Timing in chemical reaction networks

Doty, David

doi:10.1137/1.9781611973402.57

Cited by 59 publications

(73 citation statements)

References 27 publications

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“…This implies a contradiction with the correctness of the computation. Technically, our argument employs the method of bounded differences to obtain a stronger version of the main density theorem of [Dot14], and develops a new technical characterization of the stable states which can be reached by a protocol, which does not require constant bounds on state space size, generalizing upon [DS15]. The argument provides a unified analysis: the bounds for each task in turn are corollaries of the main theorem characterizing the existence of certain stable configurations.…”

Section: Introductionmentioning

confidence: 99%

Time-Space Trade-offs in Population Protocols

Alistarh

Aspnes²,

Eisenstat

et al. 2017

Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms

252

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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(log 2 n) space per node, in the case of both tasks. Overall, our results highlight a time complexity separation between O(log log n) and Θ(log 2 n) state space size for both majority and leader election in population protocols, and introduce new techniques, which should be applicable more broadly.

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Section: Introductionmentioning

confidence: 99%

Time-Space Trade-offs in Population Protocols

Alistarh

Aspnes²,

Eisenstat

et al. 2017

Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms

252

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“…The interest for this dynamics touches areas beyond the borders of computer science. It appears to play a major role in important biological processes modelled as so-called chemical reaction networks [8,17].…”

Section: Introductionmentioning

confidence: 99%

Plurality Consensus in the Gossip Model

Becchetti¹,

Clementi²,

Natale³

et al. 2014

Proceedings of the Twenty-Sixth Annual ACM-SIAM Symposium on Discrete Algorithms

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We study Plurality Consensus in the GOSSIP Model over a network of n anonymous agents. Each agent supports an initial opinion or color. We assume that at the onset, the number of agents supporting the plurality color exceeds that of the agents supporting any other color by a sufficiently-large bias, though the initial plurality itself might be very far from absolute majority. The goal is to provide a protocol that, with high probability, brings the system into the configuration in which all agents support the (initial) plurality color.We consider the Undecided-State Dynamics, a wellknown protocol which uses just one more state (the undecided one) than those necessary to store colors.We show that the speed of convergence of this protocol depends on the initial color configuration as a whole, not just on the gap between the plurality and the second largest color community. This dependence is best captured by a novel notion we introduce, namely, the monochromatic distance md(c) which measures the distance of the initial color configurationc from the closest monochromatic one. In the complete graph, we prove that, for a wide range of the input parameters, this dynamics converges within O(md(c) log n) rounds. We prove that this upper bound is almost tight in the strong sense: Starting from any color configurationc, the convergence time is Ω(md(c)).Finally, we adapt the Undecided-State Dynamics to obtain a fast, random walk-based protocol for plurality consensus on regular expanders. This protocol converges in O(md(c) polylog(n)) rounds using only polylog(n) local memory. A key-ingredient to achieve the above bounds is a new analysis of the maximum node congestion that results from performing n parallel random walks on regular expanders.All our bounds hold with high probability.

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“…Interestingly, 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 wellmixed solution (see, e.g., [41]). "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" [12,21]. However, CRNs and population protocols can only capture the dynamics of molecular counts and not of structure formation.…”

Section: Further Related Workmentioning

confidence: 99%

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

Michail

2017

Distrib. Comput.

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In this work, we consider a solution of automata (or nodes) that move passively in a well-mixed solution without being capable of controlling their movement. Nodes can cooperate by interacting in pairs and every such interaction may result in an update of their local states. Additionally, the nodes may also choose to connect to each other in order to start forming some required structure. Such nodes can be thought of as small programmable pieces of matter, like tiny nanorobots or programmable molecules. The model that we introduce here is a more applied version of network constructors, imposing physical (or geometric) constraints on the connections that the nodes are allowed to form. Each node can connect to other nodes only via a very limited number of local ports. Connections are always made at unit distance and are perpendicular to connections of neighboring ports, which makes the model capable of forming 2D or 3D shapes. We provide direct constructors for some basic shape construction problems, like spanning line, spanning square, and self-replication. We then develop new techniques for determining the computational and constructive capabilities of our model. One of the main novelties of our approach is that of exploiting the assumptions that the system is well-mixed and has a unique leader, in order to give terminating protocols that are correct with high probability. This allows us to develop terminating subroutines that can be sequentially composed to form larger modular protocols. One of our main results is a terminating protocol counting the size n of the system with high probability. We then use this protocol as a subroutine in order to develop our universal constructors, establishing that it is possible for the nodes to become self-organized with high probability into arbitrarily complex shapes while still detecting termination of the construction.

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Timing in chemical reaction networks

Cited by 59 publications

References 27 publications

Time-Space Trade-offs in Population Protocols

Time-Space Trade-offs in Population Protocols

Plurality Consensus in the Gossip Model

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

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