Abstract 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 $$\textbf{D}$$ D 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 $$\textbf{D}$$ D , 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.

We provide a complete characterization of both uniform and non-uniform deterministic consensus solvability in distributed systems with benign process and communication faults using point-set topology. More specifically, we non-trivially extend the approach introduced by Alpern and Schneider in 1985, by introducing novel fault-aware 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. Consensus is solvable in a given model if and only if the sets of admissible executions leading to different decision values is disconnected in these topologies. By applying our approach to a wide range of different applications, we provide a topological explanation of a number of existing algorithms and impossibility results and develop several new ones, including a general equivalence of the strong and weak validity conditions.

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 𝐃 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 𝐃, 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.

A 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(loglogn), where n is the number of nodes. This is an exponential improvement on the previously best known upper bound of O(logn). 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.

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.

In 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.

We consider the problem of solving consensus using deterministic algorithms in a synchronous dynamic network with unreliable, directional point-to-point links, which are under the control of a message adversary. In contrast to the large body of existing work that focuses on message adversaries that pick the communication graphs from a predefined set of candidate graphs arbitrarily, we consider message adversaries that also allow to express eventual properties, like stable periods that occur only eventually. Such message adversaries can model systems that exhibit erratic boot-up phases or recover after repeatedly occurring, massive transient faults. We precisely determine how much eventual stability is necessary and sufficient, and provide an optimal consensus algorithm. Unlike in the case of longer stability periods, where standard algorithms can be adapted for solving consensus, different algorithmic techniques are needed in the case of short-lived stability.

We study distributed agreement in synchronous directed dynamic networks, where an omniscient message adversary controls the presence/absence of communication links. We prove that consensus is impossible under a message adversary that guarantees weak connectivity only, and introduce eventually vertex-stable source components (VSSCs) as a means for circumventing this impossibility: A VSSC ( k , d ) message adversary guarantees that, eventually, there is an interval of d consecutive rounds where every communication graph contains at most k strongly connected components consisting of the same processes (with possibly varying interconnect topology), which have no incoming links from outside processes. We present a consensus algorithm that works correctly under a VSSC ( 1 , 4 E + 2 ) message adversary, where E is the dynamic network depth. Our algorithm maintains local estimates of the communication graphs, and applies techniques for detecting network stability and univalent system configurations. Several related impossibility results and lower bounds, in particular, that neither a VSSC ( 1 , E − 1 ) message adversary nor a VSSC ( 2 , ∞ ) one allow to solve consensus, reveal that there is not much hope to deal with (much) stronger message adversaries here. However, we show that gracefully degrading consensus, which degrades to general k -set agreement in case of unfavorable network conditions, allows to cope with stronger message adversaries: We provide a k -universal k -set agreement algorithm, where the number of system-wide decision values k is not encoded in the algorithm, but rather determined by the actual power of the message adversary in a run: Our algorithm guarantees at most k decision values under a VSSC ( n , d ) + MAJINF ( k ) message adversary, which combines VSSC ( n , d ) (with some small value of d , ensuring termination) with some information flow guarantee MAJINF ( k ) between certain VSSCs (ensuring k -agreement). Since related impossibility results reveal that a VSSC ( k , d ) message adversary is too strong for solving k -set agreement and that some information flow between VSSCs is mandatory for this purpose as well, our results provide a significant step towards the exact solvability/impossibility border of general k -set agreement in directed dynamic networks. Finally, we relate (the eventually-forever-variants of) our message adversaries to failure detectors. It turns out that even though VSSC ( 1 , ∞ ) allows to solve consensus and to implement the Ω failure detector, it does not allow to implement Σ. This contrasts the fact that, in asynchronous message-passing systems with a majority of process crashes, ( Σ , Ω ) is a weakest failure detector for solving consensus. Similarly, although the message adversary VSSC ( n , d ) + MAJINF ( k ) allows to solve k -set agreement, it does not allow to implement the failure detector Σ k , which is known to be necessary for k -set agreement in asynchronous message-passing systems with a majority of process crashes. Consequently, it is not possible to adapt failure-detector-based algorithms to work in conjunction with our message adversaries.

We present a k-set agreement algorithm for synchronous dynamic distributed systems with unidirectional links controlled by an omniscient adversary. Our algorithm automatically adapts to the actual network properties: If the network is sufficiently well-connected, it solves consensus, while degrading gracefully to general k-set agreement in less well-behaved runs. The algorithm is oblivious to the maximum number of system-wide decision values k, which is bounded by the number of certain strongly connected components occurring in the dynamically changing network in a run. Related impossibility results reveal that this bound is close to the solvability border for k-set agreement. To the best of our knowledge, this is the first consensus algorithm that degrades in a graceful way in a dynamic network.