Noisy Radio Network Lower Bounds via Noiseless Beeping Lower Bounds

Abstract

Much of today’s communication is carried out over large wireless systems with different input-output behaviors. In this work, we compare the power of central abstractions of wireless communication through the general notion of boolean symmetric f -channels : In every round of the f -channel, each of its n parties decides to either broadcast or not, and the channel outputs f ( m ), where m is the number of broadcasting parties. Our first result is that the well studied beeping channel , where f is the threshold-1 function, is not stronger than any other f -channel. To this end, we design a protocol over the f -channel and prove that any protocol that simulates it over the beeping channel blows up the round complexity by a factor of Ω(log n ). Our lower bound technique may be of independent interest, as it essentially generalizes the popular fooling set technique by exploiting a “local” relaxation of combinatorial rectangles. Curiously, while this result shows the limitations of a noiseless channel, namely, the beeping channel, we are able to use it to show the limitations of the noisy version of many other channels. This includes the extensively studied single-hop radio network model with collisions-as-silence (CAS), which is equivalent to the f -channel with f ( m ) = 1 iff m = 1. In particular, our second and main result, obtained from the first, shows that converting CAS protocols to noise resilient ones may incur a large performance overhead, i.e., no constant rate interactive code exists. To this end, we design a CAS protocol and prove that any protocol that simulates it over the noisy CAS model with correlated stochastic noise, blows up the round complexity by a factor of Ω(log n ). We mention that the Ω(log n ) overhead in both our results is tight.