  Given an algorithm of resolution of singularities satisfying certain
conditions (``good algorithms''), natural notions of simultaneous algorithmic
resolution, or equiresolution, for families of embedded schemes (parametrized
by a reduced scheme $T$) are proposed. It is proved that these conditions are
equivalent. Something similar is done for families of sheaves of ideals, here
the goal is algorithmic simultaneous principalization. A consequence is that
given a family of embedded schemes over a reduced $T$, this parameter scheme
can be naturally expressed as a disjoint union of locally closed sets $T_{j}$,
such that the induced family on each part $T_{j}$ is equisolvable. In
particular, this can be applied to the Hilbert scheme of a smooth projective
variety; in fact, our result shows that, in characteristic zero, the underlying
topological space of any Hilbert scheme parametrizing embedded schemes can be
naturally stratified in equiresolvable families.
