We show that for any radio network there exists a schedule of a broadcast whose time is O(D + log 5 n), where D is the diameter and n is the number of nodes. (This result implies an optimal broadcast to networks with D = Ω(log 5 n).) We present a centralized randomized polynomial time algorithm that given a network and a source, outputs a schedule for broadcasting the message from the source to the rest of the network.

Broadcasting is the process of transmitting a message from a member in a network (originator) to all other members. A line-broadcasting scheme allows two members to communicate during one time unit as long as there is a path of lines between them and no link is used in more than one call between two members. Farley [3] showed an algorithm that accomplishes line broadcasting in any tree on n vertices in minimum time, which is log 2 n n n time units. Since the structure of the tree is unknown in advance, the total number of communication link uses (cost) of his scheme is Θ(n n n − − − 1) log 2 n n n . In this paper, we present line-broadcasting schemes for complete binary trees. The cost of the algorithms is linear in the number of vertices. This answers a question raised by Fujita and Farley [5].

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