Broadcasting in m-dimensional grid graphs with a given neighborhood template

Abstract

Given a finite nonempty set F of vectors in Z m , consider the graph G=( V, E) whose vertices are the elements of Z m and such that each vertex v is connected to all vertices v+f for all f∈ F. Two models of communication have been considered in such graphs: whispering (in which a node can only call one neighbor per unit of time) and shouting (in which a node can simultaneously call all of its neighbors). Let σ( t) (resp. ω( t)) be the maximum number of nodes that can be reached in t steps by a shouting (resp. whispering) broadcast from a single source. This paper deals with the particular case where F contains only integer vectors with only one nonzero component. For m=2, we give the exact form of σ( t). For m=2 and when F contains only positive vectors we give a concrete upper bound for whispering. We believe that this upper bound can be achieved. Furthermore, when F includes the unit vector in each dimension, we describe a whispering broadcast scheme which gives a lower bound of ω( t) in that case. For m⩾3, we prove that for large t both σ( t) and ω( t) are of the form [( Πi=m i=1 (d i+c i)) m! ]t m+ O(t m−1) where d i (resp. c i ) is the largest “positive” (resp. “negative”) step in the ith dimension, i=1,…, m.