Algorithms to solve fault-tolerant consensus in asynchronous systems often rely on primitives such as crusader agreement, adopt-commit, and graded broadcast, which provide weaker agreement properties than consensus. Although these primitives have a similar flavor, they have been defined and implemented separately in ad hoc ways. We propose a new problem called connected consensus that has as special cases crusader agreement, adopt-commit, and graded broadcast, and generalizes them to handle multi-valued inputs. The generalization is accomplished by relating the problem to approximate agreement on graphs. We present three algorithms for multi-valued connected consensus in asynchronous message-passing systems, one tolerating crash failures and two tolerating malicious (unauthenticated Byzantine) failures. We extend the definition of binding, a desirable property recently identified as supporting binary consensus algorithms that are correct against adaptive adversaries, to the multi-valued input case and show that all our algorithms satisfy the property. Our crash-resilient algorithm has failure-resilience and time complexity that we show are optimal. When restricted to the case of binary inputs, the algorithm has improved time complexity over prior algorithms. Our two algorithms for malicious failures trade off failure resilience and time complexity. The first algorithm has time complexity that we prove is optimal but worse failure-resilience, while the second has failure-resilience that we prove is optimal but worse time complexity. When restricted to the case of binary inputs, the time complexity (as well as resilience) of the second algorithm matches that of prior algorithms.

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