# On the Necessary Memory to Compute the Plurality in Multi-Agent Systems

3 COATI - Combinatorics, Optimization and Algorithms for Telecommunications
CRISAM - Inria Sophia Antipolis - Méditerranée , Laboratoire I3S - COMRED - COMmunications, Réseaux, systèmes Embarqués et Distribués
Abstract : We consider theRelative-Majority Problem(also known asPlurality), in which, given amulti-agent system where each agent is initially provided an input value out of a set ofkpossible ones,each agent is required to eventually compute the input valuewith the highest frequency in the initialconfiguration. We consider the problem in the general Population Protocols model in which, givenan underlying undirected connected graph whose nodes represent the agents, edges are selected by aglobally fairscheduler.Thestate complexitythat is required for solving the Plurality Problem (i.e., the minimum number ofmemory states that each agent needs to have in order to solve the problem), has been a long-standingopen problem. The best protocol so far for the general multi-valued case requires polynomial memory:Salehkaleybar et al. (2015) devised a protocol that solves the problem by employing $O(k2k)$ states per agent, and they conjectured their upper bound to be optimal. On the other hand, under the strongassumption that agents initially agree on a total ordering of the initial input values, G֒asieniec et al.(2017), provided an elegant logarithmic-memory pluralityprotocol.In this work, we refute Salehkaleybar et al.’s conjecture, by providing a plurality protocol which em-ploys $O(k11)$ states per agent. Central to our result is an ordering protocol which allows to leverageon the plurality protocol by G֒asieniec et al., of independent interest. We also provide $aΩ(k2)$-state lower bound on the neccesary memory to solve the problem, proving thet the Plurality Problem cannot be solved within the mere memory necessary to encode the output.
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Conference papers

Cited literature [27 references]

https://hal.archives-ouvertes.fr/hal-02002448
Contributor : Emanuele Natale <>
Submitted on : Thursday, July 11, 2019 - 5:05:40 PM
Last modification on : Tuesday, May 26, 2020 - 6:50:59 PM

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### Citation

Emanuele Natale, Iliad Ramezani. On the Necessary Memory to Compute the Plurality in Multi-Agent Systems. CIAC 2019 - 11th International Conference Algorithms and Complexity, May 2019, Rome, Italy. pp.323-338, ⟨10.1007/978-3-030-17402-6_27⟩. ⟨hal-02002448⟩

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