Before revision:  This improves the previously smallest dependence on \varepsilon in the running time (1/\varepsilon)^{ \mathcal{O (1/\varepsilon^{d-1})}n \log n of the algorithm by Rao and Smith (STOC 1998).
After revision:  This improves the previously smallest dependence on \varepsilon in the running time (1/\varepsilon)^{ O (1/\varepsilon^{d-1})}n \log n of the algorithm by Rao and Smith (STOC 1998).