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 [1g n], 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. It is shown that B(n) = kelvin(L (n).n) and D(n) = kelvin(lg lg n + L(n)), and for every n we give a construction simultaneously within a constant factor of both lower bounds. For all n, graphs with O(n) edges and O(lg lg n) maxdegree requiring at most [lg n] + 1 time units to broadcast are constructed. These broadcast protocols may be implemented with local control and O(lg lg n) bits overhead per message.