The Nagata automorphism is a kind of complicated automorphism on the affine 3-space . For a long time, it remained unknown whether or not the Nagata automorphism is tame until Shestakov and Umirbaev at last proved that it is not tame in 2004, by purely algebraic methods (e.g. Poisson algebra). In this paper, we consider a certain necessary condition for a given automorphism on to be tame from the point of view of the Sarkisov program established by Corti. Furthermore, by using it, we shall give a new algebro-geometric proof of the non-tameness of the Nagata automorphism.