Consider a complete communication network of @math nodes, where the nodes receive a common clock pulse. We study the synchronous @math -counting problem: given any starting state and up to @math faulty nodes with arbitrary behaviour, the task is to eventually have all correct nodes labeling the pulses with increasing values modulo @math in agreement. Thus, we are considering algorithms that are self-stabilising despite Byzantine failures. In this work, we give new algorithms for the synchronous counting problem that (1) are deterministic, (2) have optimal resilience, (3) have a linear stabilisation time in @math (asymptotically optimal), (4) use a small number of states, and consequently, (5) communicate a small number of bits per round. Prior algorithms either resort to randomisation, use a large number of states and need high communication bandwidth, or have suboptimal resilience. In particular, we achieve an exponential improvement in both state complexity and message size for deterministic algorithms. Moreover, we present two complementary approaches for reducing the number of bits communicated during and after stabilisation.
In the case of algorithms, algorithm synthesis has been used for computer-aided design of optimal algorithms with resilience @math , but the approach does not scale due to the extremely fast-growing space of possible algorithms @cite_2 . In general, many fast-stabilising algorithms build on a connection between Byzantine consensus and synchronous counting, but require a large number of states per node @cite_11 due to, e.g., running a large number of consensus instances in parallel. Recently, in one of the preliminary conference reports @cite_0 this paper is based on, we outlined a recursive approach where each node needs to participate in only @math parallel instances of consensus. However, this approach resulted in suboptimal resilience of @math .
@cite_0 @cite_11 @cite_2
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Consider a complete communication network of n nodes, in which the nodes receive a common clock pulse. We study the synchronous c-counting problem: given any starting state and up to f faulty nodes with arbitrary behaviour, the task is to eventually have all correct nodes count modulo c in agreement. Thus, we are considering algorithms that are self-stabilising despite Byzantine failures. In this work, we give new algorithms for the synchronous counting problem that (1) are deterministic, (2) have linear stabilisation time in f, (3) use a small number of states, and (4) achieve almost-optimal resilience. Prior algorithms either resort to randomisation, use a large number of states, or have poor resilience. In particular, we achieve an exponential improvement in the state complexity of deterministic algorithms, while still achieving linear stabilisation time and almost-linear resilience. Consider a distributed network of n nodes that is connected to a global source of "beats". All nodes receive the "beats" simultaneously, and operate in lock-step. A scheme that produces a "pulse" every Cycle beats is shown. That is, the nodes agree on "special beats", which are spaced Cycle beats apart. Given such a scheme, a clock synchronization algorithm is built. The "pulsing" scheme is self-stabilized despite any transient faults and the continuous presence of up to f < n 3 Byzantine nodes. Therefore, the clock synchronization built on top of the "pulse" is highly fault tolerant. In addition, a highly fault tolerant general stabilizer algorithm is constructed on top of the "pulse" mechanism. Previous clock synchronization solutions, operating in the exact same model as this one, either support f < n 4 and converge in linear time, or support f < n 3 and have exponential convergence time that also depends on the value of max-clock (the clock wrap around value). The proposed scheme combines the best of both worlds: it converges in linear time that is independent of max-clock and is tolerant to up to f < n 3 Byzantine nodes. Moreover, considering problems in a self-stabilizing, Byzantine tolerant environment that require nodes to know the global state (clock synchronization, token circulation, agreement, etc.), the work presented here is the first protocol to operate in a network that is not fully connected. Consider a complete communication network on n nodes, each of which is a state machine with s states. In synchronous 2-counting, the nodes receive a common clock pulse and they have to agree on which pulses are "odd" and which are "even". We require that the solution is self-stabilising (reaching the correct operation from any initial state) and it tolerates f Byzantine failures (nodes that send arbitrary misinformation). Prior algorithms are expensive to implement in hardware: they require a source of random bits or a large number of statesas. We use computational techniques to construct very compact deterministic algorithms for the first non-trivial case of f = 1. While no algorithm exists for n < 4, we show that as few as 3 states are sufficient for all values n ≤ 4. We prove that the problem cannot be solved with only 2 states for n = 4, but there is a 2-state solution for all values n ≤ 6.