Determine shortest path in a 4x5 grid graph Suppose we had a graph $G = (V,E)$. 

This graph can also be seen as a $4x5$ grid graph as shown in the image. 

There is a directed edge from $v_{i,j} \rightarrow v_{i,j+1}$ for $1 \leq i \leq n$ and $1 \leq j \leq m-1$. 

Similarly, there is a directed edge from $v_{i,j} \rightarrow v_{i+1,j}$ for $1 \leq i \leq n-1$ and $1 \leq j \leq m$. 

What is the most efficient algorithm to determine the shortest path from $v_{1,1} \rightarrow v_{n,m}$? My main idea is to use the Bellman-Ford algorithm, but I am clearly not taking advantage of how this graph is laid out. Any guidance would be much appreciated. 

EDIT: The edges are weighted and may possibly be negative.
Since your graph is a DAG you can topologically sort $V$ and solve your problem via dynamic programming by considering the vertices in reverse topological order.

For each vertex $v$, let $d(v)$ be the distance from $v$ to $v_{n,m}$.
Clearly, if $v = v_{n,m}$, then $d(v)=0$.

For $v \neq  v_{n,m}$, you have:
$$
d(v) = \min_{(v,u) \in E} \{ w(v,u) + d(u) \},
$$

where $w(v,u) \in \mathbb{R}$ is the weight of edge $(v,u)$ in $G$.

Both the topological order of $V$ and all the quantities $d(v)$ can be found in $O(|E|)$ time. Therefore the overall time required is $O(|E|) = O(|V|) = O(nm)$.