Document Open Access Logo

An Almost Singularly Optimal Asynchronous Distributed MST Algorithm

Authors Fabien Dufoulon , Shay Kutten , William K. Moses Jr. , Gopal Pandurangan , David Peleg

PDF
Thumbnail PDF

File

LIPIcs.DISC.2022.19.pdf
  • Filesize: 0.88 MB
  • 24 pages

Document Identifiers

Author Details

Fabien Dufoulon
  • Department of Computer Science, University of Houston, Houston, TX, USA
Shay Kutten
  • Faculty of Industrial Engineering and Management, Technion - Israel Institute of Technology, Haifa, Israel
William K. Moses Jr.
  • Department of Computer Science, University of Houston, Houston, TX, USA
Gopal Pandurangan
  • Department of Computer Science, University of Houston, Houston, TX, USA
David Peleg
  • Department of Computer Science and Applied Mathematics, Weizmann Institute of Science, Rehovot, Israel

Funding

  • Dufoulon, Fabien: This work was supported in part by NSF grants CCF-1717075, CCF-1540512, IIS-1633720, and BSF grant 2016419.
  • Kutten, Shay: This work was supported in part by the Bi-national Science Foundation (BSF) grant 2016419 and supported in part by ISF grant 1346/22.
  • Moses Jr., William K.: This work was supported in part by NSF grants CCF1540512, IIS-1633720, CCF-1717075, and BSF grant 2016419.
  • Pandurangan, Gopal: This work was supported in part by NSF grants CCF-1717075, CCF-1540512, IIS-1633720, and BSF grant 2016419.
  • Peleg, David: This work was supported in part by the US-Israel Binational Science Foundation grant 2018043.

Cite AsGet BibTex

Fabien Dufoulon, Shay Kutten, William K. Moses Jr., Gopal Pandurangan, and David Peleg. An Almost Singularly Optimal Asynchronous Distributed MST Algorithm. In 36th International Symposium on Distributed Computing (DISC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 246, pp. 19:1-19:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
https://doi.org/10.4230/LIPIcs.DISC.2022.19

Abstract

A singularly (near) optimal distributed algorithm is one that is (near) optimal in two criteria, namely, its time and message complexities. For synchronous CONGEST networks, such algorithms are known for fundamental distributed computing problems such as leader election [Kutten et al., JACM 2015] and Minimum Spanning Tree (MST) construction [Pandurangan et al., STOC 2017, Elkin, PODC 2017]. However, it is open whether a singularly (near) optimal bound can be obtained for the MST construction problem in general asynchronous CONGEST networks. In this paper, we present a randomized distributed MST algorithm that, with high probability, computes an MST in asynchronous CONGEST networks and takes Õ(D^{1+ε} + √n) time and Õ(m) messages, where n is the number of nodes, m the number of edges, D is the diameter of the network, and ε > 0 is an arbitrarily small constant (both time and message bounds hold with high probability). Since Ω̃(D+√n) and Ω(m) are respective time and message lower bounds for distributed MST construction in the standard KT₀ model, our algorithm is message optimal (up to a polylog(n) factor) and almost time optimal (except for a D^ε factor). Our result answers an open question raised in Mashregi and King [DISC 2019] by giving the first known asynchronous MST algorithm that has sublinear time (for all D = O(n^{1-ε})) and uses Õ(m) messages. Using a result of Mashregi and King [DISC 2019], this also yields the first asynchronous MST algorithm that is sublinear in both time and messages in the KT₁ CONGEST model. A key tool in our algorithm is the construction of a low diameter rooted spanning tree in asynchronous CONGEST that has depth Õ(D^{1+ε}) (for an arbitrarily small constant ε > 0) in Õ(D^{1+ε}) time and Õ(m) messages. To the best of our knowledge, this is the first such construction that is almost singularly optimal in the asynchronous setting. This tree construction may be of independent interest as it can also be used for efficiently performing basic tasks such as verified broadcast and convergecast in asynchronous networks.

Subject Classification

ACM Subject Classification
  • Theory of computation → Distributed algorithms
  • Mathematics of computing → Probabilistic algorithms
  • Mathematics of computing → Discrete mathematics
Keywords
  • Asynchronous networks
  • Minimum Spanning Tree
  • Distributed Algorithm
  • Singularly Optimal

Metrics

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

Related Versions

References

  1. Yehuda Afek and Eli Gafni. Time and message bounds for election in synchronous and asynchronous complete networks. SICOMP, 20(2):376-394, 1991. Google Scholar
  2. Yehuda Afek and Yossi Matias. Elections in anonymous networks. Information and Computation, 113(2):312-330, 1994. Google Scholar
  3. John Augustine, Seth Gilbert, Fabian Kuhn, Peter Robinson, and Suman Sourav. Latency, capacity, and distributed minimum spanning tree. In 2020 IEEE 40th International Conference on Distributed Computing Systems (ICDCS), pages 157-167. IEEE, 2020. Google Scholar
  4. Baruch Awerbuch. Complexity of network synchronization. Journal of the ACM (JACM), 32(4):804-823, 1985. Google Scholar
  5. Baruch Awerbuch. Optimal distributed algorithms for minimum weight spanning tree, counting, leader election, and related problems. In Proceedings of the 19th ACM Symposium on Theory of Computing (STOC), pages 230-240, 1987. Google Scholar
  6. Baruch Awerbuch. Distributed shortest paths algorithms (extended abstract). In Proceedings of the twenty-first annual ACM symposium on Theory of computing, pages 490-500, 1989. Google Scholar
  7. Baruch Awerbuch, Oded Goldreich, Ronen Vainish, and David Peleg. A trade-off between information and communication in broadcast protocols. J. ACM, 37:238-256, 1990. Google Scholar
  8. Baruch Awerbuch, Oded Goldreich, Ronen Vainish, and David Peleg. A trade-off between information and communication in broadcast protocols. Journal of the ACM (JACM), 37(2):238-256, 1990. Google Scholar
  9. Baruch Awerbuch and David Peleg. Network synchronization with polylogarithmic overhead. In 31st Annual Symposium on Foundations of Computer Science (FOCS), pages 514-522, 1990. Google Scholar
  10. Yi-Jun Chang, Varsha Dani, Thomas P. Hayes, Qizheng He, Wenzheng Li, and Seth Pettie. The energy complexity of broadcast. In Proceedings of the 2018 ACM Symposium on Principles of Distributed Computing, PODC '18, pages 95-104, New York, NY, USA, 2018. Association for Computing Machinery. URL: https://doi.org/10.1145/3212734.3212774.
  11. Yi-Jun Chang, Varsha Dani, Thomas P. Hayes, and Seth Pettie. The energy complexity of bfs in radio networks. In Proceedings of the 39th Symposium on Principles of Distributed Computing, PODC '20, pages 273-282, New York, NY, USA, 2020. Association for Computing Machinery. URL: https://doi.org/10.1145/3382734.3405713.
  12. Yogen K Dalal. A Distributed Algorithm for Constructing Minimal Spanning Trees in Computer-Communication Networks. Stanford University, 1976. Google Scholar
  13. Yogen K. Dalal. A distributed algorithm for constructing minimal spanning trees. IEEE Trans. Software Eng., 13(3):398-405, 1987. Google Scholar
  14. Atish Das Sarma, Stephan Holzer, Liah Kor, Amos Korman, Danupon Nanongkai, Gopal Pandurangan, David Peleg, and Roger Wattenhofer. Distributed verification and hardness of distributed approximation. SIAM J. Comput., 41(5):1235-1265, 2012. Google Scholar
  15. Devdatt P. Dubhashi and Alessandro Panconesi. Concentration of Measure for the Analysis of Randomized Algorithms. Cambridge University Press, 2009. URL: http://www.cambridge.org/gb/knowledge/isbn/item2327542/.
  16. Michael Elkin. A faster distributed protocol for constructing minimum spanning tree. Journal of Computer and System Sciences, 72(8):1282-1308, 2006. Google Scholar
  17. Michael Elkin. An unconditional lower bound on the time-approximation trade-off for the distributed minimum spanning tree problem. SIAM J. Comput., 36(2):433-456, 2006. Google Scholar
  18. Michael Elkin. A simple deterministic distributed MST algorithm, with near-optimal time and message complexities. In Proceedings of the 2017 ACM Symposium on Principles of Distributed Computing (PODC), pages 157-163, 2017. Google Scholar
  19. Michael Elkin, Hartmut Klauck, Danupon Nanongkai, and Gopal Pandurangan. Can quantum communication speed up distributed computation? In ACM Symposium on Principles of Distributed Computing, PODC, pages 166-175. ACM, 2014. Google Scholar
  20. Michalis Faloutsos and Mart Molle. A linear-time optimal-message distributed algorithm for minimum spanning trees. Distributed Computing, 17(2):151-170, 2004. Google Scholar
  21. Pierre Fraigniaud, Amos Korman, and Emmanuelle Lebhar. Local mst computation with short advice. Theory of Computing Systems, 47(4):920-933, 2010. Google Scholar
  22. Robert G. Gallager, Pierre A. Humblet, and Philip M. Spira. A distributed algorithm for minimum-weight spanning trees. ACM Trans. Program. Lang. Syst., 5(1):66-77, 1983. Google Scholar
  23. Juan A. Garay, Shay Kutten, and David Peleg. A sublinear time distributed algorithm for minimum-weight spanning trees. SIAM J. Comput., 27(1):302-316, 1998. Google Scholar
  24. Mohsen Ghaffari and Bernhard Haeupler. Distributed algorithms for planar networks II: low-congestion shortcuts, mst, and min-cut. In Proceedings of the Twenty-Seventh Annual ACM-SIAM Symposium on Discrete Algorithms, SODA, pages 202-219. SIAM, 2016. Google Scholar
  25. Mohsen Ghaffari and Bernhard Haeupler. Distributed algorithms for planar networks II: low-congestion shortcuts, MST, and min-cut. In Proceedings of the 27th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 202-219, 2016. Google Scholar
  26. Mohsen Ghaffari and Fabian Kuhn. Distributed MST and broadcast with fewer messages, and faster gossiping. In Proceedings of the 32nd International Symposium on Distributed Computing (DISC), pages 30:1-30:12, 2018. Google Scholar
  27. Robert Gmyr and Gopal Pandurangan. Time-message trade-offs in distributed algorithms. In 32nd International Symposium on Distributed Computing, DISC 2018, New Orleans, LA, USA, October 15-19, 2018, pages 32:1-32:18, 2018. Google Scholar
  28. Robert Gmyr and Gopal Pandurangan. Time-message trade-offs in distributed algorithms. In Proceedings of the 32nd International Symposium on Distributed Computing (DISC), pages 32:1-32:18, 2018. Google Scholar
  29. Sandeep KS Gupta and Pradip K Srimani. Self-stabilizing multicast protocols for ad hoc networks. Journal of Parallel and Distributed Computing, 63(1):87-96, 2003. Google Scholar
  30. Bernhard Haeupler, D. Ellis Hershkowitz, and David Wajc. Round-and message-optimal distributed graph algorithms. In PODC, pages 119-128, 2018. Google Scholar
  31. Bernhard Haeupler and David Wajc. A faster distributed radio broadcast primitive: Extended abstract. In Proceedings of the 2016 ACM Symposium on Principles of Distributed Computing, PODC '16, pages 361-370, New York, NY, USA, 2016. Association for Computing Machinery. URL: https://doi.org/10.1145/2933057.2933121.
  32. Bernhard Haeupler, David Wajc, and Goran Zuzic. Universally-optimal distributed algorithms for known topologies. In STOC '21: 53rd Annual ACM SIGACT Symposium on Theory of Computing, pages 1166-1179. ACM, 2021. Google Scholar
  33. Valerie King, Shay Kutten, and Mikkel Thorup. Construction and impromptu repair of an MST in a distributed network with o(m) communication. In Proceedings of the 2015 ACM Symposium on Principles of Distributed Computing (PODC), pages 71-80, 2015. Google Scholar
  34. Liah Kor, Amos Korman, and David Peleg. Tight bounds for distributed MST verification. In Proc. 28th Symp. on Theoretical Aspects of Computer Science (STACS), volume 9 of LIPIcs, pages 69-80. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2011. Google Scholar
  35. Amos Korman and Shay Kutten. Distributed verification of minimum spanning trees. Distributed Computing, 20(4):253-266, 2007. Google Scholar
  36. Amos Korman, Shay Kutten, and David Peleg. Proof labeling schemes. In Proc.24th ACM Symp. on Principles of Distributed Computing (PODC), pages 9-18, 2005. Google Scholar
  37. Shay Kutten, William K. Moses Jr., Gopal Pandurangan, and David Peleg. Singularly near optimal leader election in asynchronous networks. In 35th International Symposium on Distributed Computing (DISC), pages 27:1-27:18, 2021. Google Scholar
  38. Shay Kutten, Gopal Pandurangan, David Peleg, Peter Robinson, and Amitabh Trehan. On the complexity of universal leader election. J. ACM, 62(1), 2015. Google Scholar
  39. Shay Kutten and David Peleg. Fast distributed construction of small k-dominating sets and applications. J. Algorithms, 28(1):40-66, 1998. Google Scholar
  40. Zvi Lotker, Boaz Patt-Shamir, Elan Pavlov, and David Peleg. Minimum-weight spanning tree construction in O(log log n) communication rounds. SIAM J. Comput., 35:120-131, 2005. Google Scholar
  41. Zvi Lotker, Boaz Patt-Shamir, and David Peleg. Distributed MST for constant diameter graphs. In Proc. 20th ACM Symp. on Principles of Distributed Computing (PODC), pages 63-71, 2001. Google Scholar
  42. Ali Mashreghi and Valerie King. Time-communication trade-offs for minimum spanning tree construction. In Proceedings of the 18th International Conference on Distributed Computing and Networking (ICDCN), 2017. Google Scholar
  43. Ali Mashreghi and Valerie King. Broadcast and minimum spanning tree with o(m) messages in the asynchronous CONGEST model. In 32nd International Symposium on Distributed Computing, DISC 2018, New Orleans, LA, USA, October 15-19, 2018, volume 121 of LIPIcs, pages 37:1-37:17, 2018. Google Scholar
  44. Ali Mashreghi and Valerie King. Brief announcement: Faster asynchronous MST and low diameter tree construction with sublinear communication. In Jukka Suomela, editor, 33rd International Symposium on Distributed Computing, DISC 2019, October 14-18, 2019, Budapest, Hungary, volume 146 of LIPIcs, pages 49:1-49:3, 2019. Google Scholar
  45. Ali Mashreghi and Valerie King. Broadcast and minimum spanning tree with o(m) messages in the asynchronous CONGEST model. Distributed Computing, pages 1-17, 2021. Google Scholar
  46. Gary L. Miller, Richard Peng, and Shen Chen Xu. Parallel graph decompositions using random shifts. In Proceedings of the Twenty-Fifth Annual ACM Symposium on Parallelism in Algorithms and Architectures, SPAA '13, pages 196-203, New York, NY, USA, 2013. Association for Computing Machinery. URL: https://doi.org/10.1145/2486159.2486180.
  47. Gopal Pandurangan, Peter Robinson, and Michele Scquizzato. A time- and message-optimal distributed algorithm for minimum spanning trees. In Proceedings of the 49th Annual ACM Symposium on the Theory of Computing (STOC), pages 743-756, 2017. Google Scholar
  48. Gopal Pandurangan, Peter Robinson, and Michele Scquizzato. The distributed minimum spanning tree problem. Bulletin of the EATCS, 125, 2018. Google Scholar
  49. Gopal Pandurangan, Peter Robinson, and Michele Scquizzato. A time- and message-optimal distributed algorithm for minimum spanning trees. ACM Transactions on Algorithms (TALG), 16(1):1-27, 2019. Google Scholar
  50. David Peleg. Distributed Computing: A Locality Sensitive Approach. SIAM, 2000. Google Scholar
  51. David Peleg and Vitaly Rubinovich. A near-tight lower bound on the time complexity of distributed minimum-weight spanning tree construction. SIAM J. Comput., 30(5):1427-1442, 2000. Google Scholar
  52. Deepak Rohilla, Mahendra Kumar Murmu, and Shashidhar Kulkarni. An efficient distributed approach to construct a minimum spanning tree in cognitive radio network. In First International Conference on Sustainable Technologies for Computational Intelligence, pages 397-407. Springer, 2020. Google Scholar
  53. Baruch Schieber and Marc Snir. Calling names on nameless networks. Information and Computation, 113(1):80-101, 1994. Google Scholar
  54. Adrian Segall. Distributed network protocols. IEEE transactions on Information Theory, 29(1):23-35, 1983. Google Scholar
  55. Gurdip Singh. Efficient leader election using sense of direction. Distributed Computing, 10(3):159-165, 1997. URL: https://doi.org/10.1007/s004460050033.
  56. Philip Spira. Communication complexity of distributed minimum spanning tree algorithms. In Proceedings of the second Berkeley conference on distributed data management and computer networks, 1977. Google Scholar
  57. Robert Endre Tarjan. Data Structures and Network Algorithms. Society for Industrial and Applied Mathematics, 1983. Google Scholar
  58. Gerard Tel. Introduction to Distributed Algorithms. Cambridge University Press, 1994. Google Scholar
© 2023-2024 Schloss Dagstuhl – LZI GmbH Imprint Privacy Contact