Suppose I have a DAG, $G = (V, E)$ and we know that all nodes in the DAG have at most $A$ ancestors. Let $V' \subseteq V$ be a subset of vertices of $V$. Is there a way to obtain the set of all ancestors of every vertex in $V'$ in $O(|V'|A + m)$ time? In other words, I want sets $B_i$ consisting of all ancestors of $v_i \in V'$ for all $v_i$.
The trivial algorithm does this in $O(|V'|A^2)$ time (for very dense graphs) but I would like to do this in $O(|V'|A + m)$ time or even better $O(|V'|A + E(V', B(V')))$. (Here $E(V', B(V'))$ indicates time proportional to the induced subgraph consisting of $V'$ and all ancestors of vertices in $V'$.)
If we cannot do this, are there any known lower bounds for this runtime?