On the trade-off between network connectivity, round complexity, and communication complexity of reliable message transmission

Abstract

Perfectly reliable message transmission (PRMT) is one of the fundamental problems in distributed computing. It allows a sender to reliably transmit a message to a receiver in an unreliable network, even in the presence of a computationally unbounded adversary. In this article, we study the inherent trade-off between the three important parameters of the PRMT protocols, namely, the network connectivity (n), the round complexity (r), and the communication complexity by considering the following generic question (which can be considered as the holy grail problem) in the context of the PRMT protocols.Given ann-connected network, a message of size ℓ (to be reliably communicated) and a limitcfor the total communication allowed between the sender and the receiver, what is the minimum number of communication rounds required by a PRMT protocol to send the message, such that the communication complexity of the protocol is O(c)?We answer this interesting question by deriving a nontrivial lower bound on the round complexity. Moreover, we show that the lower bound is tight in the amortized sense, by designing a PRMT protocol whose round complexity matches the lower bound. The lower bound is the first of its kind, that simultaneously captures the inherent tradeoff between the three important parameters of a PRMT protocol.