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Dynamic Averaging Load Balancing on Cycles

We consider the following dynamic load-balancing process: given an underlying graph G with n nodes, in each step [Formula: see text] , a random edge is chosen, one unit of load is created, and placed at one of the endpoints. In the same step, assuming that loads are arbitrarily divisible, the two no...

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Autores principales: Alistarh, Dan, Nadiradze, Giorgi, Sabour, Amirmojtaba
Formato: Online Artículo Texto
Lenguaje:English
Publicado: Springer US 2021
Materias:
Acceso en línea:https://www.ncbi.nlm.nih.gov/pmc/articles/PMC8927032/
https://www.ncbi.nlm.nih.gov/pubmed/35330618
http://dx.doi.org/10.1007/s00453-021-00905-9
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author Alistarh, Dan
Nadiradze, Giorgi
Sabour, Amirmojtaba
author_facet Alistarh, Dan
Nadiradze, Giorgi
Sabour, Amirmojtaba
author_sort Alistarh, Dan
collection PubMed
description We consider the following dynamic load-balancing process: given an underlying graph G with n nodes, in each step [Formula: see text] , a random edge is chosen, one unit of load is created, and placed at one of the endpoints. In the same step, assuming that loads are arbitrarily divisible, 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. Peres et al. (Random Struct Algorithms 47(4):760–775, 2015) studied the variant of this process, where the unit of load is placed in the least loaded endpoint of the chosen edge, and the averaging is not performed. In the case of dynamic load balancing on the cycle of length n the only known upper bound on the expected gap is of order [Formula: see text] , following from the majorization argument due to the same work. In this paper, we leverage the power of averaging and provide an improved upper bound of [Formula: see text] . 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 [Formula: see text] . 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 also show that our analysis can be extended to the specific instance of Harary graphs. On the other hand, we prove that the expected second moment of the gap is lower bounded by [Formula: see text] . Additionally, we provide experimental evidence that our upper bound on the gap is tight up to a logarithmic factor.
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spelling pubmed-89270322022-03-22 Dynamic Averaging Load Balancing on Cycles Alistarh, Dan Nadiradze, Giorgi Sabour, Amirmojtaba Algorithmica Article We consider the following dynamic load-balancing process: given an underlying graph G with n nodes, in each step [Formula: see text] , a random edge is chosen, one unit of load is created, and placed at one of the endpoints. In the same step, assuming that loads are arbitrarily divisible, 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. Peres et al. (Random Struct Algorithms 47(4):760–775, 2015) studied the variant of this process, where the unit of load is placed in the least loaded endpoint of the chosen edge, and the averaging is not performed. In the case of dynamic load balancing on the cycle of length n the only known upper bound on the expected gap is of order [Formula: see text] , following from the majorization argument due to the same work. In this paper, we leverage the power of averaging and provide an improved upper bound of [Formula: see text] . 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 [Formula: see text] . 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 also show that our analysis can be extended to the specific instance of Harary graphs. On the other hand, we prove that the expected second moment of the gap is lower bounded by [Formula: see text] . Additionally, we provide experimental evidence that our upper bound on the gap is tight up to a logarithmic factor. Springer US 2021-12-24 2022 /pmc/articles/PMC8927032/ /pubmed/35330618 http://dx.doi.org/10.1007/s00453-021-00905-9 Text en © The Author(s) 2021 https://creativecommons.org/licenses/by/4.0/Open AccessThis article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/ (https://creativecommons.org/licenses/by/4.0/) .
spellingShingle Article
Alistarh, Dan
Nadiradze, Giorgi
Sabour, Amirmojtaba
Dynamic Averaging Load Balancing on Cycles
title Dynamic Averaging Load Balancing on Cycles
title_full Dynamic Averaging Load Balancing on Cycles
title_fullStr Dynamic Averaging Load Balancing on Cycles
title_full_unstemmed Dynamic Averaging Load Balancing on Cycles
title_short Dynamic Averaging Load Balancing on Cycles
title_sort dynamic averaging load balancing on cycles
topic Article
url https://www.ncbi.nlm.nih.gov/pmc/articles/PMC8927032/
https://www.ncbi.nlm.nih.gov/pubmed/35330618
http://dx.doi.org/10.1007/s00453-021-00905-9
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