Broadcasting in Bounded Degree Graphs

Broadcasting is an information dissemination process in which a message is to be sent from a single originator to all members of a network by placing calls over the communication lines of the network. Several previous papers have investigated methods to construct sparse graphs (networks) in which this process can be completed in minimum time from any originator. The graphs produced by these methods contain high degree vertices. In [20] and in [4] we began an investigation of graphs with fixed maximum degree in which broadcasting can be completed in near minimum time. In this paper we continue this investigation by giving lower bounds and constructing bounded degree graphs which allow rapid broadcasting. Our constructions use ideas developed by Jerrum and Skyum [16] which allow one to move from a graph with good average case behaviour to one with good worst case behaviour. In addition, we use de Bruijn digraphs [7], minimum broadcast graphs, and sparse broadcast graphs [3]. The resulting graphs yield the best broadcasting time known for bounded degree graphs. We also obtain asymptotic upper and lower bounds for broadcasting time, as the maximum degree increases.