Document Open Access Logo

Byzantine-Tolerant Phase Clock

Authors Costas Busch , Paweł Garncarek , Dariusz R. Kowalski

PDF

File

Thumbnail PDF
PDF
LIPIcs.OPODIS.2025.30.pdf
  • Filesize: 1.01 MB
  • 19 pages

Document Identifiers

Subject Classification

ACM Subject Classification
  • Computing methodologies → Distributed algorithms
  • Theory of computation → Adversary models
Keywords
  • phase clock
  • Byzantine nodes
  • population protocols
  • balls into bins

Metrics

Abstract

A phase clock is a basic synchronization mechanism that keeps distributed nodes closely synchronized to execute the same phase of a distributed algorithm. A phase clock is typically implemented with a local logical counter that keeps track of the current phase count. Phase clocks are particularly useful in population protocols for implementing leader election and majority selection. We study phase clocks that tolerate Byzantine faults. We show that there is a phase clock that tolerates up to f < n/3 faulty nodes, where n is the number of nodes, such that the gap of the local counter values is O(n²log n). The gap can be further lowered to O(log n) when f ≤ n/8. We also show that if f > n/3, then the gap grows to infinity as time increases. While analyzing phase clock we introduce novel techniques and bounds for balls into bins processes, which might be of independent interest. Using the phase clock, we obtain a majority selection population protocol that tolerates up to f faults and decides on the majority value in O(log² n) parallel time using poly-log states per node.

Cite As Get BibTex

Costas Busch, Paweł Garncarek, and Dariusz R. Kowalski. Byzantine-Tolerant Phase Clock. In 29th International Conference on Principles of Distributed Systems (OPODIS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 361, pp. 30:1-30:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.OPODIS.2025.30

Author Details

Costas Busch
  • Department of Computer Science, Augusta University, GA, USA
Paweł Garncarek
  • Institute of Computer Science, University of Wrocław, Poland
Dariusz R. Kowalski
  • Department of Computer Science, Augusta University, GA, USA

Funding

NSF Grant 2131538

References

  1. Muhammed Abdulazeez, Pawel Garncarek, Dariusz R. Kowalski, and Prudence W. H. Wong. Lightweight robust framework for workload scheduling in clouds. In IEEE International Conference on Edge Computing, EDGE 2017, Honolulu, HI, USA, June 25-30, 2017, pages 206-209. IEEE Computer Society, 2017. URL: https://doi.org/10.1109/IEEE.EDGE.2017.36.
  2. Dan Alistarh, James Aspnes, David Eisenstat, Rati Gelashvili, and Ronald L. Rivest. Time-space trade-offs in population protocols. In Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2017, Barcelona, Spain, Hotel Porta Fira, January 16-19, pages 2560-2579. SIAM, 2017. URL: https://doi.org/10.1137/1.9781611974782.169.
  3. Dan Alistarh, James Aspnes, and Rati Gelashvili. Space-optimal majority in population protocols. In Proceedings of the 2018 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 2221-2239, 2018. URL: https://doi.org/10.1137/1.9781611975031.144.
  4. Dan Alistarh, Rati Gelashvili, and Milan Vojnovic. Fast and exact majority in population protocols. In Proceedings of the 2015 ACM Symposium on Principles of Distributed Computing, PODC 2015, Donostia-San Sebastián, Spain, July 21 - 23, 2015, pages 47-56. ACM, 2015. URL: https://doi.org/10.1145/2767386.2767429.
  5. Dana Angluin, James Aspnes, Zoë Diamadi, Michael J. Fischer, and René Peralta. Computation in networks of passively mobile finite-state sensors. Distributed Comput., 18(4):235-253, 2006. URL: https://doi.org/10.1007/s00446-005-0138-3.
  6. Dana Angluin, James Aspnes, and David Eisenstat. Fast computation by population protocols with a leader. Distributed Computing, 21(3):183-199, September 2008. URL: https://doi.org/10.1007/S00446-008-0067-Z.
  7. Dana Angluin, James Aspnes, and David Eisenstat. A simple population protocol for fast robust approximate majority. Distributed Comput., 21(2):87-102, 2008. URL: https://doi.org/10.1007/s00446-008-0059-z.
  8. Anish Arora, Shlomi Dolev, and Mohamed Gouda. Maintaining digital clocks in step. In Distributed Algorithms: 5th International Workshop, WDAG'91 Delphi, Greece, October 7-9, 1991 Proceedings 5, pages 71-79. Springer, 1992. Google Scholar
  9. Hagit Attiya and Jennifer Welch. Distributed Computing: Fundamentals, Simulations, and Advanced Topics. Wiley, Second edition, 2004. Google Scholar
  10. Yossi Azar, Andrei Z. Broder, Anna R. Karlin, and Eli Upfal. Balanced allocations. SIAM Journal on Computing, 29(1):180-200, 1999. URL: https://doi.org/10.1137/S0097539795288490.
  11. Nikhil Bansal and Ohad N. Feldheim. The power of two choices in graphical allocation. In Proceedings of the 54th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2022, pages 52-63, New York, NY, USA, 2022. Association for Computing Machinery. URL: https://doi.org/10.1145/3519935.3519995.
  12. Nikhil Bansal and William Kuszmaul. Balanced allocations: The heavily loaded case with deletions. In 2022 IEEE 63rd Annual Symposium on Foundations of Computer Science (FOCS), pages 801-812, 2022. URL: https://doi.org/10.1109/FOCS54457.2022.00081.
  13. Stav Ben-Nun, Tsvi Kopelowitz, Matan Kraus, and Ely Porat. An o(log3/2 n) parallel time population protocol for majority with o(log n) states. In Proceedings of the 39th Symposium on Principles of Distributed Computing, PODC '20, pages 191-199, New York, NY, USA, 2020. Association for Computing Machinery. URL: https://doi.org/10.1145/3382734.3405747.
  14. Michael Ben-Or, Elan Pavlov, and Vinod Vaikuntanathan. Byzantine agreement in the full-information model in o(log n) rounds. In Proceedings of the Thirty-Eighth Annual ACM Symposium on Theory of Computing, STOC '06, pages 179-186, New York, NY, USA, 2006. Association for Computing Machinery. URL: https://doi.org/10.1145/1132516.1132543.
  15. R. Beraldi and G. Mattia. Power of random choices made efficient for fog computing. IEEE Transactions on Cloud Computing, 10(02):1130-1141, April 2022. URL: https://doi.org/10.1109/TCC.2020.2968443.
  16. Petra Berenbrink, Artur Czumaj, Angelika Steger, and Berthold Vöcking. Balanced allocations: The heavily loaded case. SIAM Journal on Computing, 35(6):1350-1385, 2006. URL: https://doi.org/10.1137/S009753970444435X.
  17. Petra Berenbrink, Robert Elsässer, Tom Friedetzky, Dominik Kaaser, Peter Kling, and Tomasz Radzik. A population protocol for exact majority with O(log^5/3 n) stabilization time and Θ(log n) states. In 32nd International Symposium on Distributed Computing (DISC 2018). Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2018. URL: https://doi.org/10.4230/LIPIcs.DISC.2018.10.
  18. Maury Bramson, Yi Lu, and Balaji Prabhakar. Asymptotic independence of queues under randomized load balancing. Queueing Syst. Theory Appl., 71(3):247-292, July 2012. URL: https://doi.org/10.1007/s11134-012-9311-0.
  19. Costas Busch and Dariusz R. Kowalski. Byzantine-resilient population protocols. CoRR, abs/2105.07123, 2021. URL: https://arxiv.org/abs/2105.07123.
  20. Ran Canetti and Tal Rabin. Fast asynchronous byzantine agreement with optimal resilience. In Proceedings of the Twenty-Fifth Annual ACM Symposium on Theory of Computing, May 16-18, 1993, San Diego, CA, USA, pages 42-51. ACM, 1993. URL: https://doi.org/10.1145/167088.167105.
  21. Richard Cole, Bruce M. Maggs, Friedhelm Meyer auf der Heide, Michael Mitzenmacher, Andréa W. Richa, Klaus Schröder, Ramesh K. Sitaraman, and Berthold Vöcking. Randomized protocols for low-congestion circuit routing in multistage interconnection networks. In Proceedings of the Thirtieth Annual ACM Symposium on Theory of Computing, STOC '98, pages 378-388, New York, NY, USA, 1998. Association for Computing Machinery. URL: https://doi.org/10.1145/276698.276790.
  22. Colin Cooper, Robert Elsässer, and Tomasz Radzik. The power of two choices in distributed voting. In Automata, Languages, and Programming, pages 435-446, Berlin, Heidelberg, 2014. Springer Berlin Heidelberg. URL: https://doi.org/10.1007/978-3-662-43951-7_37.
  23. Ariel Daliot, Danny Dolev, and Hanna Parnas. Self-stabilizing pulse synchronization inspired by biological pacemaker networks. In Symposium on Self-Stabilizing Systems, pages 32-48. Springer, 2003. URL: https://doi.org/10.1007/3-540-45032-7_3.
  24. Shlomi Dolev and Jennifer L. Welch. Self-stabilizing clock synchronization in the presence of byzantine faults. J. ACM, 51(5):780-799, September 2004. URL: https://doi.org/10.1145/1017460.1017463.
  25. David Doty, Mahsa Eftekhari, Leszek Gasieniec, Eric E. Severson, Przemyslaw Uznanski, and Grzegorz Stachowiak. A time and space optimal stable population protocol solving exact majority. In 62nd IEEE Annual Symposium on Foundations of Computer Science, FOCS 2021, Denver, CO, USA, February 7-10, 2022, pages 1044-1055. IEEE, 2021. URL: https://doi.org/10.1109/FOCS52979.2021.00104.
  26. Paweł Garncarek, Costas Busch, and Dariusz Kowalski. Locally balanced allocations under strong byzantine influence. In 31st International Colloquium On Structural Information and Communication Complexity (SIROCCO 2024), 2024. Google Scholar
  27. Leszek Gąsieniec and Grzegorz Stachowiak. Enhanced phase clocks, population protocols, and fast space optimal leader election. J. ACM, 68(1), November 2020. URL: https://doi.org/10.1145/3424659.
  28. Gaston H. Gonnet. Expected length of the longest probe sequence in hash code searching. J. ACM, 28(2):289-304, April 1981. URL: https://doi.org/10.1145/322248.322254.
  29. Rachid Guerraoui and Eric Ruppert. Names trump malice: Tiny mobile agents can tolerate byzantine failures. In Automata, Languages and Programming, 36th Internatilonal Colloquium, ICALP 2009, Rhodes, Greece, July 5-12, 2009, Proceedings, Part II, volume 5556 of Lecture Notes in Computer Science, pages 484-495. Springer, 2009. URL: https://doi.org/10.1007/978-3-642-02930-1_40.
  30. T. Herman. Phase clocks for transient fault repair. IEEE Transactions on Parallel and Distributed Systems, 11(10):1048-1057, 2000. URL: https://doi.org/10.1109/71.888644.
  31. Norman L. Johnson and Samuel Kotz. Urn models and their application : an approach to modern discrete probability theory. International Statistical Review, 46:319, 1978. Google Scholar
  32. Krishnaram Kenthapadi and Rina Panigrahy. Balanced allocation on graphs. In ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 434-443, 2006. URL: http://dl.acm.org/citation.cfm?id=1109557.1109606.
  33. Leslie Lamport, Robert E. Shostak, and Marshall C. Pease. The byzantine generals problem. ACM Trans. Program. Lang. Syst., 4(3):382-401, 1982. URL: https://doi.org/10.1145/357172.357176.
  34. Dimitrios Los and Thomas Sauerwald. Balanced allocations with the choice of noise. In Proceedings of the 2022 ACM Symposium on Principles of Distributed Computing, PODC'22, pages 164-175, New York, NY, USA, 2022. Association for Computing Machinery. URL: https://doi.org/10.1145/3519270.3538428.
  35. Nancy A. Lynch. Distributed Algorithms. Morgan Kaufmann, 1st edition, 1996. URL: http://www.amazon.com/Distributed-Algorithms-Kaufmann-Management-Systems/dp/1558603484.
  36. George B. Mertzios, Sotiris E. Nikoletseas, Christoforos L. Raptopoulos, and Paul G. Spirakis. Determining majority in networks with local interactions and very small local memory. Distributed Comput., 30(1):1-16, 2017. URL: https://doi.org/10.1007/s00446-016-0277-8.
  37. C. J. Park. Random allocations (valentin f. kolchin, boris a. sevast’yanov and vladimir p. chistyakov). Siam Review, 22:104-104, 1980. Google Scholar
  38. Marshall C. Pease, Robert E. Shostak, and Leslie Lamport. Reaching agreement in the presence of faults. J. ACM, 27(2):228-234, 1980. URL: https://doi.org/10.1145/322186.322188.
  39. Yuval Peres, Kunal Talwar, and Udi Wieder. The (1+beta)-choice process and weighted balls into bins. In SODA'2010, January 2010. URL: https://www.microsoft.com/en-us/research/publication/the-1beta-choice-process-and-weighted-balls-into-bins/.
  40. Yuval Peres, Kunal Talwar, and Udi Wieder. Graphical balanced allocations and the 1 + β-choice process. Random Struct. Algorithms, 47(4):760-775, December 2015. URL: https://doi.org/10.1002/rsa.20558.
  41. Andréa Richa, Michael Mitzenmacher, and Ramesh Sitaraman. The power of two random choices: A survey of techniques and results. Handbook of Randomized Computing, 2001. URL: https://doi.org/10.1007/978-1-4615-0013-1_9.
  42. A.S. Tanenbaum and M. van Steen. Distributed Systems. CreateSpace Independent Publishing Platform, 2017. URL: https://books.google.com/books?id=c77GAQAACAAJ.
  43. Berthold Vöcking. How asymmetry helps load balancing. In Proceedings of the 40th Annual Symposium on Foundations of Computer Science, FOCS '99, page 131, USA, 1999. IEEE Computer Society. Google Scholar
Any Issues?
X

Feedback on the Current Page

CAPTCHA

Thanks for your feedback!

Feedback submitted to Dagstuhl Publishing

Could not send message

Please try again later or send an E-mail
© 2023-2025 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact