A Characterization of Consensus Solvability for Closed Message Adversaries

Distributed computations in a synchronous system prone to message loss can be modeled as a game between a (deterministic) distributed algorithm versus an omniscient message adversary. The latter determines, for each round, the directed communication graph that specifies which messages can reach their destination. Message adversary definitions range from oblivious ones, which pick the communication graphs arbitrarily from a given set of candidate graphs, to general message adversaries, which are specified by the set of sequences of communication graphs (called admissible communication patterns) that they may generate. This paper provides a complete characterization of consensus solvability for closed message adversaries, where every inadmissible communication pattern has a finite prefix that makes all (infinite) extensions of this prefix inadmissible. Whereas every oblivious message adversary is closed, there are also closed message adversaries that are not oblivious. We provide a tight non-topological, purely combinatorial characterization theorem, which reduces consensus solvability to a simple condition on prefixes of the communication patterns. Our result not only non-trivially generalizes the known combinatorial characterization of the consensus solvability for oblivious message adversaries by Coulouma, Godard, and Peters (Theor. Comput. Sci., 2015), but also provides the first combinatorial characterization for this important class of message adversaries that is formulated directly on the prefixes of the communication patterns

On the Radius of Nonsplit Graphs and Information Dissemination in Dynamic Networks

International audienceA nonsplit graph is a directed graph where each pair of nodes has a common incoming neighbor. We show that the radius of such graphs is in O(log log n), where n is the number of nodes. This is an exponential improvement on the previously best known upper bound of O(log n). We then generalize the result to products of nonsplit graphs. The analysis of nonsplit graph products has direct implications in the context of distributed systems, where processes operate in rounds and communicate via message passing in each round: communication graphs in several distributed systems naturally relate to nonsplit graphs and the graph product concisely represents relaying messages in such networks. Applying our results, we obtain improved bounds on the dynamic radius of such networks, i.e., the maximum number of rounds until all processes have received a message from a common process, if all processes relay messages in each round. We finally connect the dynamic radius to lower bounds for achieving consensus in dynamic networks

Topological Characterization of Consensus Solvability in Directed Dynamic Networks

Consensus is one of the most fundamental problems in distributed computing. This paper studies the consensus problem in a synchronous dynamic directed network, in which communication is controlled by an oblivious message adversary. The question when consensus is possible in this model has already been studied thoroughly in the literature from a combinatorial perspective, and is known to be challenging. This paper presents a topological perspective on consensus solvability under oblivious message adversaries, which provides interesting new insights. Our main contribution is a topological characterization of consensus solvability, which also leads to explicit decision procedures. Our approach is based on the novel notion of a communication pseudosphere, which can be seen as the message-passing analog of the well-known standard chromatic subdivision for wait-free shared memory systems. We further push the elegance and expressiveness of the "geometric" reasoning enabled by the topological approach by dealing with uninterpreted complexes, which considerably reduce the size of the protocol complex, and by labeling facets with information flow arrows, which give an intuitive meaning to the implicit epistemic status of the faces in a protocol complex

The Time Complexity of Consensus Under Oblivious Message Adversaries

We study the problem of solving consensus in synchronous directed dynamic networks, in which communication is controlled by an oblivious message adversary that picks the communication graph to be used in a round from a fixed set of graphs ? arbitrarily. In this fundamental model, determining consensus solvability and designing efficient consensus algorithms is surprisingly difficult. Enabled by a decision procedure that is derived from a well-established previous consensus solvability characterization for a given set ?, we study, for the first time, the time complexity of solving consensus in this model: We provide both upper and lower bounds for this time complexity, and also relate it to the number of iterations required by the decision procedure. Among other results, we find that reaching consensus under an oblivious message adversary can take exponentially longer than both deciding consensus solvability and broadcasting the input value of some unknown process to all other processes

Easy Impossibility Proofs for k-Set Agreement

Abweichender Titel laut Übersetzung der Verfasserin/des VerfassersZsfassung in dt. SpracheThis thesis is concerned with impossibility results, i.e., proofs of the fact that certain classes of algorithms cannot exist. The algorithms investigated are from the field of fault-tolerant distributed computing, which is devoted to the formal study of processing entities, modeled as communicating state machines, that may possibly fail and communicate with each other by either exchanging messages or via access to a shared memory. We investigate the problem of k-set agreement, a natural generalization of consensus. While consensus concerns itself with the task in which all processes eventually have to decide on a common value that was originally some process- input value, k-set agreement allows up to k different decision values. Hence, for k = 1, k-set agreement is equivalent to consensus. Although there exist impossibility results for deterministic consensus in systems prone to failures, relying solely on combinatoric arguments that might be considered classical today, the corresponding impossibility results for k-set agreement require complex arguments from algebraic topology. Nevertheless, there has been recent research on finding "easy" or non-topological impossibility proofs for k-set agreement, which may also provide a new handle on solving some long-standing open problems like the weakest failure detector for k-set agreement in message-passing systems. The focus of this thesis lies on such non-topological impossibilities for k-set agreement. We present and discuss existing approaches and results and provide rigorous proofs for new results regarding various models and scenarios, including the important class of dynamic systems that may evolve over time.6

Characterization of consensus solvability under message adversaries

We study characterizations of consensus solvability in dynamic networks controlled by an omniscient message adversary. This model assumes a system of n distributed agents, called processes, that execute a distributed algorithm and communicate by message passing in lockstep synchronous rounds. Communication in each round is subject to a message adversary, which determines which messages are successfully delivered and which are lost, encoded via a directed communication graph. In its most general form, the message adversary can be represented by the set of infinite sequences of communication graphs, one for each round, that it may generate. Perhaps our most important theoretical result are a precise topological characterization of consensus solvability for general message adversaries and a considerably less abstract combinatorial characterization for the important class of closed message adversaries, which encompasses most of the message adversary classes for which consensus solvability characterizations existed so far. Thanks to this, we are able to state for the first time in a precise and rigorous way exactly what is the crucial property of an arbitrary message adversary that permits a consensus solution algorithm. Our arguably most important practical result is an optimal algorithm for dynamic networks that guarantee only brief stability phases along with its correctness and optimality proof. This algorithm is relatively simple and might be used as a template for solving consensus in real systems that exhibit massive transient message loss.17

Topological Characterization of Consensus under General Message Adversaries

International audienceIn this paper, we provide a rigorous characterization of consensus solvability in synchronous directed dynamic networks controlled by an arbitrary message adversary using point-set topology: We extend the approach introduced by Alpern and Schneider in 1985 by introducing two novel topologies on the space of infinite executions: the process-view topology, induced by a distance function that relies on the local view of a given process in an execution, and the minimum topology, which is induced by a distance function that focuses on the local view of the process that is the last to distinguish two executions. We establish some simple but powerful topological results, which not only lead to a topological explanation of bivalence arguments, but also provide necessary and sufficient topological conditions on the admissible graph sequences of a message adversary for solving consensus. In particular, we characterize consensus solvability in terms of connectivity of the set of admissible graph sequences. For non-compact message adversaries, which are not limit-closed in the sense that there is a convergent sequence of graph sequences whose limit is not permitted, this requires the exclusion of all "fair" and "unfair" limit sequences that coincide with the forever bivalent runs constructed in bivalence proofs. For both compact and non-compact message adversaries, we also provide tailored characterizations of consensus solvability, i.e., tight conditions for impossibility and existence of algorithms, based on the broadcastability of the connected components of the set of admissible graph sequences