Broadcasting in one dimension

Abstract

We consider the broadcasting problem for one-dimensional grid graphs with a given neighborhood template. There are two different models that have been considered-shouting (a node informs all of its neighbors in one step) and whispering (a node informs a single neighbor in one step). Let σ( t) (respectively ω( t)) denote the maximum number of nodes that can be reached in t steps by shouting (respectively whispering) broadcast from a single source. We obtain detailed information about the benefits of shouting over whispering. We prove for the one-dimensional case a conjecture by Stout that ω( t) eventually becomes a polynomial. In particular, we show that there exist constants i and t 0 such that ω( t)= σ( t)− i for all t ≥ t 0. When the broadcast only goes in one direction (i.e., when all elements of the template are positive), we also determine that i= d −1 and t 0≤3 d for a neighborhood template with the furthest neighbor at distance d.