Before revision:  The key ingredient of the fronts propagation and the computation of the geodesic distances is the construction of the metrics. We make use of the Finsler metrics with Randers form, by which the asymmetric and anisotropic measures of the geodesic curve length are taken into account to avoid the shortcuts and leaking problems.
After revision:  The key ingredient of the fronts propagation and the computation of the geodesic distances is the geodesic metrics, which can govern the action of the geodesic distance propagation. We consider a Finsler metric with the Randers form, by which the asymmetric and anisotropic measures of the geodesic curve length are taken into account to avoid the shortcuts and leaking problems.