Before revision:  For the special case of \calS comprising of concave-in splinegons, we have devised another algorithm which does not rely on the  algorithm to compute a rectilinear shortest path in the polygonal domain.
After revision:  For the special case of \calS comprising of concave-in splinegons, we have devised another algorithm in which the reduction procedure does not rely on the algorithm to compute a rectilinear shortest path in the polygonal domain.