Plurality Consensus in the Gossip Model

Becchetti, Luca; Clementi, Andrea E. F.; Natale, Emanuele; Pasquale, Francesco; Silvestri, Riccardo

doi:10.1137/1.9781611973730.27

Cited by 41 publications

(86 citation statements)

References 41 publications

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“…The above result strongly improves over the best previous bounds [12][13][14] and it is almost tight, since the classical lower bound Ω(log n/ log log n) on the maximum load (see, e.g., [11]) clearly applies also in our repeated setting. Our result further implies that, under the FIFO queueing policy, any ball performs Ω(t/ log n) steps of its individual random walk over any sequence of t = poly(n) rounds w.h.p., so the parallel cover time is O n log 2 n w.h.p.…”

Section: Our Resultssupporting

confidence: 70%

“…In particular, in [13], a logarithmic bound is shown for the complete graph when m = O(n/ log n) random walks are performed over a logarithmic time interval, while a similar bound is also given for some families of almost-regular random graphs in [14]. A new analysis is given in [12] for regular graphs and time intervals of arbitrary length, yielding the bound O √ t . Finally, after the conference version of this paper [17], a probabilistic version of the Tetris process, where the number of new balls arriving at each round is a random variable with expectation λn, for some λ = λ(n) ∈ [0, 1], has been studied in [18].…”

Section: Related Workmentioning

confidence: 89%

“…The repeated balls-into-bins process was first considered in [13] (see there Lemma 2 in Subsection 3.1), [12] (see there Theorem 4.1 in Sect. 4), and [14] (see there Lemma 6 in Subsection 3.1), since it describes the process of performing parallel random walks in the (uniform) gossip model (also known as random phone-call model [15,16]) when every message can contain at most one token.…”

Section: Related Workmentioning

confidence: 99%

“…The previous approach in [12] relies on the fact that, in every round, the expected balance between the number of incoming and outgoing balls is always non-positive for every non-empty bin (notice that the expected number of incoming balls is always at most one). This may suggest viewing the process as a sort of parallel birth-death process [10].…”

Section: Overview Of the Analysismentioning

confidence: 99%

“…This may suggest viewing the process as a sort of parallel birth-death process [10]. Using this approach and with some further arguments, one can (only) get the "standard-deviation" bound O( √ t) in [12]. Our new analysis proving Theorem 1 proceeds along three main steps.…”

Section: Overview Of the Analysismentioning

confidence: 99%

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Self-stabilizing repeated balls-into-bins

et al. 2017

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We study the following synchronous process that we call repeated balls-into-bins. The process is started by assigning n balls to n bins in an arbitrary fashion. In every subsequent round, one ball is extracted from each non-empty bin according to some fixed strategy (random, FIFO, etc), and re-assigned to one of the n bins uniformly at random. We define a configuration legitimate if its maximum load is O(log n). We prove that, starting from any configuration, the process converges to a legitimate configuration in linear time and then only takes on legitimate configurations over a period of length bounded by any polynomial in n, with high probability (w.h.p.). This implies that the process is self-stabilizing and that every ball traverses all bins within O(n log 2 n) rounds, w.h.p.

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Section: Our Resultssupporting

confidence: 70%

Section: Related Workmentioning

confidence: 89%

Section: Related Workmentioning

confidence: 99%

Section: Overview Of the Analysismentioning

confidence: 99%

Section: Overview Of the Analysismentioning

confidence: 99%

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Self-stabilizing repeated balls-into-bins

et al. 2017

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On the Necessary Memory to Compute the Plurality in Multi-agent Systems

Natale

Ramezani

2019

Lecture Notes in Computer Science

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We consider the Relative-Majority Problem (also known as Plurality), in which, given a multi-agent system where each agent is initially provided an input value out of a set of k possible ones, each agent is required to eventually compute the input value with the highest frequency in the initial configuration. We consider the problem in the general Population Protocols model in which, given an underlying undirected connected graph whose nodes represent the agents, edges are selected by a globally fair scheduler. The state complexity that is required for solving the Plurality Problem (i.e., the minimum number of memory states that each agent needs to have in order to solve the problem), has been a long-standing open 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(k2 k ) states per agent, and they conjectured their upper bound to be optimal. On the other hand, under the strong assumption that agents initially agree on a total ordering of the initial input values, G ֒ asieniec et al. (2017), provided an elegant logarithmic-memory plurality protocol. In this work, we refute Salehkaleybar et al.'s conjecture, by providing a plurality protocol which employs O(k 11 ) states per agent. Central to our result is an ordering protocol which allows to leverage on the plurality protocol by G ֒ asieniec et al., of independent interest. We also provide a Ω(k 2 )-state lower bound on the necessary memory to solve the problem, proving that the Plurality Problem cannot be solved within the mere memory necessary to encode the output.

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Simple dynamics for plurality consensus

et al. 2016

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We study a plurality-consensus process in which each of n anonymous agents of a communication network initially supports a color chosen from the set [k]. Then, in every round, each agent can revise his color according to the colors currently held by a random sample of his neighbors. It is assumed that the initial color configuration exhibits a sufficiently large biass towards a fixed plurality color, that is, the number of nodes supporting the plurality color exceeds the number of nodes supporting any other color by s additional nodes. The goal is having the process to converge to the stable configuration in which all nodes support the initial plurality. We consider a basic model in which the network is a clique and the update rule (called here the 3-majority dynamics) of the process is the following: each agent looks at the colors of three random neighbors and then applies the majority rule (breaking ties uniformly). We prove that the process converges in time (Formula presented.) with high probability, provided that (Formula presented.). We then prove that our upper bound above is tight as long as (Formula presented.). This fact implies an exponential time-gap between the plurality-consensus process and the median process (see Doerr et al. in Proceedings of the 23rd annual ACM symposium on parallelism in algorithms and architectures (SPAAâ\u80\u9911), pp 149â\u80\u93158. ACM, 2011). A natural question is whether looking at more (than three) random neighbors can significantly speed up the process. We provide a negative answer to this question: in particular, we show that samples of polylogarithmic size can speed up the process by a polylogarithmic factor only

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Plurality Consensus in the Gossip Model

Cited by 41 publications

References 41 publications

Self-stabilizing repeated balls-into-bins

Self-stabilizing repeated balls-into-bins

On the Necessary Memory to Compute the Plurality in Multi-agent Systems

Simple dynamics for plurality consensus

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