  Computing uniformization maps for surfaces has been a challenging problem and
has many practical applications. In this paper, we provide a theoretically
rigorous algorithm to compute such maps via combinatorial Calabi flow for
vertex scaling of polyhedral metrics on surfaces, which is an analogue of the
combinatorial Yamabe flow introduced by Luo. To handle the singularies along
the combinatorial Calabi flow, we do surgery on the flow by flipping. Using the
discrete conformal theory established by Gu-Guo-Luo-Sun-Wu and Gu-Luo-Sun-Wu,
we prove that for any initial Euclidean or hyperbolic polyhedral metric on a
closed surface, the combinatorial Calabi flow with surgery exists for all time
and converges exponentially fast after finite number of surgeries. The
convergence is independent of the combinatorial structure of the initial
triangulation on the surface.
