In the population protocol model, many problems cannot be solved in a self-stabilizing way. However, global knowledge, such as the number of nodes in a network, sometimes allows us to design a self-stabilizing protocol for such problems. In this paper, we investigate the effect of global knowledge on the possibility of self-stabilizing population protocols in arbitrary graphs. Specifically, we clarify the solvability of the leader election problem, the ranking problem, the degree recognition problem, and the neighbor recognition problem by self-stabilizing population protocols with knowledge of the number of nodes and/or the number of edges in a network.
We propose a self-stabilizing leader election protocol on directed rings in the model of population protocols. Given an upper bound $N$ on the population size $n$, the proposed protocol elects a unique leader within $O(nN)$ expected steps starting from any configuration and uses $O(N)$ states. This convergence time is optimal if a given upper bound $N$ is asymptotically tight, i.e., $N=O(n)$.
We fully characterize self-stabilizing functions in population protocols for complete interaction graphs. In particular, we investigate self-stabilization in systems of $n$ finite state agents in which a malicious scheduler selects an arbitrary sequence of pairwise interactions under a global fairness condition. We show a necessary and sufficient condition for self-stabilization. Specifically we show that functions without certain set-theoretic conditions are impossible to compute in a self-stabilizing manner. Our main contribution is in the converse, where we construct a self-stabilizing prot…
We consider the leader election problem in population protocol models. 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 \emph{exact} size of a network) and rich computational resources (i.e. the number of states). Loose-stabilization, introduced by Sudo et al [Theoretical Computer Science, 2012], is a promising relaxed concept of self-stabilization to ad…
We consider the standard population protocol model, where (a priori) indistinguishable and anonymous agents interact in pairs according to uniformly random scheduling. The self-stabilizing leader election problem requires the protocol to converge on a single leader agent from any possible initial configuration. We initiate the study of time complexity of population protocols solving this problem in its original setting: with probability 1, in a complete communication graph. The only previously known protocol by Cai, Izumi, and Wada [Theor. Comput. Syst. 50] runs in expected parallel time $Θ(n^…
We extend the concept of monotonic searchability for self-stabilizing systems from one to multiple dimensions. A system is self-stabilizing if it can recover to a legitimate state from any initial illegal state. These kind of systems are most often used in distributed applications. Monotonic searchability provides guarantees when searching for nodes while the recovery process is going on. More precisely, if a search request started at some node $u$ succeeds in reaching its destination $v$, then all future search requests from $u$ to $v$ succeed as well. Although there already exists a self-sta…
We present a self-stabilizing protocol for an overlay network that constructs the Minimum Spanning Tree (MST) for an underlay that is modeled by a weighted tree. The weight of an overlay edge between two nodes is the weighted length of their shortest path in the tree. We rigorously prove that our protocol works correctly under asynchronous and non-FIFO message delivery. Further, the protocol stabilizes after $\mathcal{O}(N^2)$ asynchronous rounds where $N$ is the number of nodes in the overlay.
In this paper, we focus on the implementation of distributed programs in using a key-value store where the state of the nodes is stored in a replicated and partitioned data store to improve performance and reliability. Applications of such algorithms occur in weather monitoring, social media, etc. We argue that these applications should be designed to be stabilizing so that they recover from an arbitrary state to a legitimate state. Specifically, if we use a stabilizing algorithm then we can work with more efficient implementations that provide eventual consistency rather than sequential consi…
We consider congestion control in peer-to-peer distributed systems. The problem can be reduced to the following scenario: Consider a set $V$ of $n$ peers (called clients in this paper) that want to send messages to a fixed common peer (called server in this paper). We assume that each client $v \in V$ sends a message with probability $p(v) \in [0,1)$ and the server has a capacity of $σ\in \mathbb{N}$, i.e., it can recieve at most $σ$ messages per round and excess messages are dropped. The server can modify these probabilities when clients send messages. Ideally, we wish to converge to a state …
We present a loosely-stabilizing phase clock for population protocols. In the population model we are given a system of $n$ identical agents which interact in a sequence of randomly chosen pairs. Our phase clock is leaderless and it requires $O(\log n)$ states. It runs forever and is, at any point of time, in a synchronous state w.h.p. When started in an arbitrary configuration, it recovers rapidly and enters a synchronous configuration within $O(n\log n)$ interactions w.h.p. Once the clock is synchronized, it stays in a synchronous configuration for at least poly $n$ parallel time w.h.p. We u…
Searching for other participants is one of the most important operations in a distributed system. We are interested in topologies in which it is possible to route a packet in a fixed number of hops until it arrives at its destination. Given a constant $d$, this paper introduces a new self-stabilizing protocol for the $q$-ary $d$-dimensional de Bruijn graph ($q = \sqrt[d]{n}$) that is able to route any search request in at most $d$ hops w.h.p., while significantly lowering the node degree compared to the clique: We require nodes to have a degree of $\mathcal O(\sqrt[d]{n})$, which is asymptotic…
While a lot of research in distributed computing has covered solutions for self-stabilizing computing and topologies, there is far less work on self-stabilization for distributed data structures. Considering crashing peers in peer-to-peer networks, it should not be taken for granted that a distributed data structure remains intact. In this work, we present a self-stabilizing protocol for a distributed data structure called the hashed Patricia Trie (Kniesburges and Scheideler WALCOM'11) that enables efficient prefix search on a set of keys. The data structure has a wide area of applications inc…
From data centers to IoT devices to Internet-based applications, overlay networks have become an important part of modern computing. Many of these overlay networks operate in fragile environments where processes are susceptible to faults which may perturb a node's state and the network topology. Self-stabilizing overlay networks have been proposed as one way to manage these faults, promising to build or restore a particular topology from any initial configuration or after the occurrence of any transient fault. To date there have been several self-stabilizing protocols designed for overlay netw…
Self-stabilizing protocols enable distributed systems to recover correct behavior starting from any arbitrary configuration. In particular, when processors communicate by message passing, fake messages may be placed in communication links by an adversary. When the number of such fake messages is unknown, self-stabilization may require huge resources: (a) generic solutions (a.k.a. data link protocols) require unbounded resources, which makes them unrealistic to deploy and (b) specific solutions (e.g., census or tree construction) require $O(n\log n)$ or $O(Δ\log n)$ bits of memory per node, whe…
This paper shows that every leader election protocol requires logarithmic stabilization time both in expectation and with high probability in the population protocol model. This lower bound holds even if each agent has knowledge of the exact size of a population and is allowed to use an arbitrarily large number of agent states. This lower bound concludes that the protocol given in [Sudo et al., SSS 2019] is time-optimal in expectation.
Population protocols are a model of distributed computation intended for the study of networks of independent computing agents with dynamic communication structure. Each agent has a finite number of states, and communication opportunities occur nondeterministically, allowing the agents involved to change their states based on each other's states. Population protocols are often studied in terms of reaching a consensus on whether the input configuration satisfied some predicate. In the present paper we propose an alternative point of view. Instead of studying the properties of inputs that a prot…
For reaching fast and efficient self-stabilizing Byzantine pulse synchronization (SSBPS) upon the bounded-delay message-passing networks, we consider the peaceable SSBPS problem where the resource occupation in the stabilized system is required to be as sparse as possible and the stabilization of the system is required to be as fast as possible. To solve it, the decoupled absorption process and emergency process are investigated under a general framework. The constant-time two-stage absorption process and more than one kind of emergency process are provided in integrating the merits of tempora…
Population protocols are a model of distributed computation intended for the study of networks of independent computing agents with dynamic communication structure. Each agent has a finite number of states, and communication opportunities occur nondeterministically, allowing the agents involved to change their states based on each other's states. In the present paper we study unreliable models based on population protocols and their variations from the point of view of expressive power. We model the effects of message loss. We show that for a general definition of unreliable protocols with con…
In this paper, the leader election problem in the population protocol model is considered. A leader election protocol with logarithmic stabilization time is given. Given a rough knowledge m of the population size n such that m >= \log_2 n and m=O(log n), the proposed protocol guarantees that exactly one leader is elected from n agents within O(log n) parallel time in expectation and the unique leader is kept forever thereafter. The number of states per agent of the protocol is O(log n).
In this paper, we focus on the uniform bipartition problem in the population protocol model. This problem aims to divide a population into two groups of equal size. In particular, we consider the problem in the context of \emph{arbitrary} communication graphs. As a result, we clarify the solvability of the uniform bipartition problem with arbitrary communication graphs when agents in the population have designated initial states, under various assumptions such as the existence of a base station, symmetry of the protocol, and fairness of the execution. When the problem is solvable, we present p…
We present a new version of Peregrine, the tool for the analysis and parameterized verification of population protocols introduced in [Blondin et al., CAV'2018]. Population protocols are a model of computation, intensely studied by the distributed computing community, in which mobile anonymous agents interact stochastically to perform a task. Peregrine 2.0 features a novel verification engine based on the construction of stage graphs. Stage graphs are proof certificates, introduced in [Blondin et al., CAV'2020], that are typically succinct and can be independently checked. Moreover, unlike the…
We consider the problem of counting the population size in the population model. In this model, we are given a distributed system of $n$ identical agents which interact in pairs with the goal to solve a common task. In each time step, the two interacting agents are selected uniformly at random. In this paper, we consider so-called uniform protocols, where the actions of two agents upon an interaction may not depend on the population size $n$. We present two population protocols to count the size of the population: protocol Approximate, which computes with high probability either $\lfloor\log n…
Population protocols are a model of computation in which an arbitrary number of indistinguishable finite-state agents interact in pairs. The goal of the agents is to decide by stable consensus whether their initial global configuration satisfies a given property, specified as a predicate on the set of configurations. The state complexity of a predicate is the number of states of a smallest protocol that computes it. Previous work by Blondin \textit{et al.} has shown that the counting predicates $x \geq η$ have state complexity $\mathcal{O}(\log η)$ for leaderless protocols and $\mathcal{O}(\lo…
