Document

# Tight Bounds for Repeated Balls-Into-Bins

## File

LIPIcs.STACS.2023.45.pdf
• Filesize: 0.95 MB
• 22 pages

## Cite As

Dimitrios Los and Thomas Sauerwald. Tight Bounds for Repeated Balls-Into-Bins. In 40th International Symposium on Theoretical Aspects of Computer Science (STACS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 254, pp. 45:1-45:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)
https://doi.org/10.4230/LIPIcs.STACS.2023.45

## Abstract

We study the repeated balls-into-bins process introduced by Becchetti, Clementi, Natale, Pasquale and Posta (2019). This process starts with m balls arbitrarily distributed across n bins. At each round t = 1,2,…, one ball is selected from each non-empty bin, and then placed it into a bin chosen independently and uniformly at random. We prove the following results: - For any n ⩽ m ⩽ poly(n), we prove a lower bound of Ω(m/n ⋅ log n) on the maximum load. For the special case m = n, this matches the upper bound of 𝒪(log n), as shown in [Luca Becchetti et al., 2019]. It also provides a positive answer to the conjecture in [Luca Becchetti et al., 2019] that for m = n the maximum load is ω(log n/ log log n) at least once in a polynomially large time interval. For m ∈ [ω(n), n log n], our new lower bound disproves the conjecture in [Luca Becchetti et al., 2019] that the maximum load remains 𝒪(log n). - For any n ⩽ m ⩽ poly(n), we prove an upper bound of 𝒪(m/n ⋅ log n) on the maximum load for all steps of a polynomially large time interval. This matches our lower bound up to multiplicative constants. - For any m ⩾ n, our analysis also implies an 𝒪(m²/n) waiting time to reach a configuration with a 𝒪(m/n ⋅ log m) maximum load, even for worst-case initial distributions. - For m ⩾ n, we show that every ball visits every bin in 𝒪(m log m) rounds. For m = n, this improves the previous upper bound of 𝒪(n log² n) in [Luca Becchetti et al., 2019]. We also prove that the upper bound is tight up to multiplicative constants for any n ⩽ m ⩽ poly(n).

## Subject Classification

##### ACM Subject Classification
• Mathematics of computing → Probability and statistics
• Mathematics of computing → Discrete mathematics
• Theory of computation → Randomness, geometry and discrete structures
• Theory of computation → Design and analysis of algorithms
##### Keywords
• Repeated balls-into-bins
• self-stabilizing systems
• balanced allocations
• potential functions
• random walks

## Metrics

• Access Statistics
• Total Accesses (updated on a weekly basis)
0

## References

1. 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.
2. Jialu Bao, Marco Gaboardi, Justin Hsu, and Joseph Tassarotti. A separation logic for negative dependence. In Proceedings of the ACM on Programming Languages (POPL), volume 6, New York, NY, USA, January 2022. Association for Computing Machinery. URL: https://doi.org/10.1145/3498719.
3. Luca Becchetti, Andrea E. F. Clementi, Emanuele Natale, Francesco Pasquale, and Gustavo Posta. Self-stabilizing repeated balls-into-bins. Distributed Computing, 32(1):59-68, 2019. (earlier version in SPAA'15). URL: https://doi.org/10.1007/s00446-017-0320-4.
4. Luca Becchetti, Andrea E. F. Clementi, Emanuele Natale, Francesco Pasquale, and Luca Trevisan. Stabilizing consensus with many opinions. In Proceedings of the 27th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 620-635, Philadelphia, PA, 2016. SIAM. URL: https://doi.org/10.1137/1.9781611974331.ch46.
5. Petra Berenbrink, Artur Czumaj, Matthias Englert, Tom Friedetzky, and Lars Nagel. Multiple-choice balanced allocation in (almost) parallel. In Proceedings of the 16th International Workshop on Randomization and Computation (RANDOM), pages 411-422, Berlin Heidelberg, 2012. Springer-Verlag. URL: https://doi.org/10.1007/978-3-642-32512-0_35.
6. 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.
7. Petra Berenbrink, Tom Friedetzky, Leslie Ann Goldberg, Paul W. Goldberg, Zengjian Hu, and Russell Martin. Distributed selfish load balancing. SIAM Journal on Computing, 37(4):1163-1181, 2007. URL: https://doi.org/10.1137/060660345.
8. Petra Berenbrink, Tom Friedetzky, Peter Kling, Frederik Mallmann-Trenn, Lars Nagel, and Chris Wastell. Self-stabilizing balls and bins in batches: the power of leaky bins. Algorithmica. An International Journal in Computer Science, 80(12):3673-3703, 2018. URL: https://doi.org/10.1007/s00453-018-0411-z.
9. Petra Berenbrink, Martin Hoefer, and Thomas Sauerwald. Distributed selfish load balancing on networks. In Proceedings of the 22nd Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 1487-1497. SIAM, Philadelphia, PA, 2011.
10. Nicoletta Cancrini and Gustavo Posta. Propagation of chaos for a balls into bins model. Electronic Communications in Probability, 24:Paper No. 1, 9, 2019. URL: https://doi.org/10.1214/18-ECP204.
11. Nicoletta Cancrini and Gustavo Posta. Mixing time for the repeated balls into bins dynamics. Electronic Communications in Probability, 25:Paper No. 60, 14, 2020. URL: https://doi.org/10.1214/20-ecp338.
12. Nicoletta Cancrini and Gustavo Posta. Propagation of chaos for a general balls into bins dynamics. Electronic Journal of Probability, 26:Paper No. 23, 20, 2021. URL: https://doi.org/10.1214/21-EJP590.
13. Fan Chung and Linyuan Lu. Concentration inequalities and martingale inequalities: a survey. Internet Mathematics, 3(1):79-127, 2006. URL: http://projecteuclid.org/euclid.im/1175266369.
14. Colin Cooper. Random walks, interacting particles, dynamic networks: Randomness can be helpful. In Proceedings of the 18th International Colloquium on Structural Information and Communication Complexity (SIROCCO), volume 6796 of Lecture Notes in Computer Science, pages 1-14, Berlin, Heidelberg, 2011. Springer. URL: https://doi.org/doi.org/10.1007/978-3-642-22212-2_1.
15. Artur Czumaj, Chris Riley, and Christian Scheideler. Perfectly balanced allocation. In Proceedings of the 7th International Workshop on Randomization and Computation (RANDOM), volume 2764 of Lecture Notes in Computer Science, pages 240-251. Springer, Berlin, 2003. URL: https://doi.org/10.1007/978-3-540-45198-3_21.
16. Devdatt P. Dubhashi and Alessandro Panconesi. Concentration of Measure for the Analysis of Randomized Algorithms. Cambridge University Press, Cambridge, 2009. URL: http://www.cambridge.org/gb/knowledge/isbn/item2327542/.
17. Yehuda Hassin and David Peleg. Distributed probabilistic polling and applications to proportionate agreement. Information and Computation, 171(2):248-268, 2001. URL: https://doi.org/10.1006/inco.2001.3088.
18. Amos Israeli and Marc Jalfon. Token management schemes and random walks yield self-stabilizing mutual exclusion. In Proceedings of the 9th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 119-131, New York, NY, USA, 1990. ACM. URL: https://doi.org/10.1145/93385.93409.
19. James R. Jackson. Jobshop-like queueing systems. Management Science, 50(12 Supplement):1796-1802, December 2004. URL: https://doi.org/10.1287/mnsc.1040.0268.
20. Richard M. Karp, Michael Luby, and Friedhelm Meyer auf der Heide. Efficient PRAM simulation on a distributed memory machine. Algorithmica. An International Journal in Computer Science, 16(4-5):517-542, 1996. URL: https://doi.org/10.1007/BF01940878.
21. Frank P. Kelly. Networks of queues. Advances in Applied Probability, 8(2):416-432, 1976. URL: https://doi.org/10.2307/1425912.
22. Dimitrios Los and Thomas Sauerwald. Balanced Allocations with Incomplete Information: The Power of Two Queries. In Proceedings of the 13th Innovations in Theoretical Computer Science Conference (ITCS), volume 215 of Leibniz International Proceedings in Informatics (LIPIcs), pages 103:1-103:23, Dagstuhl, Germany, 2022. Schloss Dagstuhl - Leibniz-Zentrum für Informatik. URL: https://doi.org/10.4230/LIPIcs.ITCS.2022.103.
23. 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), pages 164-175, New York, NY, USA, 2022. Association for Computing Machinery. URL: https://doi.org/10.1145/3519270.3538428.
24. Dimitrios Los and Thomas Sauerwald. Brief announcement: Tight bounds for repeated balls-into-bins. In Proceedings of the 34th Annual ACM Symposium on Parallel Algorithms and Architectures (SPAA), pages 419-421, New York, NY, USA, 2022. ACM. URL: https://doi.org/10.1145/3490148.3538561.
25. Dimitrios Los and Thomas Sauerwald. Tight bounds for repeated balls-into-bins, 2022. URL: https://doi.org/10.48550/arXiv.2203.12400.
26. Dimitrios Los, Thomas Sauerwald, and John Sylvester. Balanced Allocations: Caching and Packing, Twinning and Thinning. In Proceedings of the 33rd Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 1847-1874, Alexandria, Virginia, 2022. SIAM. URL: https://doi.org/10.1137/1.9781611977073.74.
27. Michael Mitzenmacher, Andréa W. Richa, and Ramesh Sitaraman. The power of two random choices: a survey of techniques and results. In Handbook of randomized computing, Vol. I, II, volume 9 of Combinatorial Optimization, pages 255-312. Kluwer Acad. Publ., Dordrecht, Netherlands, 2001. URL: https://doi.org/10.1007/978-1-4615-0013-1_9.
28. Michael Mitzenmacher and Eli Upfal. Probability and computing. Cambridge University Press, Cambridge, second edition, 2017. Randomization and probabilistic techniques in algorithms and data analysis.
29. David Peleg and Eli Upfal. The token distribution problem. SIAM Journal on Computing, 18(2):229-243, 1989. URL: https://doi.org/10.1137/0218015.
30. Yuval Peres, Kunal Talwar, and Udi Wieder. Graphical balanced allocations and the (1+β)-choice process. Random Structures & Algorithms, 47(4):760-775, 2015. URL: https://doi.org/10.1002/rsa.20558.
31. Martin Raab and Angelika Steger. "Balls into bins" - a simple and tight analysis. In Proceedings of the 2nd International Workshop on Randomization and Computation (RANDOM), volume 1518, pages 159-170. Springer, Barcelona, Spain, 1998. URL: https://doi.org/10.1007/3-540-49543-6_13.
32. Udi Wieder. Hashing, load balancing and multiple choice. Foundations and Trends in Theoretical Computer Science, 12(3-4):275-379, 2017. URL: https://doi.org/10.1561/0400000070.
33. Mingxi Yin, Yuli Yang, Jen-Ming Wu, and Bingli Jiao. Opportunistic bits in short-packet communications: A finite blocklength perspective. IEEE Transactions on Communications, 69(12):8085-8099, 2021. URL: https://doi.org/10.1109/TCOMM.2021.3117606.