Logarithmic Expected-Time Leader Election in Population Protocol Model

Sudo, Yuichi; Ooshita, Fukuhito; Izumi, Taisuke; Kakugawa, Hirotsugu; Masuzawa, Toru

doi:10.1007/978-3-030-34992-9_26

Cited by 11 publications

(22 citation statements)

References 18 publications

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“…This lower bound holds even if we can use an arbitrarily large number of agent states and each agent knows the exact size of a population. Thus, by this lower bound, we can say that the protocols of [MST18] and [Sud+19] are optimal in terms of convergence time.…”

Section: Our Contributionmentioning

confidence: 99%

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Leader Election Requires Logarithmic Time in Population Protocols

Sudo

Masuzawa

2020

Parallel Process. Lett.

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In this paper, it is shown that any leader election problem requires logarithmic stabilization time in the population protocol model. This lower bound holds even if each agent has knowledge of the exact size of a population and we can use arbitrarily large number of agent states. This lower bound concludes that the protocol given in [Sudo et al., PODC 2019] is optimal in terms of stabilization time.

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Section: Our Contributionmentioning

confidence: 99%

“…Michail et al [MST18] gave a protocol with O(log n) parallel time but with a linear number of states. Our previous work [Sud+19] gave a protocol with O(log n) parallel time and O(log n) states. Those protocols with non-constant number…”

Section: Related Workmentioning

confidence: 99%

Leader Election Requires Logarithmic Time in Population Protocols

Sudo

Masuzawa

2020

Parallel Process. Lett.

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“…The work of this paper was inspired by recent work on nonuniform polylog time leader election/majority [4,6,2,17,15,3,39]; the fact that those protocols require an approximate size estimate is the direct motivation for seeking a protocol that can compute such an estimate (though unfortunately due to Theorem 4.1, composition of our protocol with these is not totally straightforward). Some nonuniform protocols crucially rely on an estimate of log n (e.g.…”

Section: Related Workmentioning

confidence: 99%

“…Some nonuniform protocols crucially rely on an estimate of log n (e.g. [4,2,17,15,3,39]) for correctness. Other nonuniform protocols are more robust, using the estimate merely to allow the protocol to have a finite number of states.…”

Section: Related Workmentioning

confidence: 99%

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Efficient Size Estimation and Impossibility of Termination in Uniform Dense Population Protocols

Doty

Eftekhari

2019

Proceedings of the 2019 ACM Symposium on Principles of Distributed Computing

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We study uniform population protocols: networks of anonymous agents whose pairwise interactions are chosen at random, where each agent uses an identical transition algorithm that does not depend on the population size n. Many existing polylog(n) time protocols for leader election and majority computation are nonuniform: to operate correctly, they require all agents to be initialized with an approximate estimate of n (specifically, the value log n ).Our first main result is a uniform protocol for calculating log(n) ± O(1) with high probability in O(log 2 n) time and O(log 4 n) states (O(log log n) bits of memory). The protocol is not terminating: it does not signal when the estimate is close to the true value of log n. If it could be made terminating with high probability, this would allow composition with protocols requiring a size estimate initially. We do show how our main protocol can be indirectly composed with others in a simple and elegant way, based on leaderless phase clocks, demonstrating that those protocols can in fact be made uniform.However, our second main result implies that the protocol cannot be made terminating, a consequence of a much stronger result: a uniform protocol for any task requiring more than constant time cannot be terminating even with probability bounded above 0, if infinitely many initial configurations are dense: any state present initially occupies Ω(n) agents. (In particular no leader is allowed.) Crucially, the result holds no matter the memory or time permitted.Finally, we show that with an initial leader, our size-estimation protocol can be made terminating with high probability, with the same asymptotic time and space bounds.(storing the value in every agent), using O(log 2 n) time and O(log 4 n) states. 5 This answers affirmatively open question 5 of [25]. This is done primarily by generating a sequence of geometric random variables, 6 and propagating the maximum to each agent. However, before the maximum reaches all agents they begin computation; thus we use a restart scheme similar to [29] to reset an agent's computation when it updates to a higher estimate of the max.One might hope to use this protocol as a subroutine to "uniformize" existing nonuniform protocols for leader election and majority [4,2,17,15,3]. 7 Suppose the size-estimating protocol could be made terminating, eventually producing a termination "signal" that with high probability does not appear until the size estimate has converged. This would allow composition with other protocols requiring the size estimate. It has been known since the beginning of the population protocol model [7] that termination cannot be guaranteed with probability 1. However, leader-driven protocols can be made terminating with high probability, including simulation of register machines [9] or exact population size counting [32].Our second main result, Theorem 4.1, shows that this is impossible to do with our leaderless size-estimation protocol and a very wide range of others. This answers negatively open questions 1-3 of [25]. Th...

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