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 $\lceil \lg n\rceil$, 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 $B( n ) = \Theta ( L( n )\cdot n )$ and $D( n ) = \Theta ( \lg \lg n + L ( n ) )$ 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 $\lceil \lg n \rceil + 1$ time units to broadcast are constructed. These broadcast protocols may be implemented with local control and $O( \lg \lg n )$ bits overhead per message.