  In this paper we consider the problem of computing all possible order ideals
and also sets connected to 1, and the corresponding border bases, for the
vanishing ideal of a given finite set of points. In this context two different
approaches are discussed: based on the Buchberger-M\"oller Algorithm, we first
propose a new algorithm to compute all possible order ideals and the
corresponding border bases for an ideal of points. The second approach involves
adapting the Farr-Gao Algorithm for finding all sets connected to 1, as well as
the corresponding border bases, for an ideal of points. It should be noted that
our algorithms are term ordering free. Therefore they can compute successfully
all border bases for an ideal of points. Both proposed algorithms have been
implemented and their efficiency is discussed via a set of benchmarks.
