We show an $O(n\cdot L \cdot d^{L-1} + m + M_{\kappa,L})$-time algorithm finding all $\kappa$-MEMs between $Q$ and $G$ spanning exactly $L$ nodes in $G$, where $n$ is the total length of node labels, $d$ is the maximum degree of a node in $G$, $m = |Q|$, and $M_{\kappa,L}$ is the number of output MEMs.
We obtain a new MPC parameterized algorithm for DAGs running in time $O(k^2|V| + |E|)$. Our algorithm is the first solving the problem in parameterized linear time. Additionally, we obtain an edge sparsification algorithm preserving the width of a DAG but reducing $|E|$ to less than $2|V|$. This algorithm runs in time $O(k^2|V|)$ and requires an MPC of a DAG as input, thus its total running time is the same as the running time of our MPC algorithm.