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.

enter image description here

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    $\begingroup$ To find a shortest path from $v_{i,j}$ to $v_{h,k}$ in this particular graph (assuming $h \ge i$ and $k \ge j$, otherwise there is no directed path) you can pick any path that uses a vertical edge $h-i$ times and an horizontal edge $k-j$ times. E.g., the one going from $v_{i,j}$, to $v_{h,j}$, to $v_{h,k}$. Clearly this can be done in constant time if you are fine with just the distance or with a compact description of the path. Otherwise you can explicitly report the path's edges in time proportional to the path's length (which is constant anyway since this is at most $7$ in this graph). $\endgroup$ – Steven Nov 13 '19 at 8:24
  • $\begingroup$ I apologize, I forgot to mention that we are trying to determine the most efficient way to find the shortest path from $v_{1,1} \rightarrow v_{n,m}$ $\endgroup$ – flutterbug98 Nov 13 '19 at 8:25
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    $\begingroup$ All the considerations of my previous comment still apply. If you don't have the vertices labelled as in the figure, and you're just given the id of $v_{1,1}$ then you can just use a DFS search from $v_{1,1}$. You'll reach a sink in $7$ (in general $n+m-2$) steps and that'll be exactly $v_{4,5}$ (in general $v_{n,m})$. (Notice also that each traversed vertex takes a constant time to process). $\endgroup$ – Steven Nov 13 '19 at 8:30
  • $\begingroup$ Would it matter if the edges could possibly have negative weights? $\endgroup$ – flutterbug98 Nov 13 '19 at 8:48
  • $\begingroup$ It would. In fact it matters even if there are non-negative weights. Since the graph is a DAG you can solve your problem with possibly negative weights in $O(|V|+|E|) = O(|V|)$ time by using dynamic programming. $\endgroup$ – Steven Nov 13 '19 at 9:15

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 $O(|E|) = O(|V|) = O(nm)$.


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