12,189 research outputs found

    Time-Optimal Loosely-Stabilizing Leader Election in Population Protocols

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    We consider the leader election problem in the population protocol model. In pragmatic settings of population protocols, self-stabilization is a highly desired feature owing to its fault resilience and the benefit of initialization freedom. However, the design of self-stabilizing leader election is possible only under a strong assumption (i.e., the knowledge of the exact size of a network) and rich computational resource (i.e., the number of states). Loose-stabilization is a promising relaxed concept of self-stabilization to address the aforementioned issue. Loose-stabilization guarantees that starting from any configuration, the network will reach a safe configuration where a single leader exists within a short time, and thereafter it will maintain the single leader for a long time, but not necessarily forever. The main contribution of this paper is giving a time-optimal loosely-stabilizing leader election protocol. The proposed protocol with design parameter ? ? 1 attains O(? log n) parallel convergence time and ?(n^?) parallel holding time (i.e., the length of the period keeping the unique leader), both in expectation. This protocol is time-optimal in the sense of both the convergence and holding times in expectation because any loosely-stabilizing leader election protocol with the same length of the holding time is known to require ?(? log n) parallel time

    Fast Space Optimal Leader Election in Population Protocols

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    The model of population protocols refers to the growing in popularity theoretical framework suitable for studying pairwise interactions within a large collection of simple indistinguishable entities, frequently called agents. In this paper the emphasis is on the space complexity in fast leader election via population protocols governed by the random scheduler, which uniformly at random selects pairwise interactions within the population of n agents. The main result of this paper is a new fast and space optimal leader election protocol. The new protocol utilises O(log^2 n) parallel time (which is equivalent to O(n log^2 n) sequential pairwise interactions), and each agent operates on O(log log n) states. This double logarithmic space usage matches asymptotically the lower bound 1/2 log log n on the minimal number of states required by agents in any leader election algorithm with the running time o(n/polylog n). Our solution takes an advantage of the concept of phase clocks, a fundamental synchronisation and coordination tool in distributed computing. We propose a new fast and robust population protocol for initialisation of phase clocks to be run simultaneously in multiple modes and intertwined with the leader election process. We also provide the reader with the relevant formal argumentation indicating that our solution is always correct, and fast with high probability.Comment: 21 pages, 2 figures, published in SODA 2018 proceeding

    Almost Logarithmic-Time Space Optimal Leader Election in Population Protocols

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    The model of population protocols refers to a large collection of simple indistinguishable entities, frequently called {\em agents}. The agents communicate and perform computation through pairwise interactions. We study fast and space efficient leader election in population of cardinality nn governed by a random scheduler, where during each time step the scheduler uniformly at random selects for interaction exactly one pair of agents. We propose the first o(log2n)o(\log^2 n)-time leader election protocol. Our solution operates in expected parallel time O(lognloglogn)O(\log n\log\log n) which is equivalent to O(nlognloglogn)O(n \log n\log\log n) pairwise interactions. This is the fastest currently known leader election algorithm in which each agent utilises asymptotically optimal number of O(loglogn)O(\log\log n) states. The new protocol incorporates and amalgamates successfully the power of assorted {\em synthetic coins} with variable rate {\em phase clocks}

    Fast Space Optimal Leader Election in Population Protocols

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    The model of population protocols refers to the growing in popularity theoretical framework suitable for studying pairwise interactions within a large collection of simple indistinguishable entities, frequently called agents. In this paper the emphasis is on the space complexity in fast leader election via population protocols governed by the random scheduler, which uniformly at random selects pairwise interactions from the population of n agents. The main result of this paper is a new fast and space optimal leader election protocol. The new protocol operates in parallel time O(log2n) equivalent to O(n log2n) sequential pairwise interactions, in which each agent utilises O(log log n) states. This double logarithmic space utilisation matches asymptotically the lower bound [Equation] log log n on the number of states utilised by agents in any leader election algorithm with the running time [Equation], see [7]. Our solution relies on the concept of phase clocks, a fundamental synchronisation and coordination tool in the field of Distributed Computing. We propose a new fast and robust population protocol for initialisation of phase clocks to be run simultaneously in multiple modes and intertwined with the leader election process. We also provide the reader with the relevant formal argumentation indicating that our solution is always correct and fast with high probability

    Efficient size estimation and impossibility of termination in uniform dense population protocols

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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 nn. Many existing polylog(n)(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 nn (specifically, the exact value logn\lfloor \log n \rfloor). Our first main result is a uniform protocol for calculating log(n)±O(1)\log(n) \pm O(1) with high probability in O(log2n)O(\log^2 n) time and O(log4n)O(\log^4 n) states (O(loglogn)O(\log \log n) bits of memory). The protocol is converging but not terminating: it does not signal when the estimate is close to the true value of logn\log n. If it could be made terminating, this would allow composition with protocols, such as those for leader election or majority, that require a size estimate initially, to make them uniform (though with a small probability of failure). We do show how our main protocol can be indirectly composed with others in a simple and elegant way, based on the leaderless phase clock, 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)\Omega(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.Comment: Using leaderless phase cloc

    Uniform Partition in Population Protocol Model Under Weak Fairness

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    We focus on a uniform partition problem in a population protocol model. The uniform partition problem aims to divide a population into k groups of the same size, where k is a given positive integer. In the case of k=2 (called uniform bipartition), a previous work clarified space complexity under various assumptions: 1) an initialized base station (BS) or no BS, 2) weak or global fairness, 3) designated or arbitrary initial states of agents, and 4) symmetric or asymmetric protocols, except for the setting that agents execute a protocol from arbitrary initial states under weak fairness in the model with an initialized base station. In this paper, we clarify the space complexity for this remaining setting. In this setting, we prove that P states are necessary and sufficient to realize asymmetric protocols, and that P+1 states are necessary and sufficient to realize symmetric protocols, where P is the known upper bound of the number of agents. From these results and the previous work, we have clarified the solvability of the uniform bipartition for each combination of assumptions. Additionally, we newly consider an assumption on a model of a non-initialized BS and clarify solvability and space complexity in the assumption. Moreover, the results in this paper can be applied to the case that k is an arbitrary integer (called uniform k-partition)

    Stable Leader Election in Population Protocols Requires Linear Time

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    A population protocol *stably elects a leader* if, for all nn, starting from an initial configuration with nn agents each in an identical state, with probability 1 it reaches a configuration y\mathbf{y} that is correct (exactly one agent is in a special leader state \ell) and stable (every configuration reachable from y\mathbf{y} also has a single agent in state \ell). We show that any population protocol that stably elects a leader requires Ω(n)\Omega(n) expected "parallel time" --- Ω(n2)\Omega(n^2) expected total pairwise interactions --- to reach such a stable configuration. Our result also informs the understanding of the time complexity of chemical self-organization by showing an essential difficulty in generating exact quantities of molecular species quickly.Comment: accepted to Distributed Computing special issue of invited papers from DISC 2015; significantly revised proof structure and intuitive explanation

    Space-Optimal Majority in Population Protocols

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    Population protocols are a model of distributed computing, in which nn agents with limited local state interact randomly, and cooperate to collectively compute global predicates. An extensive series of papers, across different communities, has examined the computability and complexity characteristics of this model. Majority, or consensus, is a central task, in which agents need to collectively reach a decision as to which one of two states AA or BB had a higher initial count. Two complexity metrics are important: the time that a protocol requires to stabilize to an output decision, and the state space size that each agent requires. It is known that majority requires Ω(loglogn)\Omega(\log \log n) states per agent to allow for poly-logarithmic time stabilization, and that O(log2n)O(\log^2 n) states are sufficient. Thus, there is an exponential gap between the upper and lower bounds. We address this question. We provide a new lower bound of Ω(logn)\Omega(\log n) states for any protocol which stabilizes in O(n1c)O( n^{1-c} ) time, for any c>0c > 0 constant. This result is conditional on basic monotonicity and output assumptions, satisfied by all known protocols. Technically, it represents a significant departure from previous lower bounds. Instead of relying on dense configurations, we introduce a new surgery technique to construct executions which contradict the correctness of algorithms that stabilize too fast. Subsequently, our lower bound applies to general initial configurations. We give an algorithm for majority which uses O(logn)O(\log n) states, and stabilizes in O(log2n)O(\log^2 n) time. Central to the algorithm is a new leaderless phase clock, which allows nodes to synchronize in phases of Θ(nlogn)\Theta(n \log{n}) consecutive interactions using O(logn)O(\log n) states per node. We also employ our phase clock to build a leader election algorithm with O(logn)O(\log n ) states, which stabilizes in O(log2n)O(\log^2 n) time
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