Broadcasting in an n-grid with a given neighborhood template

In a broadcasting process, a particular vertex called the originator broadcasts information by mean of calls to all the vertices of the network. Each call requires a time unit, and a vertex can call only its neighbors. The process is called shouting if a vertex can call all of its neighbors at one time, or whispering if a vertex can call only one of its neighbors at a time. Q.F. Stout (1981) defined σ(t) and ω(t) as the maximum number of vertices that may be informed at time t by any shouting or whispering scheme, respectively. In this paper, we consider the particular case when the network is an infinite n-dimensional grid with a given neighborhood template F. Without restricting the form of the set F, we determine σ(t) and an equivalent to ω(t). We also give a whispering scheme that is nearly optimal. Our proofs mainly use techniques from lattice theory and combinatorics to determine the number of vertices at a distance t from 0