On Counting the Population Size

Berenbrink, Petra; Kaaser, Dominik; Radzik, Tomasz

doi:10.1145/3293611.3331631

Cited by 15 publications

(35 citation statements)

References 12 publications

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“…Berenbrink, Kaaser, and Radzik [16] independently studied the same size estimation problem as ours, obtaining stronger bounds on additive error and number of states: computing the value log n or log n (i.e., additive error < 1) with high probability, using O(log 2 n) time and O(log n log log n) states. They also show a protocol with probability 1 of correctness, using O(log 2 n) time and O(log 2 n log log n) states.…”

Section: Related Workmentioning

confidence: 82%

“…Let P be a protocol with a set I of "valid" initial configurations, where each agent's memory has a Boolean field terminated set to False in every configuration in I. 16 A configuration c of P is terminated if at least one agent in c has terminated = True. (Note the distinction with a silent configuration, where no transition can change any agent's state [13].)…”

Section: Terminationmentioning

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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Section: Related Workmentioning

confidence: 82%

Section: Terminationmentioning

confidence: 99%

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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“…Both exact [22,23] (computing n) and approximate [2,6,23,24] (computing ⌈log n⌉ or ⌊log n⌋, which gives 2 ⌈log n⌉ or 2 ⌊log n⌋ as a multiplicative factor-2 estimate of n) counting have been considered in the literature. Considering the size counting problem in a nonuniform model of population protocol is trivial since we can provide the agents the values of n or ⌈log n⌉ as advice.…”

Section: Population Size Countingmentioning

confidence: 99%

“…These algorithmic advances provide composable building blocks that simplify the (uniform) solution of other problems: compute an estimate of log n, and use this value where a nonuniform protocol would use the hardcoded constant ⌈log n⌉. We can adopt a counting technique as a black box and compose it with a nonuniform protocol through a restarting scheme [10,23,24] to obtain a uniform protocol. We explain the composition scheme in Section 4.…”

Section: Population Size Countingmentioning

confidence: 99%

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A survey of size counting in population protocols

Doty,

Eftekhari

2021

Preprint

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The population protocol model describes a network of n anonymous agents who cannot control with whom they interact. The agents collectively solve some computational problem through random pairwise interactions, each agent updating its own state in response to seeing the state of the other agent. They are equivalent to the model of chemical reaction networks, describing abstract chemical reactions such as A + B → C + D, when the latter is subject to the restriction that all reactions have two reactants and two products, and all rate constants are 1. The counting problem is that of designing a protocol so that n agents, all starting in the same state, eventually converge to states where each agent encodes in its state an exact or approximate description of population size n. In this survey paper, we describe recent algorithmic advances on the counting problem.

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A survey of size counting in population protocols

Doty

Eftekhari

2021

Theoretical Computer Science

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On Counting the Population Size

Cited by 15 publications

References 12 publications

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

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

A survey of size counting in population protocols

A survey of size counting in population protocols

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