  This paper introduces the concept of the dimension of a triangulated category
with respect to a fixed full subcategory. For the bounded derived category of
an abelian category, upper bounds of the dimension with respect to a
contravariantly finite subcategory and a resolving subcategory are given. Our
methods not only recover some known results on the dimensions of derived
categories in the sense of Rouquier, but also apply to various commutative and
non-commutative noetherian rings.
