On the structure of minimum broadcast digraphs

Broadcasting is a problem of information dissemination described in a group of individuals connected by a communication network, where one individual has an item of information and needs to communicate it to everyone else. This communication pattern nds its main applications in the eld of interconnection networks for parallel and distributed architecture. Numerous previous papers have investigated ways to construct sparse undirected graphs (networks) in which this process can be completed in minimum time. In this paper, we consider the broadcast problem in directed graphs. We describe some techniques to construct sparse di-graphs on n vertices in which broadcasting can be completed in minimum time. For n = 2 p ?1 and n = 2 p ? 2, we show that these techniques produce the sparsest possible digraphs of this type (called minimum broadcast digraphs, or M BDs). In the case n = 2 p ? 1, we give one class of M BDs, as for the case n = 2 p ? 2, we give two non isomorphic classes of M BDs. We show that these techniques also produce a class of M BDs on n = 2 p vertices which is non isomorphic to the one given in LP92]. For some other innnite classes of values of n, we give techniques that produce the sparsest known digraphs of this type, and we also give some lower bounds on the size of M BDs. Finally, in the range 1::32, we give new M BDs which are not isomorphic to the ones given in LP92] (namely n = 6, n = 9, n = 14 and n = 30).