Bit complexity of breaking and achieving symmetry in chains and rings

We consider a failure-free, asynchronous message passing network with <i>n</i> links, where the processors are arranged on a ring or a chain. The processors are identically programmed but have distinct identities, taken from {0, 1,… ,<i>M</i> − 1}. We investigate the communication costs of three well studied tasks: Consensus, Leader, and MaxF (finding the maximum identity). We show that in chain and ring topologies, the message complexities of all three tasks are the same. Hence, we study a finer measure of complexity: the number of transmitted <i>bits</i> required to solve a task <i>T</i>, denoted <i>BitC</i>(<i>T</i>). We prove several new lower bounds (and some simple upper bounds) that imply the following results: For the two processors case, <i>BitC</i>(Consensus) = 2 and <i>BitC</i>(Leader) = <i>BitC</i>(MaxF) = 2log₂ <i>M</i> ± <i>O</i>(1), where the gap between the lower and upper bounds is almost always 1. For a chain, <i>BitC</i>(Consensus) = Θ(<i>n</i>), <i>BitC</i>(Leader) = Θ(<i>n</i> + log <i>M</i>), and <i>BitC</i>(MaxF) = Θ(<i>n</i> log <i>M</i>). For the ring topology, we prove the lower bound of Ω(<i>n</i> log <i>M</i>) for Leader, and (hence) MaxF. We consider also a chain where the intermediate processors have no identities. We prove that <i>BitC</i>(Leader) = Θ(<i>n</i> log <i>M</i>), which is equal to <i>n</i> times the bit complexity of the problem for two processors. For the specific case when the chain length is even, we prove that <i>BitC</i>(Leader) = Θ(<i>n</i>), for both above settings. In addition, we show that for any algorithm solving MaxF, there exists an input, for which <i>every</i> execution has the bit complexity Ω(<i>n</i> log <i>M</i>) (this is not the case for Leader). In our proofs, we use both methods of distributed computing and of communication complexity theory, establishing new links between the two areas.