  For the task of moving a group of indistinguishable agents on a connected
graph with unit edge lengths into an arbitrary goal formation, it was
previously shown that distance optimal paths can be scheduled to complete with
a tight convergence time guarantee, using a fully centralized algorithm. In
this study, we show that the problem formulation in fact induces a more
fundamental ordering of the vertices on the underlying graph network, which
directly leads to a more intuitive scheduling algorithm that assures the same
convergence time and runs faster. More importantly, this structure enables a
distributed scheduling algorithm once individual paths are assigned to the
agents, which was not possible before. The vertex ordering also readily extends
to more general graphs - those with non-unit capacities and edge lengths - for
which we again guarantee the convergence time until the desired formation is
achieved.
