Mael Luce, Frederic Magniez, and Ioan Todinca. 2024. Even-Cycle Detection in the Randomized and Quantum CONGEST Model. Technical Report. arXiv. arXiv:2402.12018 [cs.DC]
[22]
Pierre Fraigniaud and Dennis Olivetti. 2019. Distributed Detection of Cycles. ACM Trans. Parallel Comput. 6, 3 (2019), 12:1--12:20.
[23]
Silvio Frischknecht, Stephan Holzer, and Roger Wattenhofer. 2012. Networks cannot compute their diameter in sublinear time. In Proceedings of the Twenty-Third Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2012. SIAM, Kyoto, Japan, 1150--1162.
[24]
Mohsen Ghaffari. 2015. Near-Optimal Scheduling of Distributed Algorithms. In Proceedings of the 2015 ACM Symposium on Principles of Distributed Computing, PODC 2015. ACM, Donostia-San Sebastián, Spain, 3--12.
[25]
Mohsen Ghaffari and Rajan Udwani. 2015. Brief Announcement: Distributed Single-Source Reachability. In Proceedings of the 2015 ACM Symposium on Principles of Distributed Computing, PODC 2015. ACM, Donostia-San Sebastián, Spain, 163--165.
[26]
Alina Harbuzova, Ce Jin, Virginia Vassilevska Williams, and Zixuan Xu. 2024. Improved Roundtrip Spanners, Emulators, and Directed Girth Approximation. In Proceedings of the 2024 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA). SIAM, Alexandria, Virginia, USA, 4641--4669.
[27]
Loc Hoang, Matteo Pontecorvi, Roshan Dathathri, Gurbinder Gill, Bozhi You, Keshav Pingali, and Vijaya Ramachandran. 2019. A round-efficient distributed betweenness centrality algorithm. In Proceedings of the 24th ACM SIGPLAN Symposium on Principles and Practice of Parallel Programming, PPoPP 2019. ACM, Washington, DC, USA, 272--286.
[28]
Stephan Holzer and Roger Wattenhofer. 2012. Optimal distributed all pairs shortest paths and applications. In ACM Symposium on Principles of Distributed Computing, PODC '12. ACM, Madeira, Portugal, 355--364.
[29]
Chien-Chung Huang, Danupon Nanongkai, and Thatchaphol Saranurak. 2017. Distributed Exact Weighted All-Pairs Shortest Paths in Õ(n5/4) Rounds. In 58th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2017. IEEE Computer Society, Berkeley, CA, USA, 168--179.
[30]
Taisuke Izumi and François Le Gall. 2017. Triangle Finding and Listing in CONGEST Networks. In Proceedings of the ACM Symposium on Principles of Distributed Computing, PODC 2017. ACM, Washington, DC, USA, 381--389.
[31]
Taisuke Izumi, Naoki Kitamura, Takamasa Naruse, and Gregory Schwartzman. 2022. Fully Polynomial-Time Distributed Computation in Low-Treewidth Graphs. In SPAA '22: 34th ACM Symposium on Parallelism in Algorithms and Architectures. ACM, Philadelphia, PA, USA, 11--22.
[32]
Avi Kadria, Liam Roditty, Aaron Sidford, Virginia Vassilevska Williams, and Uri Zwick. 2022. Algorithmic trade-offs for girth approximation in undirected graphs. In Proceedings of the 2022 ACM-SIAM Symposium on Discrete Algorithms, SODA 2022. SIAM, Virtual Conference / Alexandria, VA, USA, 1471--1492.
[33]
Avi Kadria, Liam Roditty, Aaron Sidford, Virginia Vassilevska Williams, and Uri Zwick. 2023. Improved girth approximation in weighted undirected graphs. In Proceedings of the 2023 ACM-SIAM Symposium on Discrete Algorithms, SODA 2023. SIAM, Florence, Italy, 2242--2255.
[34]
Philip N Klein and Sairam Subramanian. 1997. A randomized parallel algorithm for single-source shortest paths. Journal of Algorithms 25, 2 (1997), 205--220.
[35]
Eyal Kushilevitz and Noam Nisan. 1996. Communication Complexity. Cambridge University Press, Cambridge.
[36]
Frank Thomson Leighton, Bruce M Maggs, and Satish B Rao. 1994. Packet routing and job-shop scheduling in O (congestion+ dilation) steps. Combinatorica 14, 2 (1994), 167--186.
[37]
Christoph Lenzen, Boaz Patt-Shamir, and David Peleg. 2019. Distributed distance computation and routing with small messages. Distributed Computing 32, 2 (2019), 133--157.
[38]
Lev B. Levitin, Mark G. Karpovsky, and Mehmet Mustafa. 2010. Minimal Sets of Turns for Breaking Cycles in Graphs Modeling Networks. IEEE Trans. Parallel Distributed Syst. 21, 9 (2010), 1342--1353.
[39]
Vignesh Manoharan and Vijaya Ramachandran. 2024. Computing Replacement Paths in the CONGEST Model. In Structural Information and Communication Complexity (SIROCCO 2024). (to appear).
[40]
Vignesh Manoharan and Vijaya Ramachandran. 2024. Improved Approximation Bounds for Minimum Weight Cycle in the CONGEST Model. Technical Report 2308.08670. arXiv. https://arxiv.org/abs/2308.08670 Full version of this paper.
[41]
Danupon Nanongkai. 2014. Distributed approximation algorithms for weighted shortest paths. In Symposium on Theory of Computing, STOC 2014. ACM, New York, NY, USA, 565--573.
[42]
Gabriele Oliva, Roberto Setola, Luigi Glielmo, and Christoforos N. Hadjicostis. 2018. Distributed Cycle Detection and Removal. IEEE Trans. Control. Netw. Syst. 5, 1 (2018), 194--204.
[43]
David Peleg. 2000. Distributed computing: a locality-sensitive approach. SIAM, USA.
[44]
David Peleg, Liam Roditty, and Elad Tal. 2012. Distributed Algorithms for Network Diameter and Girth. In Automata, Languages, and Programming - 39th International Colloquium, ICALP 2012 (Lecture Notes in Computer Science, Vol. 7392). Springer, Warwick, UK, 660--672.
[45]
Seth Pettie. 2022. Personal Communication.
[46]
A. A. Razborov. 1992. On the Distributional Complexity of Disjointness. Theor. Comput. Sci. 106, 2 (Dec. 1992), 385--390.
[47]
Liam Roditty and Roei Tov. 2013. Approximating the Girth. ACM Trans. Algorithms 9, 2, Article 15 (mar 2013), 13 pages.
[48]
Václav Rozhon, Bernhard Haeupler, Anders Martinsson, Christoph Grunau, and Goran Zuzic. 2023. Parallel Breadth-First Search and Exact Shortest Paths and Stronger Notions for Approximate Distances. In Proceedings of the 55th Annual ACM Symposium on Theory of Computing, STOC 2023. ACM, Orlando, FL, USA, 321--334.
[49]
Atish Das Sarma, Stephan Holzer, Liah Kor, Amos Korman, Danupon Nanongkai, Gopal Pandurangan, David Peleg, and Roger Wattenhofer. 2012. Distributed verification and hardness of distributed approximation. SIAM J. Comput. 41, 5 (2012), 1235--1265.
[50]
Virginia Vassilevska Williams and R. Ryan Williams. 2018. Subcubic Equivalences Between Path, Matrix, and Triangle Problems. J. ACM 65, 5 (2018), 27:1--27:38.

Cited By

View all
  • (2024)Computing Replacement Paths in the CONGEST ModelStructural Information and Communication Complexity10.1007/978-3-031-60603-8_23(420-437)Online publication date: 27-May-2024

Recommendations

Comments

Information & Contributors

Information

Published In

cover image ACM Conferences
PODC '24: Proceedings of the 43rd ACM Symposium on Principles of Distributed Computing
June 2024
570 pages
ISBN:9798400706684
DOI:10.1145/3662158
This work is licensed under a Creative Commons Attribution International 4.0 License.

Sponsors

Publisher

Association for Computing Machinery

New York, NY, United States

Publication History

Published: 17 June 2024

Check for updates

Author Tags

  1. distributed algorithms
  2. graph algorithms

Qualifiers

  • Research-article

Funding Sources

Conference

PODC '24
Sponsor:

Acceptance Rates

Overall Acceptance Rate 740 of 2,477 submissions, 30%

Contributors

Other Metrics

Bibliometrics & Citations

Bibliometrics

Article Metrics

  • Downloads (Last 12 months)165
  • Downloads (Last 6 weeks)43
Reflects downloads up to 25 Feb 2025

Other Metrics

Citations

Cited By

View all
  • (2024)Computing Replacement Paths in the CONGEST ModelStructural Information and Communication Complexity10.1007/978-3-031-60603-8_23(420-437)Online publication date: 27-May-2024

View Options

View options

PDF

View or Download as a PDF file.

PDF

eReader

View online with eReader.

eReader

Login options

Figures

Tables

Media

Share

Share

Share this Publication link

Share on social media