In this paper we present a resolution strategy that uses a
modification of Villamayor's algorithm as a black box and combines
resolutions of (irreducible or at least equidimensional) components
of a given algebraic set $X\subset W$ to compute an embedded
resolution of singularities of $X$. The arising algorithm extends
the scope of Villamayor's algorithm from equidimensional algebraic
sets to the general case. The ideas also serve well in improving
the efficiency of resolutions, using the prime ideal decomposition
of the (radical) vanishing ideal of $X$.
