Conformal mapping in linear time

Given any $\epsilon >0$ and any planar region $\Omega$ bounded by a simple n-gon $P$ we construct a ($1 + \epsilon)$-quasiconformal map between $\Omega$ and the unit disk in time $C(\epsilon)n$. One can take $C(\epsilon) = C + C \log (1/\epsilon) \log \log (1/\epsilon)$.