Population protocol

Abstract: Population protocols are used as a theoretical model for a collection (or population) of tiny mobile agents that interact with one another to carry out a computation. The agents are identically programmed finite state machines. Input values are initially distributed to the agents, and pairs of agents can exchange state information with other agents when they are close together. The movement pattern of the agents is unpredictable, but subject to some fairness constraints, and computations must eventually converge to the correct output value in any schedule that results from that movement. This framework can be used to model mobile ad hoc networks of tiny devices or collections of molecules undergoing chemical reactions. This chapter surveys results that describe what can be computed in various versions of the population protocol model.

Abstract: This paper considers the self-stabilizing leader-election problem in a model of interacting anonymous finite-state agents. Leader election is a fundamental problem in distributed systems; many distributed problems are easily solved with the help of a central coordinator. Self-stabilizing algorithms do not require initialization in order to operate correctly and can recover from transient faults that obliterate all state information in the system. Anonymous finite-state agents model systems of identical simple computational nodes such as sensor networks and biological computers. Self-stabilizing leader election is easily shown to be impossible in such systems without additional structure. An eventual leader detector Ω? is an oracle that eventually detects the presence or absence of a leader. With the help of Ω?, uniform self-stabilizing leader election algorithms are presented for two natural classes of network graphs: complete graphs and rings. The first algorithm works under either a local or global fairness condition, whereas the second requires global fairness. With only local fairness, uniform self-stabilizing leader election in rings is impossible, even with the help of Ω?.

Abstract: We consider a scenario in which anonymous, finite-state sensing devices are deployed in an ad-hoc communication network of arbitrary size and unknown topology, and explore what properties of the network graph can be stably computed by the devices. We show that they can detect whether the network has degree bounded by a constant d, and, if so, organize a computation that achieves asymptotically optimal linear memory use. We define a model of stabilizing inputs to such devices and show that a large class of predicates of the multiset of final input values are stably computable in any weakly-connected network. We also show that nondeterminism in the transition function does not increase the class of stably computable predicates.

Abstract: 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.