Leader Election in Anonymous Rings: Franklin Goes Probabilistic

We present a probabilistic leader election algorithm for anonymous, bidirectional, asynchronous rings. It is based on an algorithm from Franklin, augmented with random identity selection, hop counters to detect identity clashes, and round numbers modulo 2. As a result, the algorithm is finite-state, so that various model checking techniques can be employed to verify its correctness, that is, eventually a unique leader is elected with probability one. We also sketch a formal correctness proof of the algorithm for rings with arbitrary size

Determining Majority in Networks with Local Interactions and Very Small Local Memory

We study here the problem of determining the majority type in an arbitrary connected network, each vertex of which has initially two possible types (states). The vertices may have a few additional possible states and can interact in pairs only if they share an edge. Any (population) protocol is required to stabilize in the initial majority, i.e. its output function must interpret the local state of each vertex so that each vertex outputs the initial majority type. We first provide a protocol with 4 states per vertex that always computes the initial majority value, under any fair scheduler. Under the uniform probabilistic scheduler of pairwise interactions, we prove that our protocol stabilizes in expected polynomial time for any network and is quite fast on the clique. As we prove, this protocol is optimal, in the sense that there does not exist any population protocol that always computes majority with fewer than 4 states per vertex. However this does not rule out the existence of a protocol with 3 states per vertex that is correct with high probability (whp). To this end, we examine an elegant and very natural majority protocol with 3 states per vertex, introduced in [2] where its performance has been analyzed for the clique graph. In particular, it determines the correct initial majority type in the clique very fast and whp under the uniform probabilistic scheduler. We study the performance of this protocol in arbitrary networks. We prove that, when the two initial states are put uniformly at random on the vertices, the protocol of [2] converges to the initial majority with probability higher than the probability of converging to the initial minority. In contrast, we present an infinite family of graphs, on which the protocol of [2] can fail, i.e. it can converge to the initial minority type whp, even when the difference between the initial majority and the initial minority is n − Θ(ln n). We also present another infinite family of graphs in which the protocol of [2] takes an expected exponential time to converge. These two negative results build upon a very positive result concerning the robustness of the protocol of [2] on the clique, namely that if the initial minority is at most n7, the protocol fails with exponentially small probability. Surprisingly, the resistance of the clique to failure causes the failure in general graphs. Our techniques use new domination and coupling arguments for suitably defined processes whose dynamics capture the antagonism between the states involved

Secretive Birds: Privacy in Population Protocols

We study private computations in a system of tiny mobile agents. We consider the mobile population protocol model of Angluin et al. and ask what can be computed without ever revealing any input to a curious adversary. We show that any predicate that can be computed in the original population model can be made private through an procedure that exploits the inherent non-determinism of the mobility pattern. In short, the idea is for every mobile agent to generate, besides its actual input value, a set of wrong input values to confuse the curious adversary. To converge to the correct result, the procedure has the agents eventually eliminate the wrong values; however, the moment when this happens i s hidden from the adversary. This is achieved without jeopardizing the tiny nature of the agents: they still have very small storage size that is independent of the cardinality of the system. We present three variants of this obfuscation procedure that help compute resp ectively, remainder, threshold, and or predicates which, when com posed, cover all those that can be computed in the population protocol model

Eventual election of multiple leaders for solving consensus in anonymous systems

In classical distributed systems, each process has a unique identity. Today, new distributed systems have emerged where a unique identity is not always possible to be assigned to each process. For example, in many sensor networks a unique identity is not possible to be included in each device due to its small storage capacity, reduced computational power, or the huge number of devices to be identified. In these cases, we have to work with anonymous distributed systems where processes cannot be identified. Consensus cannot be solved in classical and anonymous asynchronous distributed systems where processes can crash. To bypass this impossibility result, failure detectors are added to these systems. It is known that ? is the weakest failure detector class for solving consensus in classical asynchronous systems when amajority of processes never crashes. Although A? was introduced as an anonymous version of ?, to find the weakest failure detector in anonymous systems to solve consensus when amajority of processes never crashes is nowadays an open question. Furthermore, A? has the important drawback that it is not implementable. Very recently, A? has been introduced as a counterpart of ? for anonymous systems. In this paper, we show that the A? failure detector class is strictly weaker than A? (i.e., A? provides less information about process crashes than A?). We also present in this paper the first implementation of A? (hence, we also show that A? is implementable), and, finally, we include the first implementation of consensus in anonymous asynchronous systems augmented with A? and where a majority of processes does not crash

Stabilizing leader election in population protocols

In this paper we address the stabilizing leader election problem in the population protocols model augmented with oracles. Population protocols is a recent model of computation that captures the interactions of biological systems. In this model emergent global behavior is observed while anonymous finite-state agents(nodes) perform local peer interactions. Uniform self-stabilizing leader election is impossible in such systems without additional assumptions. Therefore, the classical model has been augmented with the eventual leader detector, Omega?, that eventually detects the presence or absence of a leader. In the augmented model several solutions for leader election in rings and complete networks have been proposed. In this work we extend the study to trees and arbitrary topologies. We propose deterministic and probabilistic solutions. All the proposed algorithms are memory optimal --- they need only one memory bit per agent. Additionally, we prove the necessity of the eventual leader detector even in environments helped by randomization

Probabilistic verification of hierarchical leader election protocols in dynamic systems

Leader election protocols are fundamental for coordination problems—such as consensus—in distributed computing. Recently, hierarchical leader election protocols have been proposed for dynamic systems where processes can dynamically join and leave, and no process has global information. However, quantitative analysis of such protocols is generally lacking. In this paper, we present a probabilistic model checking based approach to verify quantitative properties of these protocols. Particularly, we employ the compositional technique in the style of assume-guarantee reasoning such that the sub-protocols for each of the two layers are verified separately and the correctness of the whole protocol is guaranteed by the assume-guarantee rules. Moreover, within this framework we also augment the proposed model with additional features such as rewards. This allows the analysis of time or energy consumption of the protocol. Experiments have been conducted to demonstrate the effectiveness of our approach

Loosely-Stabilizing Leader Election with Polylogarithmic Convergence Time

A loosely-stabilizing leader election protocol with polylogarithmic convergence time in the population protocol model is presented in this paper. In the population protocol model, which is a common abstract model of mobile sensor networks, it is known to be impossible to design a self-stabilizing leader election protocol. Thus, in our prior work, we introduced the concept of loose-stabilization, which is weaker than self-stabilization but has similar advantage as self-stabilization in practice. Following this work, several loosely-stabilizing leader election protocols are presented. The loosely-stabilizing leader election guarantees that, starting from an arbitrary configuration, the system reaches a safe configuration with a single leader within a relatively short time, and keeps the unique leader for an sufficiently long time thereafter. The convergence times of all the existing loosely-stabilizing protocols, i.e., the expected time to reach a safe configuration, are polynomial in n where n is the number of nodes (while the holding times to keep the unique leader are exponential in n). In this paper, a loosely-stabilizing protocol with polylogarithmic convergence time is presented. Its holding time is not exponential, but arbitrarily large polynomial in n

Population protocols with faulty interactions: The impact of a leader

We consider the problem of simulating traditional popula-tion protocols under weaker models of communication, which include one-way interactions (as opposed to two-way interactions) and omission faults (i.e., failure by an agent to read its partner’s state during an inter-action), which in turn may be detectable or undetectable. We focus on the impact of a leader, and we give a complete characterization of the models in which the presence of a unique leader in the system allows the construction of simulators: when simulations are possible, we give explicit protocols; when they are not, we give proofs of impossibility. Specifically, if each agent has only a finite amount of memory, the simulation is pos-sible only if there are no omission faults. If agents have an unbounded amount of memory, the simulation is possible as long as omissions are detectable. If an upper bound on the number of omissions involving the leader is known, the simulation is always possible, except in the one-way model in which one side is unable to detect the interaction