Algorithms for Noisy Broadcast under Erasures

TL;DR

This paper introduces new algorithms for noisy broadcast communication where erasures occur, achieving faster rounds than previous bounds by leveraging relaxed erasure models and larger alphabets.

Contribution

It presents the first sub-logarithmic round algorithms for noisy broadcast with erasures, surpassing earlier lower bounds, especially with large alphabet sizes.

Findings

Achieves $O( ext{log}^* n)$ rounds for all processors to learn input.

Develops an $O(1)$-round algorithm with large alphabet size.

Breaks previous $ ext{log} ext{log} n$ lower bounds in erasure models.

Abstract

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 $\Omega(\mathrm{poly}(n))$.