Dynamic Averaging Load Balancing on Cycles

Authors Dan Alistarh, Giorgi Nadiradze, Amirmojtaba Sabour



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Author Details

Dan Alistarh
  • IST Austria, Klosterneuburg, Austria
Giorgi Nadiradze
  • IST Austria, Klosterneuburg, Austria
Amirmojtaba Sabour
  • IST Austria, Klosterneuburg, Austria

Acknowledgements

The authors sincerely thank Thomas Sauerwald and George Giakkoupis for insightful discussions, and Mohsen Ghaffari, Yuval Peres, and Udi Wieder for feedback on earlier versions of this draft. We also thank the ICALP anonymous reviewers for their very useful comments.

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Dan Alistarh, Giorgi Nadiradze, and Amirmojtaba Sabour. Dynamic Averaging Load Balancing on Cycles. In 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 168, pp. 7:1-7:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
https://doi.org/10.4230/LIPIcs.ICALP.2020.7

Abstract

We consider the following dynamic load-balancing process: given an underlying graph G with n nodes, in each step t≥ 0, one unit of load is created, and placed at a randomly chosen graph node. In the same step, the chosen node picks a random neighbor, and the two nodes balance their loads by averaging them. We are interested in the expected gap between the minimum and maximum loads at nodes as the process progresses, and its dependence on n and on the graph structure. Variants of the above graphical balanced allocation process have been studied previously by Peres, Talwar, and Wieder [Peres et al., 2015], and by Sauerwald and Sun [Sauerwald and Sun, 2015]. These authors left as open the question of characterizing the gap in the case of cycle graphs in the dynamic case, where weights are created during the algorithm’s execution. For this case, the only known upper bound is of 𝒪(n log n), following from a majorization argument due to [Peres et al., 2015], which analyzes a related graphical allocation process. In this paper, we provide an upper bound of 𝒪 (√n log n) on the expected gap of the above process for cycles of length n. We introduce a new potential analysis technique, which enables us to bound the difference in load between k-hop neighbors on the cycle, for any k ≤ n/2. We complement this with a "gap covering" argument, which bounds the maximum value of the gap by bounding its value across all possible subsets of a certain structure, and recursively bounding the gaps within each subset. We provide analytical and experimental evidence that our upper bound on the gap is tight up to a logarithmic factor.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Combinatorics
Keywords
  • Algorithms
  • Load Balancing

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References

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