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Approximate Majority with Catalytic Inputs

Authors Talley Amir, James Aspnes, John Lazarsfeld



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Talley Amir
  • Yale University, New Haven, CT, USA
James Aspnes
  • Yale University, New Haven, CT, USA
John Lazarsfeld
  • Yale University, New Haven, CT, USA

Acknowledgements

The authors would like to thank Anne Condon, Monir Hajiaghayi, David Kirkpatrick, and J{á}n Maňuch for a helpful discussion regarding the analysis of the DBAM protocol. The authors are also grateful for a discussion with Francesco d’Amore, Andrea Clementi, and Emanuele Natale, who pointed out the connection between catalytic agents in population protocols and stubborn agents in other types of multi-agent systems. We also thank the anonymous reviewers for their helpful feedback.

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Talley Amir, James Aspnes, and John Lazarsfeld. Approximate Majority with Catalytic Inputs. In 24th International Conference on Principles of Distributed Systems (OPODIS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 184, pp. 19:1-19:16, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)
https://doi.org/10.4230/LIPIcs.OPODIS.2020.19

Abstract

Population protocols [Dana Angluin et al., 2006] are a class of algorithms for modeling distributed computation in networks of finite-state agents communicating through pairwise interactions. Their suitability for analyzing numerous chemical processes has motivated the adaptation of the original population protocol framework to better model these chemical systems. In this paper, we further the study of two such adaptations in the context of solving approximate majority: persistent-state agents (or catalysts) and spontaneous state changes (or leaks). Based on models considered in recent protocols for populations with persistent-state agents [Bartlomiej Dudek and Adrian Kosowski, 2018; Alistarh et al., 2017; Dan Alistarh et al., 2020], we assume a population with n catalytic input agents and m worker agents, and the goal of the worker agents is to compute some predicate over the states of the catalytic inputs. We call this model the Catalytic Input (CI) model. For m = Θ(n), we show that computing the parity of the input population with high probability requires at least Ω(n²) total interactions, demonstrating a strong separation between the CI model and the standard population protocol model. On the other hand, we show that the simple third-state dynamics [Angluin et al., 2008; Perron et al., 2009] for approximate majority in the standard model can be naturally adapted to the CI model: we present such a constant-state protocol for the CI model that solves approximate majority in O(n log n) total steps with high probability when the input margin is Ω(√{n log n}). We then show the robustness of third-state dynamics protocols to the transient leaks events introduced by [Alistarh et al., 2017; Dan Alistarh et al., 2020]. In both the original and CI models, these protocols successfully compute approximate majority with high probability in the presence of leaks occurring at each step with probability β ≤ O(√{n log n}/n). The resilience of these dynamics to leaks exhibits similarities to previous work involving Byzantine agents, and we define and prove a notion of equivalence between the two.

Subject Classification

ACM Subject Classification
  • Computing methodologies → Distributed algorithms
Keywords
  • population protocols
  • approximate majority
  • catalysts
  • leaks
  • lower bound

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