Algorithms for Noisy Broadcast under Erasures
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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( n) rounds with high probability. Later, a matching lower bound of Ω( 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(^* 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 Ω(poly(n)).
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