Document Open Access Logo

Dynamic Averaging Load Balancing on Arbitrary Graphs

Authors Petra Berenbrink , Lukas Hintze, Hamed Hosseinpour , Dominik Kaaser , Malin Rau

PDF
Thumbnail PDF

File

LIPIcs.ICALP.2023.18.pdf
  • Filesize: 0.84 MB
  • 18 pages

Document Identifiers

Author Details

Petra Berenbrink
  • Universität Hamburg, Germany
Lukas Hintze
  • Universität Hamburg, Germany
Hamed Hosseinpour
  • Universität Hamburg, Germany
Dominik Kaaser
  • TU Hamburg, Germany
Malin Rau
  • Universität Hamburg, Germany

Funding

  • Berenbrink, Petra: Supported by DFG Research Group ADYN (FOR 2975) under grant 411362735, and by DFG grant 427756233.
  • Hosseinpour, Hamed: Supported by DFG grant 427756233.
  • Rau, Malin: Supported by DFG Research Group ADYN (FOR 2975) under grant DFG 411362735.

Acknowledgements

We thank the anonymous reviewers for their comments.

Cite AsGet BibTex

Petra Berenbrink, Lukas Hintze, Hamed Hosseinpour, Dominik Kaaser, and Malin Rau. Dynamic Averaging Load Balancing on Arbitrary Graphs. In 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 261, pp. 18:1-18:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)
https://doi.org/10.4230/LIPIcs.ICALP.2023.18

Abstract

In this paper we study dynamic averaging load balancing on general graphs. We consider infinite time and dynamic processes, where in every step new load items are assigned to randomly chosen nodes. A matching is chosen, and the load is averaged over the edges of that matching. We analyze the discrete case where load items are indivisible, moreover our results also carry over to the continuous case where load items can be split arbitrarily. For the choice of the matchings we consider three different models, random matchings of linear size, random matchings containing only single edges, and deterministic sequences of matchings covering the whole graph. We bound the discrepancy, which is defined as the difference between the maximum and the minimum load. Our results cover a broad range of graph classes and, to the best of our knowledge, our analysis is the first result for discrete and dynamic averaging load balancing processes. As our main technical contribution we develop a drift result that allows us to apply techniques based on the effective resistance in an electrical network to the setting of dynamic load balancing.

Subject Classification

ACM Subject Classification
  • Theory of computation → Random walks and Markov chains
  • Mathematics of computing → Stochastic processes
  • Theory of computation → Distributed algorithms
Keywords
  • Dynamic Load Balancing
  • Distributed Computing
  • Randomized Algorithms
  • Drift Analysis

Metrics

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

Related Versions

References

  1. Heiner Ackermann, Simon Fischer, Martin Hoefer, and Marcel Schöngens. Distributed algorithms for QoS load balancing. Distributed Comput., 23(5-6):321-330, 2011. URL: https://doi.org/10.1007/s00446-010-0125-1.
  2. Dan Alistarh, Giorgi Nadiradze, and Amirmojtaba Sabour. Dynamic averaging load balancing on cycles. In 47th International Colloquium on Automata, Languages, and Programming, ICALP 2020, volume 168 of LIPIcs, pages 7:1-7:16. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2020. URL: https://doi.org/10.4230/LIPIcs.ICALP.2020.7.
  3. Aris Anagnostopoulos, Adam Kirsch, and Eli Upfal. Load balancing in arbitrary network topologies with stochastic adversarial input. SIAM Journal on Computing, 34(3):616-639, 2005. URL: https://doi.org/10.1137/S0097539703437831.
  4. Elliot Anshelevich, David Kempe, and Jon M. Kleinberg. Stability of load balancing algorithms in dynamic adversarial systems. SIAM J. Comput., 37(5):1656-1673, 2008. URL: https://doi.org/10.1137/050639272.
  5. Friedhelm Meyer auf der Heide, Brigitte Oesterdiekhoff, and Rolf Wanka. Strongly adaptive token distribution. Algorithmica, 15(5):413-427, 1996. URL: https://doi.org/10.1007/BF01955042.
  6. Nikhil Bansal and Ohad N. Feldheim. The power of two choices in graphical allocation. 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 52-63. ACM, 2022. URL: https://doi.org/10.1145/3519935.3519995.
  7. Petra Berenbrink, Colin Cooper, Tom Friedetzky, Tobias Friedrich, and Thomas Sauerwald. Randomized diffusion for indivisible loads. J. Comput. Syst. Sci., 81(1):159-185, 2015. URL: https://doi.org/10.1016/j.jcss.2014.04.027.
  8. Petra Berenbrink, Tom Friedetzky, and Zengjian Hu. A new analytical method for parallel, diffusion-type load balancing. J. Parallel Distributed Comput., 69(1):54-61, 2009. URL: https://doi.org/10.1016/j.jpdc.2008.05.005.
  9. Petra Berenbrink, Tom Friedetzky, Dominik Kaaser, and Peter Kling. Tight & simple load balancing. In 2019 IEEE International Parallel and Distributed Processing Symposium, IPDPS 2019, pages 718-726. IEEE, 2019. URL: https://doi.org/10.1109/IPDPS.2019.00080.
  10. Petra Berenbrink, Tom Friedetzky, and Russell A. Martin. On the stability of dynamic diffusion load balancing. Algorithmica, 50(3):329-350, 2008. URL: https://doi.org/10.1007/s00453-007-9081-y.
  11. Petra Berenbrink, Peter Kling, Christopher Liaw, and Abbas Mehrabian. Tight load balancing via randomized local search. In 2017 IEEE International Parallel and Distributed Processing Symposium, IPDPS 2017, pages 192-201. IEEE Computer Society, 2017. URL: https://doi.org/10.1109/IPDPS.2017.52.
  12. Leran Cai and Thomas Sauerwald. Randomized load balancing on networks with stochastic inputs. In 44th International Colloquium on Automata, Languages, and Programming, ICALP 2017, volume 80 of LIPIcs, pages 139:1-139:14, 2017. URL: https://doi.org/10.4230/LIPIcs.ICALP.2017.139.
  13. Fan R. K. Chung and Lincoln Lu. Survey: Concentration inequalities and martingale inequalities: A survey. Internet Math., 3(1):79-127, 2006. URL: https://doi.org/10.1080/15427951.2006.10129115.
  14. Ralf Diekmann, Andreas Frommer, and Burkhard Monien. Efficient schemes for nearest neighbor load balancing. Parallel Comput., 25(7):789-812, 1999. URL: https://doi.org/10.1016/S0167-8191(99)00018-6.
  15. Peter G. Doyle and J. Laurie Snell. Random Walks and Electric Networks. Number Book 22 in Carus Mathematical Monographs. Mathematical Association of America, Washington, DC, 1984. Google Scholar
  16. Simon Fischer, Harald Räcke, and Berthold Vöcking. Fast convergence to wardrop equilibria by adaptive sampling methods. SIAM J. Comput., 39(8):3700-3735, 2010. URL: https://doi.org/10.1137/090746720.
  17. Tobias Friedrich and Thomas Sauerwald. Near-perfect load balancing by randomized rounding. In Proceedings of the 41st Annual ACM Symposium on Theory of Computing, STOC 2009, pages 121-130. ACM, 2009. URL: https://doi.org/10.1145/1536414.1536433.
  18. Bhaskar Ghosh, Frank Thomson Leighton, Bruce M. Maggs, S. Muthukrishnan, C. Greg Plaxton, Rajmohan Rajaraman, Andréa W. Richa, Robert Endre Tarjan, and David Zuckerman. Tight analyses of two local load balancing algorithms. SIAM J. Comput., 29(1):29-64, 1999. URL: https://doi.org/10.1137/S0097539795292208.
  19. Bhaskar Ghosh and S. Muthukrishnan. Dynamic load balancing by random matchings. J. Comput. Syst. Sci., 53(3):357-370, 1996. URL: https://doi.org/10.1006/jcss.1996.0075.
  20. Bhaskar Ghosh, S. Muthukrishnan, and Martin H. Schultz. First and second order diffusive methods for rapid, coarse, distributed load balancing (extended abstract). In Proceedings of the 8th Annual ACM Symposium on Parallel Algorithms and Architectures, SPAA '96, pages 72-81. ACM, 1996. URL: https://doi.org/10.1145/237502.237509.
  21. Martin Hoefer and Thomas Sauerwald. Threshold load balancing in networks. CoRR, abs/1306.1402, 2013. URL: https://arxiv.org/abs/1306.1402.
  22. David Kempe, Alin Dobra, and Johannes Gehrke. Gossip-based computation of aggregate information. In 44th Symposium on Foundations of Computer Science (FOCS 2003), pages 482-491. IEEE Computer Society, 2003. URL: https://doi.org/10.1109/SFCS.2003.1238221.
  23. Johannes Lengler. Drift analysis. In Benjamin Doerr and Frank Neumann, editors, Theory of Evolutionary Computation - Recent Developments in Discrete Optimization, Natural Computing Series, pages 89-131. Springer, 2020. URL: https://doi.org/10.1007/978-3-030-29414-4_2.
  24. David Levin and Yuval Peres. Markov Chains and Mixing Times. AMS, 2017. URL: https://doi.org/10.1090/mbk/107.
  25. László Lovász. Random walks on graphs. Combinatorics, Paul Erdős is Eighty, 2:1-46, 1993. Google Scholar
  26. Henning Meyerhenke. Shape optimizing load balancing for mpi-parallel adaptive numerical simulations. In Graph Partitioning and Graph Clustering, 10th DIMACS Implementation Challenge Workshop, volume 588 of Contemporary Mathematics, pages 67-82. American Mathematical Society, 2012. URL: http://www.ams.org/books/conm/588/11699.
  27. Vahid Mohammadian, Nima Jafari Navimipour, Mehdi Hosseinzadeh, and Aso Mohammad Darwesh. Fault-tolerant load balancing in cloud computing: A systematic literature review. IEEE Access, 10:12714-12731, 2022. URL: https://doi.org/10.1109/ACCESS.2021.3139730.
  28. S. Muthukrishnan, Bhaskar Ghosh, and Martin H. Schultz. First- and second-order diffusive methods for rapid, coarse, distributed load balancing. Theory Comput. Syst., 31(4):331-354, 1998. URL: https://doi.org/10.1007/s002240000092.
  29. Borek Patzák and Daniel Rypl. Object-oriented, parallel finite element framework with dynamic load balancing. Adv. Eng. Softw., 47(1):35-50, 2012. URL: https://doi.org/10.1016/j.advengsoft.2011.12.008.
  30. David Peleg and Eli Upfal. The token distribution problem. SIAM J. Comput., 18(2):229-243, 1989. URL: https://doi.org/10.1137/0218015.
  31. Yuval Peres, Kunal Talwar, and Udi Wieder. Graphical balanced allocations and the (1 + β)-choice process. Random Struct. Algorithms, 47(4):760-775, 2015. URL: https://doi.org/10.1002/rsa.20558.
  32. Yuval Rabani, Alistair Sinclair, and Rolf Wanka. Local divergence of markov chains and the analysis of iterative load balancing schemes. In 39th Annual Symposium on Foundations of Computer Science, FOCS '98, pages 694-705. IEEE Computer Society, 1998. URL: https://doi.org/10.1109/SFCS.1998.743520.
  33. Thomas Sauerwald and He Sun. Tight bounds for randomized load balancing on arbitrary network topologies. In 53rd Annual IEEE Symposium on Foundations of Computer Science, FOCS 2012, pages 341-350. IEEE Computer Society, 2012. URL: https://doi.org/10.1109/FOCS.2012.86.
  34. Gengbin Zheng, Abhinav Bhatele, Esteban Meneses, and Laxmikant V. Kalé. Periodic hierarchical load balancing for large supercomputers. Int. J. High Perform. Comput. Appl., 25(4):371-385, 2011. URL: https://doi.org/10.1177/1094342010394383.
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail
© 2023-2024 Schloss Dagstuhl – LZI GmbH Imprint Privacy Contact