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 $X$.