  We give optimally fast $O(\log p)$ time (per processor) algorithms for
computing round-optimal broadcast schedules for message-passing parallel
computing systems. This affirmatively answers the questions posed in Tr\"aff
(2022). The problem is to broadcast $n$ indivisible blocks of data from a given
root processor to all other processors in a (subgraph of a) fully connected
network of $p$ processors with fully bidirectional, one-ported communication
capabilities. In this model, $n-1+\lceil\log_2 p\rceil$ communication rounds
are required. Our new algorithms compute for each processor in the network
receive and send schedules each of size $\lceil\log_2 p\rceil$ that determine
uniquely in $O(1)$ time for each communication round the new block that the
processor will receive, and the already received block it has to send. Schedule
computations are done independently per processor without communication. The
broadcast communication subgraph is the same, easily computable, directed,
$\lceil\log_2 p\rceil$-regular circulant graph used in Tr\"aff (2022) and
elsewhere. We show how the schedule computations can be done in optimal time
and space of $O(\log p)$, improving significantly over previous results of
$O(p\log^2 p)$ and $O(\log^3 p)$. The schedule computation and broadcast
algorithms are simple to implement, but correctness and complexity are not
obvious. All algorithms have been implemented, compared to previous algorithms,
and briefly evaluated on a small $36\times 32$ processor-core cluster.
