  Let $\Lambda$ be an artin algebra. We give an upper bound for the dimension
of the bounded derived category of the category $\mod \Lambda$ of finitely
generated right $\Lambda$-modules in terms of the projective and injective
dimensions of certain class of simple right $\Lambda$-modules as well as the
radical layer length of $\Lambda$. In addition, we give an upper bound for the
dimension of the singularity category of $\mod \Lambda$ in terms of the radical
layer length of $\Lambda$.
