  We present an algorithm to compute the automorphism group of a Mori dream
space. As an example calculation, we determine the automorphism groups of
singular cubic surfaces with general parameters. The strategy is to study
graded automorphisms of affine algebras graded by a finitely generated abelian
groups and apply the results to the Cox ring. Besides the application to Mori
dream spaces, our results could be used for symmetry based computing, e.g. for
Gr\"obner bases or tropical varieties.
