  Boson realizations map operators and states of groups to transformations and
states of bosonic systems. We devise a graph-theoretic algorithm to construct
the boson realizations of the canonical SU$(n)$ basis states, which reduce the
canonical subgroup chain, for arbitrary $n$. The boson realizations are
employed to construct $\mathcal{D}$-functions, which are the matrix elements of
arbitrary irreducible representations, of SU$(n)$ in the canonical basis. We
demonstrate that our $\mathcal{D}$-function algorithm offers significant
advantage over the two competing procedures, namely factorization and
exponentiation.
