Let be a set of vectors on which we are going to perform point-based DP update. As mentioned earlier, we can assume that we know a witness point for each vector in . Denote the witness point for a vector by . Point-based DP update first backs up on these points and thereby obtains a new set of vectors. To be more specific, it begins with the following subroutine:
2. for each
7. return .
In this subroutine, line 4 makes sure that the resulting set contains no duplicates and line 5 takes note of the fact that is also a witness point for (w.r.t ).