Algorithms for Noisy Broadcast under Erasures

The noisy broadcast model was first studied in [Gallager, TranInf'88] where an $n$-character input is distributed among $n$ processors, so that each processor receives one input bit. Computation proceeds in rounds, where in each round each processor broadcasts a single character, and each reception is corrupted independently at random with some probability $p$. [Gallager, TranInf'88] gave an algorithm for all processors to learn the input in $O(\log\log n)$ rounds with high probability. Later, a matching lower bound of $\Omega(\log\log n)$ was given in [Goyal, Kindler, Saks; SICOMP'08]. We study a relaxed version of this model where each reception is erased and replaced with a `?' independently with probability $p$. In this relaxed model, we break past the lower bound of [Goyal, Kindler, Saks; SICOMP'08] and obtain an $O(\log^* n)$-round algorithm for all processors to learn the input with high probability. We also show an $O(1)$-round algorithm for the same problem when the alphabet size is $\Omega(\mathrm{poly}(n))$.