Randomized Load Balancing on Networks with Stochastic Inputs

Authors Leran Cai, Thomas Sauerwald

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Leran Cai
Thomas Sauerwald

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Leran Cai and Thomas Sauerwald. Randomized Load Balancing on Networks with Stochastic Inputs. In 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 80, pp. 139:1-139:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)


Iterative load balancing algorithms for indivisible tokens have been studied intensively in the past. Complementing previous worst-case analyses, we study an average-case scenario where the load inputs are drawn from a fixed probability distribution. For cycles, tori, hypercubes and expanders, we obtain almost matching upper and lower bounds on the discrepancy, the difference between the maximum and the minimum load. Our bounds hold for a variety of probability distributions including the uniform and binomial distribution but also distributions with unbounded range such as the Poisson and geometric distribution. For graphs with slow convergence like cycles and tori, our results demonstrate a substantial difference between the convergence in the worst- and average-case. An important ingredient in our analysis is a new upper bound on the t-step transition probability of a general Markov chain, which is derived by invoking the evolving set process.
  • random walks
  • randomized algorithms
  • parallel computing


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  1. Milton Abramowitz, Irene A Stegun, et al. Handbook of mathematical functions. Applied mathematics series, 55:62, 1966. Google Scholar
  2. Dan Alistarh, Keren Censor-Hillel, and Nir Shavit. Are lock-free concurrent algorithms practically wait-free? J. ACM, 63(4):31:1-31:20, 2016. Google Scholar
  3. Aris Anagnostopoulos, Adam Kirsch, and Eli Upfal. Load balancing in arbitrary network topologies with stochastic adversarial input. SIAM J. Comput., 34(3):616-639, 2005. Google Scholar
  4. Luca Becchetti, Andrea E. F. Clementi, Emanuele Natale, Francesco Pasquale, and Luca Trevisan. Find your place: Simple distributed algorithms for community detection. In Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms (SODA'17), pages 940-959, 2017. Google Scholar
  5. Andrew C Berry. The accuracy of the gaussian approximation to the sum of independent variates. Transactions of the American Mathematical Society, 49(1):122-136, 1941. Google Scholar
  6. J. E. Boillat. Load balancing and poisson equation in a graph. Concurrency: Pract. Exper., 2:289-313, 1990. Google Scholar
  7. S. Boyd, A. Ghosh, B. Prabhakar, and D. Shah. Randomized Gossip Algorithms. IEEE Transactions on Information Theory and IEEE/ACM Transactions on Networking, 52:2508-2530, 2006. Google Scholar
  8. Leran Cai and Thomas Sauerwald. Randomized load balancing on networks with stochastic inputs, 2017. URL: http://arxiv.org/abs/1703.08702.
  9. G. Cybenko. Load balancing for distributed memory multiprocessors. J. Parallel and Distributed Comput., 7:279-301, 1989. Google Scholar
  10. Carl-Gustaf Esseen. On the Liapounoff limit of error in the theory of probability. Almqvist &Wiksell, 1942. Google Scholar
  11. 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 Journal on Computing, 29(1):29-64, 1999. Google Scholar
  12. Ashish Goel and Piotr Indyk. Stochastic load balancing and related problems. In 40th Annual Symposium on Foundations of Computer Science (FOCS), pages 579-586, 1999. Google Scholar
  13. Kenneth H. Huebner, Donald L. Dewhirst, Douglas E. Smith, and Ted G. Byrom. The Finite Element Methods for Engineers. Wiley, 2001. Google Scholar
  14. David Asher Levin, Yuval Peres, and Elizabeth Lee Wilmer. Markov chains and mixing times. American Mathematical Soc., 2009. Google Scholar
  15. Russell Lyons. Asymptotic enumeration of spanning trees. Combinatorics, Probability & Computing, 14(4):491-522, 2005. Google Scholar
  16. Marios Mavronicolas and Thomas Sauerwald. The impact of randomization in smoothing networks. Distributed Computing, 22(5-6):381-411, 2010. Google Scholar
  17. S. Muthukrishnan and Bhaskar Ghosh. Dynamic load balancing by random matchings. J. Comput. Syst. Sci., 53:357-370, 1996. Google Scholar
  18. S. Muthukrishnan, Bhaskar Ghosh, and Martin H. Schultz. First- and second-order diffusive methods for rapid, coarse, distributed load balancing. Theory Comput. Syst., 31:331-354, 1998. Google Scholar
  19. A. Panconesi and A. Srinivasan. Randomized distributed edge coloring via an extension of the Chernoff-Hoeffding bounds. SIAM Journal on Computing, 26(2):350-368, 1997. Google Scholar
  20. Alessandro Panconesi and Aravind Srinivasan. Improved distributed algorithms for coloring and network decomposition problems. In Proc. 24th Symp. Theory of Computing (STOC), pages 581-592, 1992. Google Scholar
  21. Yuval Rabani, Alistair Sinclair, and Rolf Wanka. Local divergence of Markov chains and the analysis of iterative load balancing schemes. In Proc. 39th Symp. Foundations of Computer Science (FOCS), pages 694-705, 1998. Google Scholar
  22. Peter Sanders. Analysis of nearest neighbor load balancing algorithms for random loads. Parallel Computing, 25(8):1013-1033, 1999. Google Scholar
  23. Atish Das Sarma, Danupon Nanongkai, Gopal Pandurangan, and Prasad Tetali. Distributed random walks. J. ACM, 60(1):2:1-2:31, 2013. URL: http://dx.doi.org/10.1145/2432622.2432624.
  24. Thomas Sauerwald and He Sun. Tight bounds for randomized load balancing on arbitrary network topologies. In Proc. 53rd Symp. Foundations of Computer Science (FOCS), pages 341-350, 2012. Google Scholar
  25. A.J. Sinclair and M.R. Jerrum. Approximate counting, uninform generation and rapidly mixing markov chains. Information and Computation, 82(1):93-133, 1989. Google Scholar
  26. Raghu Subramanian and Isaac D. Scherson. An analysis of diffusive load-balancing. In Proc. 6th Symp. Parallelism in Algorithms and Architectures (SPAA), pages 220-225, 1994. Google Scholar
  27. Sonesh Surana, Brighten Godfrey, Karthik Lakshminarayanan, Richard Karp, and Ion Stoica. Load balancing in dynamic structured peer-to-peer systems. Performance Evaluation, 63(3):217-240, 2006. Google Scholar
  28. Roy D. Williams. Performance of dynamic load balancing algorithms for unstructured mesh calculations. Concurrency: Practice and Experience, 3(5):457-481, 1991. Google Scholar
  29. Dongliang Zhanga, Changjun Jianga, and Shu Li. A fast adaptive load balancing method for parallel particle-based simulations. Simulation Modelling Practice and Theory, 17(6):1032-1042, 2009. Google Scholar
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