Proceedings of the 34th ACM Symposium on Parallelism in Algorithms and Architectures
We are considering the following problem. In a network of p processors, one designated root processor has n indivisible blocks of data that have to be broadcast (transmitted to) all other processors. The processors are fully connected, and in one communication operation, a processor can simultaneously receive one block of data from one other processor and send a block of data to one other, possibly different processor. This is the 1-ported, fully connected, bidirectional send-receive model 
... for which the well-known broadcast lower bound is n − 1 + ⌈log 2 p⌉ communication rounds. A round-optimal broadcast schedule reaches this lower bound, and for each processor explicitly specifies, for each round which of the n blocks are to be sent and received to and from which other processors. In a homogeneous, linear-cost communication model, where transferring a divisible message of m units between any two processors takes α + βm units of time, this lower bound gives a broadcast time of α ⌈log 2 p − 1⌉ + 2 ⌈log 2 p − 1⌉α βm + βm by dividing m appropriately into n blocks. It is well-known that the problem can be solved optimally when p is a power of 2, and many hypercube and butterfly algorithms exist, e.g.  . These algorithms are typically difficult to generalize to arbitrary numbers of processors; an exception is the appealing hypercube-based construction in  . The preprocessing required for the schedule construction is O(log p) time steps per processor, but different processors have different roles and different communication patterns in the construction. A different, explicit, optimal construction with a symmetric communication pattern was given Permission to make digital or hard copies of part or all of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for third-party components of this work must be honored. For all other uses, contact the owner/author(s).