Abstract: When faced with the problem of broadcasting a message reliably, there are tradeoffs to be made in terms of scalability, the ability to provide delivery guarantees in the face of node and link failures, per-machine message overhead, and time to delivery. Common approaches include reliable multicast, gossip-based protocols, and constrained flooding on superimposed communication graphs. Recent work has presented a new parameterized family of communication graphs that provide an explicit tradeoff between strength of delivery guarantees and reduced message overhead. In this paper, we present a modified family of communication graphs that still provide this tradeoff between reliability and message overhead, but which also have the added benefit of dissemination time that is logarithmic in the size of the network.