François Bonnet

Anonymous asynchronous systems: the case of failure detectors

Due to the multiplicity of loci of control, a main issue distributed systems have to cope with lies in the uncertainty on the system state created by the adversaries that are asynchrony, failures, dynamicity, mobility, etc. Considering message-passing systems, this paper considers the uncertainty created by the net effect of asynchrony and process crash failures in systems where the processes are anonymous (i.e., processes have no identity and locally execute the same algorithm). Trivially, agreement problems such as consensus, that cannot be solved in non-anonymous asynchronous systems prone to process failures, cannot be solved either if the system is anonymous. The paper investigates failure detectors that allow processes to circumvent this impossibility. It has several contributions. It first presents four failure detectors (denoted AP,

Anonymous graph exploration without collision by mobile robots

{overline{AP}}

On the road to the weakest failure detector for k-set agreement in message-passing systems☆

, AΩ, and AΣ) and show that they are the “identity-free” counterparts of perfect failure detectors, eventual leader failure detectors, and quorum failure detectors, respectively. AΣ is new and showing that AΣ and Σ have the same computability power in a non-anonymous system is not trivial. The paper also shows that the notion of failure detector reduction is related to the computation model. Then, the paper presents and proves correct a uniform anonymous consensus algorithm based on the failure detector pair (AΩ, AΣ) (“uniform” means here that not only processes have no identity, but no process is aware of the total number of processes). This new algorithm is not a simple “straightforward extension” of an algorithm designed for non-anonymous systems. To benefit from AΣ, it uses a novel broadcast facility which encapsulates an AΣ-based message exchange pattern that provides the processes with an interesting intersection property on the set of messages they have exchanged. Finally, the paper discusses the notions of failure detector hierarchy, weakest failure detector for anonymous consensus, and the implementation of identity-free failure detectors in anonymous systems.

Looking for the Weakest Failure Detector for k-Set Agreement in Message-Passing Systems: Is

Given an arbitrary partial anonymous grid (a finite grid with possibly missing vertices or edges), this paper focuses on the exploration of such a grid by a set of mobile anonymous agents (called robots). Assuming that the robots can move synchronously, but cannot communicate with each other, the aim is to design an algorithm executed by each robot that allows, as many robots as possible (let k be this maximal number), to visit infinitely often all the vertices of the grid, in such a way that no vertex hosts more than one robot at a time, and each edge is traversed by at most one robot at a time. n nThe paper addresses this problem by considering a central parameter, denoted ρ , that captures the view of each robot. More precisely, it is assumed that each robot sees the part of the grid (and its current occupation by other robots, if any) centered at the vertex it currently occupies and delimited by the radius ρ . Based on such a radius notion, a previous work has investigated the cases ρ = 0 and ρ = + ***, and shown that, while there is no solution for ρ = 0, k ≤ p *** q is a necessary and sufficient requirement when ρ = + ***, where p is the number of vertices of the grid, and q a parameter whose value depends on the actual topology of the partial grid. This paper completes our previous results by addressing the more difficult case, namely ρ = 1. It shows that k ≤ p *** 1 when q = 0, and k ≤ p *** q otherwise, is a necessary and sufficient requirement for solving the problem. More generally, the paper shows that this case is the borderline from which the considered problem can be solved.

{\it \Pi}_k

Considering autonomous mobile robots moving on a finite anonymous graph, this paper focuses on the Constrained Perpetual Graph Exploration problem (CPGE). That problem requires each robot to perpetually visit all the vertices of the graph, in such a way that no vertex hosts more than one robot at a time, and each edge is traversed by at most one robot at a time. The paper states an upper bound k on the number of robots that can be placed in the graph while keeping CPGE solvability. To make the impossibility result as strong as possible (no more than k robots can be initially placed in the graph), this upper bound is established under a strong assumption, namely, there is an omniscient daemon that is able to coordinate the robots movements at each round of the synchronous system. Interestingly, this upper bound is related to the topology of the graph. More precisely, the paper associates with each graph a labeled tree that captures the paths that have to be traversed by a single robot at a time (as if they were a simple edge). The length of the longest of these labeled paths reveals to be the key parameter to determine the upper bound k on the number of robots.

the End of the Road?

In small-world networks, each peer is connected to its closest neighbors in the network topology, as well as to additional long-range contact(s), also called shortcut(s). In 2000, Kleinberg provided asymptotic lower bounds on routing performances and showed that greedy routing in an n-peer small-world network performs in Ω(n1/3) steps when the distance to shortcuts is chosen uniformly at random, and in Θ(log2 n) when the distance to shortcuts is chosen according to a harmonic distribution in a d-dimensional mesh. Yet, we observe through experimental results that peer to peer gossip-based protocols achieving small-world topologies where shortcuts are randomly chosen, perform reasonably well in practice. n nKleinberg results are relevant for extremely large systems while systems considered in practice are usually of smaller size (they are typically made up of less than one million of peers). This paper explores the impact of Kleinberg results in the context of practical systems and small-world networks. More precisely, based on the observation that, despite the fact that the routing complexity of gossipbased small-world overlay networks is not polylogarithmic (as proved by Kleinberg), this type of networks ultimately provide reasonable results in practice. This leads us to think that the asymptotic big O() complexity alone might not always be sufficient to assess the practicality of a system whose size is typically smaller that what the one theory targets. The paper consequently proposes a refined routing complexity measure for small-world networks (namely, a recurrence formula that can be easily computed). Yet, given that Kleinberg proved that the distribution of shortcuts has a strong impact on the routing complexity (when extremely large networks are considered), arises the question of leveraging this result to improve upon current gossip-based protocols. We show that gossip-based protocols (designed for less than one million of peers) can benefit from a good approximation of Kleinberg-like small-world topologies (designed for extremely large networks). Along, are presented simulation results that demonstrate the relevance of the proposed approach.

A simple proof of the necessity of the failure detector Σ to implement an atomic register in asynchronous message-passing systems

Abstract In the k -set agreement problem, each process (in a set of n processes) proposes a value and has to decide a proposed value in such a way that at most k different values are decided. While this problem can easily be solved in asynchronous systems prone to t process crashes when k > t , it cannot be solved when k ≤ t . For several years, the failure-detector-based approach has been investigated to circumvent this impossibility. While the weakest failure detector class to solve the k -set agreement problem in read/write shared memory systems has recently been discovered (PODC 2009), the situation is different in message-passing systems where the weakest failure detector classes are known only for the extreme cases k = 1 (consensus) and k = n − 1 (set agreement). This paper presents four contributions whose aim is to help pave the way to discover the weakest failure detector class for k -set agreement in message-passing systems. These contributions are the following. (a) The first is a new failure detector class, denoted Π k , that is such that Π 1 = Σ × Ω (the weakest class for k = 1 ), and Π n − 1 = L (the weakest class for k = n − 1 ). (b) The second is an investigation of the structure of Π k that shows that Π k is the combination of two failure detector classes Σ k (that is new) and Ω k (they generalize the previous “quorums” and “eventual leaders” failure detector classes, respectively). (c) The third contribution concerns Σ k that is shown to be a necessary requirement (as far as information on failure is concerned) to solve the k -set agreement problem in message-passing systems. (d) Finally, the last contribution is a Π n − 1 -based algorithm that solves the ( n − 1 ) -set agreement problem. This algorithm provides us with a new algorithmic insight on the way the ( n − 1 ) -set agreement problem can be solved in asynchronous message-passing systems. It is hoped that these contributions will help discover the weakest failure detector class for k -set agreement in message-passing systems.

Geo-registers: An Abstraction for Spatial-Based Distributed Computing

In the k -set agreement problem, each process (in a set of n processes) proposes a value and has to decide a proposed value in such a way that at most k different values are decided. While this problem can easily be solved in asynchronous systems prone to t process crashes when k > t , it cannot be solved when k ≤ t . Since several years, the failure detector-based approach has been investigated to circumvent this impossibility. While the weakest failure detector class to solve the k -set agreement problem in read/write shared-memory systems has recently been discovered (PODC 2009), the situation is different in message-passing systems where the weakest failure detector classes are known only for the extreme cases k = 1 (consensus) and k = n *** 1 (set agreement). This paper introduces a candidate for the general case. It presents a new failure detector class, denoted