Grigni: [3] Tight Bounds on Minimum Broadcast Networks

A broadcast graph is an n-vertex communication network that supports a broadcast from any one vertex to all other vertices in optimal time dlgne, given that each message transmission takes one time unit and a vertex participates in at most one transmission per time step. This paper establishes tight bounds for B(n), the minimum number of edges of a broadcast graph, and D(n), the minimum maxdegree of a broadcast graph. Let L(n) denote the number of consecutive leading 1's in the binary representation of integer n 1. We show B(n) = (L(n) n) and D(n) = (lg lgn+L(n)), and for every n we give a construction simultaneously within a constant factor of both lower bounds. For all n we also construct graphs with O(n) edges and O(lg lg n) maxdegree requiring at most dlg ne + 1 time units to broadcast. Our broadcast protocols may be implemented with local control and O(lg lg n) bits overhead per message. This material is based upon work supported under a National Science Foundation Graduate Fellowship. Work done partly at AT&T Bell Labs, Murray Hill, NJ 07974. Part of this work was carried out while this author was visiting Stanford university. Supported in part by a Weizmann fellowship, by contracts ONR N00014-85-C-0731 and ONR N00014-88-K-0166 and by a grant of Stanford Center for Integrated Systems.