An asynchronous message-passing distributed algorithm for the global critical section problem

This paper considers the global $(l,k)$-CS problem which is the problem of controlling the system in such a way that, at least $l$ and at most $k$ processes must be in the CS at a time in the network. In this paper, a distributed solution is proposed in the asynchronous message-passing model. Our solution is a versatile composition method of algorithms for $l$-mutual inclusion and $k$-mutual exclusion. Its message complexity is $O(|Q|)$, where $|Q|$ is the maximum size for the quorum of a coterie used by the algorithm, which is typically $|Q| = \sqrt{n}$.Comment: This is a modified version of the conference paper in PDAA201

Brief Announcement: Loosely-stabilizing Leader Election with Polylogarithmic Convergence Time

We present a fast loosely-stabilizing leader election protocol in the population protocol model. It elects a unique leader in a poly-logarithmic time and holds the leader for a polynomial time with arbitrarily large degree in terms of parallel time, i.e, the number of steps per the population size

Move-optimal partial gathering of mobile agents in asynchronous trees

In this paper, we consider the partial gathering problem of mobile agents in asynchronous tree networks. The partial gathering problem is a generalization of the classical gathering problem, which requires that all the agents meet at the same node. The partial gathering problem requires, for a given positive integer g, that each agent should move to a node and terminate so that at least g agents should meet at each of the nodes they terminate at. The requirement for the partial gathering problem is weaker than that for the (well-investigated) classical gathering problem, and thus, we clarify the difference on the move complexity between them. We consider two multiplicity detection models: weak multiplicity detection and strong multiplicity detection models. In the weak multiplicity detection model, each agent can detect whether another agent exists at the current node or not but cannot count the exact number of the agents. In the strong multiplicity detection model, each agent can count the number of agents at the current node. In addition, we consider two token models: non-token model and removable token model. In the non-token model, agents cannot mark the nodes or the edges in any way. In the removable-token model, each agent initially leaves a token on its initial node, and agents can remove the tokens. Our contribution is as follows. First, we show that for the non-token model agents require Ω(kn) total moves to solve the partial gathering problem, where n is the number of nodes and k is the number of agents. Second, we consider the weak multiplicity detection and non-token model. In this model, for asymmetric trees, by a previous result agents can achieve the partial gathering in O(kn) total moves, which is asymptotically optimal in terms of total moves. In addition, for symmetric trees we show that there exist no algorithms to solve the partial gathering problem. Third, we consider the strong multiplicity detection and non-token model. In this model, for any trees we propose an algorithm to achieve the partial gathering in O(kn) total moves, which is asymptotically optimal in terms of total moves. At last, we consider the weak multiplicity detection and removable-token model. In this model, we propose an algorithm to achieve the partial gathering in O(gn) total moves. Note that in this model, agents require Ω(gn) total moves to solve the partial gathering problem. Hence, the second proposed algorithm is also asymptotically optimal in terms of total moves

Loosely-Stabilizing Leader Election on Arbitrary Graphs in Population Protocols Without Identifiers nor Random Numbers

In the population protocol model Angluin et al. proposed in 2004, there exists no self-stabilizing leader election protocol for complete graphs, arbitrary graphs, trees, lines, degree-bounded graphs and so on unless the protocol knows the exact number of nodes. To circumvent the impossibility, we introduced the concept of loose-stabilization in 2009, which relaxes the closure requirement of self-stabilization. A loosely-stabilizing protocol guarantees that starting from any initial configuration a system reaches a safe configuration, and after that, the system keeps its specification (e.g. the unique leader) not forever, but for a sufficiently long time (e.g. exponentially large time with respect to the number of nodes). Our previous works presented two loosely-stabilizing leader election protocols for arbitrary graphs; One uses agent identifiers and the other uses random numbers to elect a unique leader. In this paper, we present a loosely-stabilizing protocol that solves leader election on arbitrary graphs without agent identifiers nor random numbers. By the combination of virus-propagation and token-circulation, the proposed protocol achieves polynomial convergence time and exponential holding time without such external entities. Specifically, given upper bounds N and Delta of the number of nodes n and the maximum degree of nodes delta respectively, it reaches a safe configuration within O(m*n^3*d + m*N*Delta^2*log(N)) expected steps, and keeps the unique leader for Omega(N*e^N) expected steps where m is the number of edges and d is the diameter of the graph. To measure the time complexity of the protocol, we assume the uniformly random scheduler which is widely used in the field of the population protocols

Uniform deployment of mobile agents in asynchronous rings

In this paper, we consider the uniform deployment problem of mobile agents in asynchronous unidirectional rings, which requires the agents to uniformly spread in the ring. The uniform deployment problem is in striking contrast to the rendezvous problem which requires the agents to meet at the same node. While rendezvous aims to break the symmetry, uniform deployment aims to attain the symmetry. It is well known that the symmetry breaking is difficult in distributed systems and the rendezvous problem cannot be solved from some initial configurations. Hence, we are interested in clarifying what difference the uniform deployment problem has on the solvability and the number of agent moves compared to the rendezvous problem. We consider two problem settings, with knowledge of k (or n) and without knowledge of k or n where k is the number of agents and n is the number of nodes. First, we consider agents with knowledge of k (or n since k and n can be easily obtained if one of them is given). In this case, we propose two algorithms. The first algorithm solves the uniform deployment problem with termination detection. This algorithm requires O(k log n) memory space per agent, O(n) time, and O(kn) total moves. The second algorithm also solves the uniform deployment problem with termination detection. This algorithm reduces the memory space per agent to O(log n), but uses O(n log k) time, and requires O(kn) total moves. Both algorithms are asymptotically optimal in terms of total moves since there are some initial configurations such that agents re- quire Ω(kn) total moves to solve the problem. Next, we consider agents with no knowledge of k or n. In this case, we show that, when termination detection is required, there exists no algorithm to solve the uniform deployment problem. For this reason, we consider the relaxed uniform deployment problem that does not require termination detection, and we propose an algorithm to solve the relaxed uniform deployment problem. This algorithm requires O((k/l) log(n/l)) memory space per agent, O(n/l) time, and O(kn/l) total moves when the initial configuration has symmetry degree l. This means that the algorithm can solve the problem more efficiently when the initial configuration has higher symmetric degree (i.e., is closer to uniform deployment). Note that all the proposed algorithms achieve uniform deployment from any initial configuration, which is a striking difference from the rendezvous problem because the rendezvous problem is not solvable from some initial configurations