Synchronous

We study the number of rounds needed to solve consensus in a synchronous network G where at most t nodes may fail by crashing. This problem has been thoroughly studied when G is a complete graph, but very little is known when G is arbitrary. We define a notion of radius ( G , t ), that extends the standard graph theoretical notion of radius, for considering all the ways in which t nodes may crash, and we present an algorithm that solves consensus in radius ( G , t ) rounds. Then we derive a lower bound showing that, among oblivious algorithms, our algorithm is optimal for a large family of graphs including all vertex-transitive graphs.