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 vertex-stable root components (VSRCs) as a means for circumventing this impossibility: A VSRC(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 at most out-going links to the remaining processes. We present a consensus algorithm that works correctly under a VSRC(1, 4H + 2) message adversary, where H is the dynamic causal network diameter. 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 VSRC(1, H − 1) message adversary nor a VSRC(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-uniform 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 VSRC(n, d) + MAJINF(k) message adversary, which combines VSRC(n, d) (with some small value of d, ensuring termination) with some information flow guarantee MAJINF(k) between certain VSRCs (ensuring k-agreement).Since related impossibility results reveal that a VSRC(k, d) message adversary is too strong for solving k-set agreement and that some information flow between VSRCs 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.Dynamic networks, instantiated, e.g., by wireless sensor networks, mobile ad-hoc networks and vehicle area networks, are becoming ubiquitous nowadays. The primary properties of such networks are sets of participants (called processes in the sequel) that are a priori unknown and potentially changing, timevarying connectivity between processes, and the absence of a central control. Dynamic networks is an important and very active area of research [37].Accurately modeling dynamic networks is challenging, for several reasons: First, process mobility, process crashes/recoveries, deliberate joins/leaves, and peculiarities in the low-level system design like duty-cycling (used to save energy in wireless sensor networks) make static communication topologies, as typically used in class...
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 a large body of existing work that focuses on oblivious message adversaries where the communication graphs are picked from a predefined set, we consider message adversaries where guarantees about stable periods that occur only eventually can be expressed. We reveal to what extent such eventual stability is necessary and sufficient, that is, we present the shortest period of stability that permits solving consensus, a result that should prove quite useful in systems that exhibit erratic boot-up phases or recover after repeatedly occurring, massive transient faults. Contrary to the case of longer stability periods, where we show how standard algorithmic techniques for solving consensus can be employed, the short-lived nature of the stability phase forces us to use more unusual algorithmic methods that avoid waiting explicitly for the stability period to occur. KeywordsDynamic networks, consensus, message adversary, eventual stability, short stability periods, rooted directed graphs such a system has reached normal operation mode, algorithms that just terminate when a reasonably stable period has been reached are obviously advantageous. Algorithms that work correctly under short-lived stable periods are particularly interesting, since they have higher coverage and terminate earlier in systems where longer stable periods occur only rarely or even not at all. Note that the occurrence of short-lived stability periods could be confirmed in the case of a prototype wireless sensor network [17].Last but not least, stabilizing algorithms require less reliable and, in our case, not inherently bidirectional communication underneath, hence work with cheaper and/or more energy-efficient network communication interfaces. After all, guaranteeing reliable bidirectional communication links typically incurs significant costs and/or delays and might even be impossible in adverse environments. We hence conjecture that our findings may turn out useful for applications such as mobile adhoc networks [12] with heavy interference or disasterrelief applications [15].In view of such applications, our core assumption of a synchronous system may appear somewhat unreasonable. However, it is not thanks to modern communication technology [24]: As synchronized clocks are typically required for basic communication in wireless systems anyway, e.g., for transmission scheduling and sender/receiver synchronization, global synchrony is reasonably easy to achieve: It can be integrated directly at low system levels as in 802.11 MAC+PHY [1], provided by GPS receivers, or implemented by means of network time synchronization protocols like IEEE 1588 or FTSP [16].Main contributions and paper organization: In this paper, we thoroughly answer the question of the minimal stability required for solving consensus under eventual stabilizing m...
This paper is devoted to deterministic consensus in synchronous dynamic networks with unidirectional links, which are under the control of an omniscient message adversary. Motivated by unpredictable node/system initialization times and long-lasting periods of massive transient faults, we consider message adversaries that guarantee periods of less erratic message loss only eventually: We present a tight bound of 2D +1 for the termination time of consensus under a message adversary that eventually guarantees a single vertexstable root component with dynamic network diameter D, as well as a simple algorithm that matches this bound. It effectively halves the termination time 4D + 1 achieved by an existing consensus algorithm, which also works under our message adversary. We also introduce a generalized, considerably stronger variant of our message adversary, and show that our new algorithm, unlike the existing one, still works correctly under it.
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