Multiparty Interactive Coding over Networks of Intersecting Broadcast Links

We consider computations over networks with multiple broadcast channels that intersect at a single party. Each broadcast link suffers from random bit-flip noise that affects the receivers independently. We design interactive coding schemes that successfully perform any computation over these noisy networks and strive to reduce their communication overhead with respect to the original (noiseless) computation. A simple variant of a coding scheme by Rajagopalan and Schulman (STOC 1994) shows that any (noiseless) protocol of $R$ rounds can be reliably simulated in $O(R\log n)$ rounds over a network with $n=n_{1}n_{2}+1$ parties in which a single party is connected to $n_{2}$ noisy broadcast channels, each of which connects $n_{1}$ distinct parties. We design new coding schemes with improved overheads. Our approach divides the network into four regimes according to the relationship between $n_{1}$ and $n_{2}$. We employ a two-layer coding where the inner code protects each broadcast channel and is tailored to the specific conditions of the regime in consideration. The outer layer protects the computation in the network and is generally based on the scheme of Rajagopalan and Schulman, adapted to the case of broadcast channels. The overhead we obtain ranges from $O(\log \log n_{2})$ to $O\left({\log n_{2} \frac {\log \log n_{1}}{\log n_{1}}}\right)$ and beats the trivial $O(\log n)$ overhead in all four regimes.