Bit complexity of breaking and achieving symmetry in chains and rings

We consider a failure-free, asynchronous message passing network, with n processors arranged on a ring or a chain. The processes axe identically programmed but have distinct identities, taken from {1,...,M}. We investigate the communication costs of three well studied tasks: Consensus, Leader, and Maxr (finding the maximum identity, a restricted version of Leader). We show that in both chain and ring topologies, somewhat surprisingly, the message complexities of all three tasks are the same. Hence, we suggest as a finer measure of complexity the number of bits transmitted, BitC(·). We show that in chains, with respect to this measure, Consensus is easier than Leader, which is easier than MaxF. More specifically, we prove several new lower bounds (and some simple upper bounds) that imply the following results: For the two processors case, BitC(Consensus) = 2 and BitC(Leader) = BitC(MaxF) = 2log₂ M-O(1).