A Combinatorial Characterization of the Distributed 1-Solvable Tasks

Fischer, Lynch, and Paterson showed in a fundamental paper that achieving a distributed agreement is impossible in the presence of one faulty processor. This result was later extended by Moran and Wolfstahl who showed that it holds for any task with a connected input graph and a disconnected decision graph. In this paper we extend that latter result, and in fact we set an exact borderline between solvable and unsolvable tasks, by giving a necessary and sufficient condition for a task to be l-solvable (that is, solvable in the presence of one faulty processor). Our characterization is purely combinatorial and involves only relations between the input graph and the output graph, defined by the given task. It provides easy proofs for the non-solvability of tasks and also provides a universal protocol which solves any task which is found to be solvable by our condition. Using the above characterization, we also derive a novel technique to prove lower bounds on the number of messages that must be sent due to processor failure; specifically, we provide a simple proof that for each fiied N > 2 there exist distributed tasks for N processors, that can be solved in the presence of a faulty processor, but any protocol that solves them must send arbitrarily many messages in the worst case.