Before revision:  In the second part of this work, we restrict again our attention to weighted Delaunay configurations and the associated spline bases, and we capitalize on our combinatorial viewpoint to extend, in this restricted case, the well-known bivariate construction algorithm to higher dimensions and generic point multisets. Finally, we employ this machinery to propose algorithms for the determination and evaluation of all multivariate basis spline functionssupported on a given point .
After revision:  In the second part of this work, we restrict again our attention to regular fine zonotopal tilings, specializing our previous results and exploiting the adjacency graph of the tiling to propose a set of practical algorithms for the determination and evaluation of all multivariate basis spline functionssupported on a given point .