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## Section: New Results

### Distributed algorithms for dynamic networks

Participants : Luciana Bezerra Arantes [correspondent] , Marjorie Bournat, Swan Dubois, Denis Jeanneau, Mohamed Hamza Kaaouachi, Sébastien Monnet, Franck Petit [correspondent] , Pierre Sens, Julien Sopena.

Nowadays, distributed systems are more and more heterogeneous and versatile. Computing units can join, leave or move inside a global infrastructure. These features require the implementation of dynamic systems, that is to say they can cope autonomously with changes in their structure in terms of physical facilities and software. It therefore becomes necessary to define, develop, and validate distributed algorithms able to managed such dynamic and large scale systems, for instance mobile ad hoc networks, (mobile) sensor networks, P2P systems, Cloud environments, robot networks, to quote only a few.

We have obtained results both on fundamental aspects of distributed algorithms and on specific emerging large-scale applications.

We study various key topics of distributed algorithms: agreement, failure detection, data dissemination and data finding in large scale systems, self-stabilization and self-* services.

#### Agreement and failure detection in dynamic Distributed Systems

Distributed systems should provide reliable and continuous services despite the failures of some of their components. A classical way for a distributed system to tolerate failures is to detect them and then to recover. It is now well recognized that the dominant factor in system unavailability lies in the failure detection phase. In 2015, we obtain the following results on failure detection:

Assuming a message-passing environment with a majority of correct processes, the necessary and sufficient information about failures for implementing a general state machine replication scheme ensuring consistency is captured by the $\Omega$ failure detector. We show in [46] that in such a message-passing environment, $\Omega$ is also the weakest failure detector to implement an eventually consistent replicated service, where replicas are expected to agree on the evolution of the service state only after some (a priori unknown) time.

We also study the k-set agreement problem is a generalization of the consensus problem where processes can decide up to k different values. Very few papers have tackled this problem in dynamic networks. Exploiting the formalism of the Time Varying Graph model, we propose in [70] a new quorum-based failure detector for solving k-set agreement in dynamic networks with asynchronous communications. We present two algorithms that implement this new failure detector using graph connectivity and message pattern assumptions. We also provide an algorithm for solving k-set agreement using our new failure detector.

We propose several algorithms to implement efficient failure detection services. We introduce in [60] the Two Windows Failure Detector (2WFD), an algorithm that provides QoS and is able to react to sudden changes in network conditions, a property that currently existing algorithms do not satisfy. We ran tests on real traces and compared the 2W-FD to state-of-the-art algorithms. Our results show that our algorithm presents the best performance in terms of speed and accuracy in unstable scenarios. In [62] , we propose a new approach towards the implementation of failure detectors for large and dynamic networks: we study reputation systems as a means to detect failures. The reputation mechanism allows efficient node cooperation via the sharing of views about other nodes. Our experimental results show that a simple prototype of a reputation-based detection service performs better than other known adaptive failure detectors, with improved flexibility. It can thus be used in a dynamic environment with a large and variable number of nodes.

#### Probabilistic Byzantine Tolerance allocation strategies in Hybrid Cloud Environments

We explore the node allocation challenges in providing probabilistic Byzantine fault tolerance in a hybrid cloud environment, consisting of nodes with varying reliability levels, compute power, and monetary cost. We consider hybrid computing architectures that combine edge nodes with cloud hosted computing. In such a system, a large fraction of the computation is performed by donated machines at the edge of the network, which significantly reduces the cost to the owner of the computation.

Considering “bag of tasks” (BoT) applications where a large computational problem is broken into a large number of independent tasks, the probabilistic Byzantine fault tolerance guarantee refers to the confidence level that the result of a given computation is correct despite potential Byzantine failures. In [36] we explore probabilistic Byzantine tolerance, in which computation tasks are replicated on dynamic replication sets whose size is determined based on ensuring probabilistic thresholds of correctness.

#### Covering problems in dynamic systems

We study covering problems (such as minimal dominating set or maximal matching) in the context of highly dynamic distributed systems. We first obtain some general results. In [48] , we first propose a new definition of this family of problems since classical ones are meaningless in such systems. We generalize the classical definition of time complexity (for static systems) to our setting. We also provided in [40] a generic tool to help the writing of impossibility proofs in dynamic distributed systems. Then, we focus on the particular case of the minimal dominating set problem. We characterize the necessary and sufficient condition to construct deterministically a minimal dominating set in a dynamic system according to our definition.

#### Self-Stabilization

Self-stabilization is a generic paradigm to tolerate transient faults (i.e., faults of finite duration) in distributed systems. Results obtained in this area by Regal members in 2015 follow.

Spanning tree construction is a well-studied problem in distributed computing for its numerous applications like routing, broadcast...Properties of the obtained trees, efficiency of the construction, and fault-tolerance guarantees are naturally at the heart of many researches. In this context, we propose in [39] a new self-stabilizing algorithm for the minimum diameter spanning tree that achieves better time and space complexity than existing solutions. Moreover, our solution tolerates a fully asynchronous adversary.

A classical way to endowed self-stabilization with (permanent) fault tolerance is confinement. That is, we ensure that the self-stabilizing system moreover ensures that the effect of permanent faults is limited to some topological areas of the system. In [27] , we propose a characterization of optimal confinement areas for a large set of spanning tree metrics in presence of Byzantine faults. In [24] , we propose a stabilizing implementation of an atomic register in presence of crash faults. By avoiding the propagation of fault effects further than a given radius, confinement is clearly a spatial approach. Another approach, called temporal, consists in recovering as quick as possible to a configuration from which some forms of safety are satisfied.

In [68] , we introduce the notion of gradual stabilization and provide a gradually self-stabilizing algorithm that solves the unison problem, i.e., the problem that consists in synchronizing logical clocks locally maintained by the processes.

#### Team of Mobile Robots

Swarm of autonomous mobile sensor devices (or, robots) recently emerged as an attractive issue in the study of dynamic distributed systems permits to assess the intrinsic difficulties of many fundamentals tasks, such as exploring or gathering in a discrete space. We consider autonomous robots that are endowed with visibility sensors (but that are otherwise unable to communicate) and motion actuators. The robots we consider are weak, i.e., they are anonymous, uniform, unable to explicitly communicate, and oblivious (they do not remember any of their past actions). Despite their weakness, those robots must collaborate to solve a collective tasks such as exploration, gathering, flocking, to quote only a few.

In [45] , we first show that it is impossible to explore any simple torus of arbitrary size with (strictly) less than four robots, even if the algorithm is probabilistic. Next, we propose an optimal (w.r.t. the number of robots) solution for the terminating exploration of torus-shaped networks by a team of $k$ such robots in the SSYNC model. The proposed algorithm is probabilistic and works for any simple torus of size $\ell ×L$, where $7\le \ell \le L$. Since the optimal number of robots is also four in rings, our result shows that increasing the number of possible symmetries in the network (due to increasing dimensions) does not necessarily come at an extra cost w.r.t. the number of robots that are necessary to solve the problem.