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On the Hierarchy of Distributed Majority Protocols

Authors Petra Berenbrink , Amin Coja-Oghlan , Oliver Gebhard , Max Hahn-Klimroth , Dominik Kaaser , Malin Rau

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ACM Subject Classification
  • Theory of computation → Distributed algorithms
  • Theory of computation → Random walks and Markov chains
  • Mathematics of computing → Stochastic processes
Keywords
  • Consensus
  • Majority
  • Hierarchy
  • Stochastic Dominance
  • Population Protocols
  • Gossip Model
  • Strassen’s Theorem

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Abstract

We study the consensus problem among n agents, defined as follows. Initially, each agent holds one of two possible opinions. The goal is to reach a consensus configuration in which every agent shares the same opinion. To this end, agents randomly sample other agents and update their opinion according to a simple update function depending on the sampled opinions.
We consider two communication models: the gossip model and a variant of the population model. In the gossip model, agents are activated in parallel, synchronous rounds. In the population model, one agent is activated after the other in a sequence of discrete time steps. For both models we analyze the following natural family of majority processes called j-Majority: when activated, every agent samples j other agents uniformly at random (with replacement) and adopts the majority opinion among the sample (breaking ties uniformly at random). As our main result we show a hierarchy among majority protocols: (j+1)-Majority (for j > 1) converges stochastically faster than j-Majority for any initial opinion configuration. In our analysis we use Strassen’s Theorem to prove the existence of a coupling. This gives an affirmative answer for the case of two opinions to an open question asked by Berenbrink et al. [PODC 2017].

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Petra Berenbrink, Amin Coja-Oghlan, Oliver Gebhard, Max Hahn-Klimroth, Dominik Kaaser, and Malin Rau. On the Hierarchy of Distributed Majority Protocols. In 26th International Conference on Principles of Distributed Systems (OPODIS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 253, pp. 23:1-23:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023) https://doi.org/10.4230/LIPIcs.OPODIS.2022.23

Author Details

Petra Berenbrink
  • Universität Hamburg, Germany
Amin Coja-Oghlan
  • TU Dortmund University, Germany
Oliver Gebhard
  • TU Dortmund University, Germany
Max Hahn-Klimroth
  • TU Dortmund University, Germany
Dominik Kaaser
  • TU Hamburg, Germany
Malin Rau
  • Universität Hamburg, Germany

Funding

  • Berenbrink, Petra: DFG FOR 2975
  • Coja-Oghlan, Amin: DFG FOR 2975
  • Gebhard, Oliver: DFG CO 646/3
  • Hahn-Klimroth, Max: DFG FOR 2975
  • Rau, Malin: DFG FOR 2975

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