  We give a fast(er), communication-free, parallel construction of optimal
communication schedules that allow broadcasting of $n$ distinct blocks of data
from a root processor to all other processors in $1$-ported, $p$-processor
networks with fully bidirectional communication. For any $p$ and $n$,
broadcasting in this model requires $n-1+\lceil\log_2 p\rceil$ communication
rounds. In contrast to other constructions, all processors follow the same,
circulant graph communication pattern, which makes it possible to use the
schedules for the allgather (all-to-all-broadcast) operation as well. The new
construction takes $O(\log^3 p)$ time steps per processor, each of which can
compute its part of the schedule independently of the other processors in
$O(\log p)$ space. The result is a significant improvement over the sequential
$O(p \log^2
  p)$ time and $O(p\log p)$ space construction of Tr\"aff and Ripke (2009) with
considerable practical import. The round-optimal schedule construction is then
used to implement communication optimal algorithms for the broadcast and
(irregular) allgather collective operations as found in MPI (the
\emph{Message-Passing Interface}), and significantly and practically improves
over the implementations in standard MPI libraries (\texttt{mpich}, OpenMPI,
Intel MPI) for certain problem ranges. The application to the irregular
allgather operation is entirely new.
