2,012 research outputs found

    Time-space trade-offs in population protocols

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    Population protocols are a popular model of distributed computing, in which randomly-interacting agents with little computational power cooperate to jointly perform computational tasks. Inspired by developments in molecular computation, and in particular DNA computing, recent algorithmic work has focused on the complexity of solving simple yet fundamental tasks in the population model, such as leader election (which requires convergence to a single agent in a special "leader" state), and majority (in which agents must converge to a decision as to which of two possible initial states had higher initial count). Known results point towards an inherent trade-off between the time complexity of such algorithms, and the space complexity, i.e. size of the memory available to each agent. In this paper, we explore this trade-off and provide new upper and lower bounds for majority and leader election. First, we prove a unified lower bound, which relates the space available per node with the time complexity achievable by a protocol: for instance, our result implies that any protocol solving either of these tasks for n agents using O(log log n) states must take Ω(n/polylogn) expected time. This is the first result to characterize time complexity for protocols which employ super-constant number of states per node, and proves that fast, poly-logarithmic running times require protocols to have relatively large space costs. On the positive side, we give algorithms showing that fast, poly-logarithmic convergence time can be achieved using O(log²n) space per node, in the case of both tasks. Overall, our results highlight a time complexity separation between O (log log n) and Θ(log²n) state space size for both majority and leader election in population protocols, and introduce new techniques, which should be applicable more broadly

    Time-space trade-offs in population protocols

    Get PDF
    Population protocols are a popular model of distributed computing, in which randomly-interacting agents with little computational power cooperate to jointly perform computational tasks. Inspired by developments in molecular computation, and in particular DNA computing, recent algorithmic work has focused on the complexity of solving simple yet fundamental tasks in the population model, such as leader election (which requires convergence to a single agent in a special "leader" state), and majority (in which agents must converge to a decision as to which of two possible initial states had higher initial count). Known results point towards an inherent trade-off between the time complexity of such algorithms, and the space complexity, i.e. size of the memory available to each agent. In this paper, we explore this trade-off and provide new upper and lower bounds for majority and leader election. First, we prove a unified lower bound, which relates the space available per node with the time complexity achievable by a protocol: for instance, our result implies that any protocol solving either of these tasks for n agents using O(log log n) states must take Ω(n/polylogn) expected time. This is the first result to characterize time complexity for protocols which employ super-constant number of states per node, and proves that fast, poly-logarithmic running times require protocols to have relatively large space costs. On the positive side, we give algorithms showing that fast, poly-logarithmic convergence time can be achieved using O(log²n) space per node, in the case of both tasks. Overall, our results highlight a time complexity separation between O (log log n) and Θ(log²n) state space size for both majority and leader election in population protocols, and introduce new techniques, which should be applicable more broadly

    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

    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)

    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

    Simple and Efficient Local Codes for Distributed Stable Network Construction

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    In this work, we study protocols so that populations of distributed processes can construct networks. In order to highlight the basic principles of distributed network construction we keep the model minimal in all respects. In particular, we assume finite-state processes that all begin from the same initial state and all execute the same protocol (i.e. the system is homogeneous). Moreover, we assume pairwise interactions between the processes that are scheduled by an adversary. The only constraint on the adversary scheduler is that it must be fair. In order to allow processes to construct networks, we let them activate and deactivate their pairwise connections. When two processes interact, the protocol takes as input the states of the processes and the state of the their connection and updates all of them. Initially all connections are inactive and the goal is for the processes, after interacting and activating/deactivating connections for a while, to end up with a desired stable network. We give protocols (optimal in some cases) and lower bounds for several basic network construction problems such as spanning line, spanning ring, spanning star, and regular network. We provide proofs of correctness for all of our protocols and analyze the expected time to convergence of most of them under a uniform random scheduler that selects the next pair of interacting processes uniformly at random from all such pairs. Finally, we prove several universality results by presenting generic protocols that are capable of simulating a Turing Machine (TM) and exploiting it in order to construct a large class of networks.Comment: 43 pages, 7 figure
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