Constant-Rate Interactive Coding Is Impossible, Even in Constant-Degree Networks

Multiparty interactive coding allows a network of <inline-formula> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula> parties to perform distributed computations when the communication channels suffer from noise. Previous results (Rajagopalan and Schulman, STOC 1994) obtained a multiparty interactive coding protocol, resilient to random noise, with a blowup of <inline-formula> <tex-math notation="LaTeX">$O(\log (\Delta +1))$ </tex-math></inline-formula> for networks whose topology has a maximal degree <inline-formula> <tex-math notation="LaTeX">$\Delta $ </tex-math></inline-formula>. Vitally, the communication model in their work forces all the parties to send one message at every round of the protocol, even if they have nothing to send. We re-examine the question of multiparty interactive coding, lifting the requirement that forces all the parties to communicate at each and every round. We use the recently developed information-theoretic machinery of Braverman <italic>et al.</italic> (J. ACM 2018) to show that if the network’s topology is a cycle, then there is a specific cycle task for which any coding scheme has a communication blowup of <inline-formula> <tex-math notation="LaTeX">$\Omega (\log n)$ </tex-math></inline-formula>. This is quite surprising since the cycle has a maximal degree of <inline-formula> <tex-math notation="LaTeX">$\Delta =2$ </tex-math></inline-formula>, implying a coding with a <italic>constant blowup</italic> when all parties are forced to speak at all rounds. We complement our lower bound with a matching coding scheme for the cycle task that has a communication blowup of <inline-formula> <tex-math notation="LaTeX">$\Theta (\log n)$ </tex-math></inline-formula>. This makes our lower bound for the cycle task tight.