This paper proposes algorithms for computing the reduced Gröbner basis of the vanishing ideal of a finite set of points in the frame of ideal interpolation. We also consider the case that the points have multiplicity conditions. First, we introduce the definition of "reverse" reduced team and compute the interpolation monomial basis of a single point ideal interpolation problem; then we translate the interpolation condition functionals into formal power series via Taylor expansion; this will help convert the general ideal interpolation problem to a single point ideal interpolation problem; and finally, the reduced Gröbner basis is read from formal power series by Gaussian elimination. Our algorithm has a polynomial time complexity, and an example is given to illustrate its effectiveness.