Terminating distributed construction of shapes and patterns in a fair solution of automata

Michail, Othon

doi:10.1007/s00446-017-0309-z

Cited by 12 publications

(8 citation statements)

References 46 publications

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“…In the less restrictive setting in which all nodes start from the same state (apart possibly from a unique leader and/or unique ids), not much is known. In a recent work, Michail [28] proposed a terminating protocol in which a pre-elected leader equipped with two n-counters computes an approximate count between n/2 and n in O(n log n) parallel time with high probability. The idea is to have the leader implement two competing processes, running in parallel.…”

Section: Related Workmentioning

confidence: 99%

Simple and Fast Approximate Counting and Leader Election in Populations

Michail

Spirakis

Theofilatos

2018

Lecture Notes in Computer Science

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We study the problems of leader election and population size counting for population protocols: networks of finite-state anonymous agents that interact randomly under a uniform random scheduler. We show a protocol for leader election that terminates in O(log m (n) · log 2 n) parallel time, where m is a parameter, using O(max{m, log n}) states. By adjusting the parameter m between a constant and n, we obtain a single leader election protocol whose time and space can be smoothly traded off between O(log 2 n) to O(log n) time and O(log n) to O(n) states. Finally, we give a protocol which provides an upper boundn of the size n of the population, wheren is at most n a for some a > 1. This protocol assumes the existence of a unique leader in the population and stabilizes in Θ(log n) parallel time, using constant number of states in every node, except the unique leader which is required to use Θ(log 2 n) states.information in O(log n) expected parallel time. We use this property to construct an algorithm that solves the Leader Election problem. In addition, by observing the rate of the epidemic spreading under the uniform random scheduler, we can extract valuable information about the population. This is the key idea of our Approximate Counting algorithm. Related WorkThe framework of population protocols was first introduced by Angluin et al. [1] in order to model the interactions in networks between small resource-limited mobile agents. When operating under a uniform random scheduler, population protocols are formally equivalent to a restricted version of stochastic Chemical Reaction Networks (CRNs), which model chemistry in a well-mixed solution [4]. "CRNs are widely used to describe information processing occurring in natural cellular regulatory networks, and with upcoming advances in synthetic biology, CRNs are a promising programming language for the design of artificial molecular control circuitry" [5,6]. Results in both population protocols and CRNs can be transfered to each other, owing to a formal equivalence between these models.Angluin et al. [7] showed that all predicates stably computable in population protocols (and certain generalizations of it) are semilinear. Semilinearity persists up to o(log log n) local space but not more than this [8]. Moreover, the computational power of population protocols can be increased to the commutative subclass of NSPACE(n 2 ), if we allow the processes to form connections between each other that can hold a state from a finite domain [9], or by equipping them with unique identifiers, as in [10]. For introductory texts to population protocols the interested reader is encouraged to consult [11,9] and [12] (the latter discusses population protocols and related developments as part of a more general overview of the emerging theory of dynamic networks).Optimal algorithms, regarding the time complexity of fundamental tasks in distributed networks, for example leader election and majority, is the key for many distributed problems. For instance, the help of a central coordinator ca...

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Section: Related Workmentioning

confidence: 99%

Simple and Fast Approximate Counting and Leader Election in Populations

Michail

Spirakis

Theofilatos

2018

Lecture Notes in Computer Science

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“…In the less restrictive setting in which all nodes start from a pre-determined state, Michail [23] proposed a terminating protocol in which a pre-elected leader equipped with two ncounters computes an approximate count between n/2 and n in O(n log n) parallel time with high probability. Regarding approximation rather than exact counting, Alistarh, Aspnes, Eisenstat, Gelashvili, Rivest [1] have shown a uniform protocol that in O(log n) expected time converges to an approximation n of the true size n such that with high probability 1/2 log n ≤ log n ≤ 9 log n, i.e., √ n ≤ n ≤ n 9 .…”

Section: Related Workmentioning

confidence: 99%

Exact size counting in uniform population protocols in nearly logarithmic time

Doty¹,

Eftekhari²,

Michail³

et al. 2018

Preprint

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We study population protocols: networks of anonymous agents whose pairwise interactions are chosen uniformly at random. The size counting problem is that of calculating the exact number n of agents in the population, assuming no leader (each agent starts in the same state). We give the first protocol that solves this problem in sublinear time.The protocol converges in O(log n log log n) time and uses O(n 60 ) states (O(1)+60 log n bits of memory per agent) with probability 1−O( log log n n ). The time to converge is also O(log n log log n) in expectation. Crucially, unlike most published protocols with ω(1) states, our protocol is uniform: it uses the same transition algorithm for any population size, so does not need an estimate of the population size to be embedded into the algorithm. A sub-protocol is the first uniform sublinear-time leader election population protocol, taking O(log n log log n) time and O(n 18 ) states. The state complexity of both the counting and leader election protocols can be reduced to O(n 30 ) and O(n 9 ) respectively, while increasing the time to O(log 2 n).

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“…It was shown that very complex global networks can be formed stably, despite the dynamicity of the environment. Then, Michail [15] studied a geometric variant of network constructors in which the entities can only form geometrically constrained shapes in 2D or 3D space. Another interesting hybrid dynamic network model is the one by Gmyr et al [16] in which the entities have partial control over the connections of an otherwise worst-case passively dynamic network, following the model of Kuhn et al [7].…”

Section: Introductionmentioning

confidence: 99%

On the Distributed Construction of Stable Networks in Polylogarithmic Parallel Time

2021

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We study the class of networks, which can be created in polylogarithmic parallel time by network constructors: groups of anonymous agents that interact randomly under a uniform random scheduler with the ability to form connections between each other. Starting from an empty network, the goal is to construct a stable network that belongs to a given family. We prove that the class of trees where each node has any k≥2 children can be constructed in O(logn) parallel time with high probability. We show that constructing networks that are k-regular is Ω(n) time, but a minimal relaxation to (l,k)-regular networks, where l=k−1, can be constructed in polylogarithmic parallel time for any fixed k, where k>2. We further demonstrate that when the finite-state assumption is relaxed and k is allowed to grow with n, then k=loglogn acts as a threshold above which network construction is, again, polynomial time. We use this to provide a partial characterisation of the class of polylogarithmic time network constructors.

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Terminating distributed construction of shapes and patterns in a fair solution of automata

Cited by 12 publications

References 46 publications

Simple and Fast Approximate Counting and Leader Election in Populations

Simple and Fast Approximate Counting and Leader Election in Populations

Exact size counting in uniform population protocols in nearly logarithmic time

On the Distributed Construction of Stable Networks in Polylogarithmic Parallel Time

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