  On the holographic complexity dual to the bulk action, we investigate the
action growth for a shock wave geometry in a massive gravity theory within the
Wheeler-De Witt (WDW) patch at the late time limit. For a global shock wave,
the graviton mass does not affect the action growth in the bulk, i.e. the
complexity on the boundary, showing that the action growth (complexity) is the
same for both the Einstein gravity and the massive gravity. Nevertheless, for a
local shock wave that depends on transverse coordinates, the action growth
(complexity) is proportional to the butterfly velocity for the two gravity
theories, but the butterfly velocity of the massive gravity theory is smaller
than that of the Einstein gravity theory, indicating that the action growth
(complexity) of the massive gravity is depressed by the graviton mass. In
addition, we extend the black hole thermodynamics of the massive gravity and
obtain the right Smarr formula.
