respectively, to ${{\mathscr I}}^+$ and ${{\mathscr I}}^-$.

In order to compute the complexity growth, we need to consider evolution of the gray region. Since we want to compare our results with which is given by taking the flat-space limit from the AdS case, we assume that the BMSFT on the right- and left-hand sides develop symmetrically. This requires a symmetric evolution of the advanced and retarded coordinates on the different null infinities as $$u^{+}_{R}= - v^{+}_{L} =\mu^+,
\qquad  v^{-}_{R}= - u^{-}_{L} = \mu^-.$$

There is a critical time when the gray region leaves the cutoff near the past singularity. For the simplicity of calculation and avoiding unnecessary shifts in the origin of boundary times, we assume that this cross occurs at $t=0$. The symmetric evolution guarantees that the last crossing point remains permanently on $t=0$. The initial times are those before this time. It is clear from the Penrose diagram that for the symmetric evolution the initial and late times are, respectively, given by $\mu^{\pm} < \mu^{\pm}_{c}$ and $\mu^{\pm} > \mu^{\pm}_{c}$ where $$\label{criticaltime}
\mu^+_{c} =- r^{\ast}(\epsilon_{0}).$$

All of the relevant points in the Penrose diagram are collected in the next table:\

[ |p[3cm]{}|p[3cm]{}|p[3cm]{}|p[2cm]{}|p[3.7cm]{}| ]{}

\
& && [[Sign of $f(r)$]{}]{}\
