Interplay between (im)perfectness, synchrony and connectivity: The case of reliable message transmission

Reliable message transmission is a fundamental problem in distributed communication networks. Of late, several interesting results have been obtained by modelling the network as a directed graph. An important result among them was due to Bhavani et al. [Unconditionally reliable message transmission in directed networks, 18 in: SODA'08: Proceedings of the Nineteenth Annual ACM-SIAM Symposium on Discrete Algorithms, SIAM, Philadelphia, PA, USA, 2008, pp. 1048-1055], where it was shown that if a negligible probability of error is allowed, the connectivity required in a synchronous network for unconditionally reliable message transmission (URMT) is significantly reduced. We refer to the variant of URMT studied by them as Monte Carlo URMT; here the receiver is allowed to output an incorrect message with small probability. Another interesting variant which has received little attention so far in the literature is the Las Vegas variant, where the receiver is allowed to abort with a small probability, but never to output an incorrect message. We show that the minimum connectivity requirements for the existence of Las Vegas URMT protocols over synchronous networks are the same as that of Monte Carlo URMT protocols over asynchronous networks-a surprising equivalence between two very different models. Furthermore, we show that the higher connectivity requirements of Las Vegas URMT over asynchronous networks match exactly with that of zero-error (perfect) variant over (a)synchronous networks. We also show that there exists a family of graphs where in the number of critical edges for the 'easier' randomized variants are asymptotically higher than that for the perfect variants. Hence, our results demonstrate an interesting interplay between (im)perfectness, synchrony and connectivity for the case of reliable message transmission.