Disconnected Agreement in Networks Prone to Link Failures

We consider deterministic distributed algorithms for reaching agreement in synchronous networks of arbitrary topologies. Links are bi-directional and prone to failures while nodes stay non-faulty at all times. A faulty link may omit messages. Agreement among nodes is understood as holding in each connected component of a network obtained by removing faulty links – we call it a “disconnected agreement”. We introduce the concept of stretch, which is the number of connected components of a network, obtained by removing faulty links, minus 1 plus the sum of diameters of connected components. We define the concepts of “fast” and “early-stopping” algorithms for disconnected agreement by referring to stretch. We consider trade-offs between the knowledge of nodes, the size of messages, and the running times of algorithms. A network has n nodes and m links. We give a general disconnected agreement algorithm operating in n+ 1 rounds that uses messages of O(log n) bits. Let λ be an unknown stretch occurring in an execution; we give an algorithm working in time (λ+ 2 )³ and using messages of O(nlog n) bits. We show that disconnected agreement can be solved in the optimal O(λ) time, but at the cost of increasing message size to O(mlog n). We also design an algorithm that uses only O(n) non-faulty links and works in time O(nm), while nodes start with their ports mapped to neighbors and messages carry O(mlog n) bits. We prove lower bounds on the performance of disconnected-agreement solutions that refer to the parameters of evolving network topologies and the knowledge available to nodes.