Let $G$ be a weighted directed acyclic graph with $m$ edges and $n$ vertices. Given $G$ and a source vertex $s$ in $G$, which one of the following options gives the worst case time complexity of the fastest algorithm to find the lengths of shortest paths from $s$ to all vertices that are reachable from $s$ in $G$?

For a weighted Directed Acyclic Graph (DAG), single-source shortest paths can be computed efficiently by processing vertices in topologically sorted order [1]:

1. Topologically sort the vertices of the DAG in $\Theta(n + m)$ time using Depth First Search (DFS) or Kahn's algorithm [1].

2. Initialize the distance to the source $s$ as $0$ and to all other vertices as $\infty$ in $\Theta(n)$ time [1].

3. Iterate through the vertices in topological order and relax all outgoing edges from each vertex [1]. Since each edge and vertex is relaxed/visited once, the total time for edge relaxation across the entire graph is $\Theta(m)$ [1].

Thus, the total worst-case running time is:

Hence, (A) is the correct option [1].