We present a nominal unification algorithm that runs in $O(n \times log(n) \times G(n))$ time, where $G$ is the functional inverse of Ackermann's function. Nominal unification generates a set of variable assignments if there exists one, that makes terms involving binding operations $\alpha$-equivalent. We preserve names while using special representations of de Bruijn numbers to enable efficient name management. We use Martelli-Montanari style multi-equation reduction to generate these name management problems from arbitrary unification terms.