  With the help of hyper-ideal circle pattern theory, we have developed a
discrete version of the classical uniformization theorems for surfaces
represented as finite branched covers over the Riemann sphere as well as
compact polyhedral surfaces with non-positive curvature. We show that in the
case of such surfaces discrete uniformization via hyper-ideal circle patterns
always exists and is unique. We also propose a numerical algorithm, utilizing
convex optimization, that constructs the desired discrete uniformization.
