In earlier work, we showed that it is possible to achieve O(logN)
streaming delay with high probability in a peer-to-peer network, where each
peer has as little as four neighbors, while achieving any arbitrary fraction of
the maximum possible streaming rate. However, the constant in the O(logN)
delay term becomes rather large as we get closer to the maximum streaming rate.
In this paper, we design an alternative pairing and chunk dissemination
algorithm that allows us to transmit at the maximum streaming rate while
ensuring that all, but a negligible fraction of the peers, receive the data
stream with O(logN) delay with high probability. The result is established
by examining the properties of graph formed by the union of two or more random
1-regular digraphs, i.e., directed graphs in which each node has an incoming
and an outgoing node degree both equal to one