MPI inter-group collective communication patterns can be viewed as bipartite graphs that divide processes into two disjoint groups in which messages are transferred between but not within the groups. Such communication patterns can serve as basic operations for scientific application workflows. In this paper, we present parallel algorithms for inter-group all-to-all broadcast (Allgather) communication with optimal bandwidth for any message size and process number under single-port communication constraints. We implement the algorithms using MPI point-to-point and intra-group collective communication functions and evaluate their performance on the Cori supercomputer at NERSC. Using message sizes ranging from 256B to 64MB, the experiments show a significant performance improvement achieved by our algorithm, which is up to 9.27 times faster than production MPI libraries that adopt the so called root-gathering algorithm.