#### Consensus in Networks Prone to Link Failures

##### Bogdan S. Chlebus, Dariusz R. Kowalski, Jan Olkowski, Jedrzej Olkowski

We consider deterministic distributed algorithms solving Consensus in synchronous networks of arbitrary topologies. Links are prone to failures. Agreement is understood as holding in each connected component of a network obtained by removing faulty links. We introduce the concept of stretch, which is a function of the number of connected components of a network and their respective diameters. Fast and early-stopping algorithms solving Consensus are defined by referring to stretch resulting in removing faulty links. We develop algorithms that rely only on nodes knowing their own names and the ability to associate communication with local ports. A network has $n$ nodes and it starts with $m$ functional links. We give a general algorithm operating in time $n$ that uses messages of $O(\log n)$ bits. If we additionally restrict executions to be subject to a bound $\Lambda$ on stretch, then there is a fast algorithm solving Consensus in time $O(\Lambda)$ using messages of $O(\log n)$ bits. Let $\lambda$ be an unknown stretch occurring in an execution; we give an algorithm working in time $(\lambda+2)^3$ and using messages of $O(n\log n)$ bits. We show that Consensus can be solved in the optimal $O(\lambda)$ time, but at the cost of increasing message size to $O(m\log n)$. We also demonstrate how to solve Consensus by an algorithm that uses only $O(n)$ non-faulty links and works in time $O(n m)$, while nodes start with their ports mapped to neighbors and messages carry $O(m\log n)$ bits. We prove lower bounds on performance of Consensus solutions that refer to parameters of evolving network topologies and the knowledge available to nodes.

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