Incoming edges for an adjacency list. $[20$ points$]$

Recall that in the adjacency list representation of a directed graph $G=(V,E),$ we are given, for each vinV, a list of the vertices in $a_v=\{uinV$:$(v,u)inE\},$ i$.$e$.,$ the vertices that can be reached by following an edge starting from $v.$ For some applications, it might be useful to also have a list, for each vinV, of the vertices from which one can reach $v$ by following an edge, i$.$e$.,$ the vertices in $b_v=\{winV$:$(w,v)inE\}.$ Design an algorithm that, given the adjacency list representation of a digraph $G,$ produces the lists $b_v$ for all vinV. Prove that your algorithm is correct and analyze its asymptotic running time as a function of $n=|V|$ and $m=|E|.$ To receive full credit, your algorithm should have the lowest possible asymptotic running time.