The BellmanFordAlgorithm uses the Bellman-Ford algorithm to find the weighted path of minimum weight from s to t.

If G is an unweighted graph, the edges are assumed all to have weight 1.

If G is a weighted graph, BellmanFordAlgorithm(G,s,t) returns the cheapest weighted path from vertex s to vertex t in the graph G. If a path from s to t exists, the output is a list of the form $\left[\left[s\,\mathrm{...}\,t\right]\,w\right]$ where $\left[s\,\mathrm{...}\,t\right]$ is the path and w is the weight of that path.  If no such path exists the output is $\left[\left[\right]\,\mathrm{\infty }\right]$.

In the second calling sequence where T is a list of vertices of G, this is short for $\left[\mathrm{seq}\left(\mathrm{BellmanFordAlgorithm}\left(G,s,t\right),t=T\right)\right]$ , except that the algorithm does not need to recompute cheapest paths.

In the third calling sequence where no destination vertices are given, this is short for BellmanFordAlgorithm(G,s, Vertices(G)), and the cheapest path from s to every vertex in G is output.

To compute distances between all pairs of vertices simultaneously, use the AllPairsDistance command.  To ignore edge weights (and use a faster breadth-first search), use the ShortestPath command.  

Note that G can have no negative cycles, which also means that any edges with negative weights must be directed (as otherwise the undirected negative weight edge forms a negative weight cycle between the vertices it connects). If G has no negative edge weights, DijkstrasAlgorithm may be able to find the cheapest paths more efficiently.

$\mathrm{with}\left(\mathrm{GraphTheory}\right)\:$

$\mathrm{C6}≔\mathrm{Graph}\left(\left\{\left[\left[1\,2\right]\,1\right]\,\left[\left[1\,5\right]\,4\right]\,\left[\left[2\,3\right]\,-3\right]\,\left[\left[3\,4\right]\,7\right]\,\left[\left[5\,6\right]\,-1\right]\,\left[\left[6\,4\right]\,3\right]\right\}\right)$

$\mathrm{C6}≔\mathrm{Graph\; 1:\; a\; directed\; weighted\; graph\; with\; 6\; vertices\; and\; 6\; arcs}$

$\mathrm{BellmanFordAlgorithm}\left(\mathrm{C6}\,1\,4\right)$

$\left[\left[1\,2\,3\,4\right]\,5\right]$

$\mathrm{BellmanFordAlgorithm}\left(\mathrm{C6}\,2\,5\right)$

$\left[\left[\right]\,\mathrm{\infty }\right]$

$\mathrm{DrawGraph}\left(\mathrm{C6}\right)$

$\mathrm{BellmanFordAlgorithm}\left(\mathrm{C6}\,1\right)$

$\left[\left[\left[1\right]\,0\right]\,\left[\left[1\,2\right]\,1\right]\,\left[\left[1\,2\,3\right]\,\mathrm{-2}\right]\,\left[\left[1\,2\,3\,4\right]\,5\right]\,\left[\left[1\,5\right]\,4\right]\,\left[\left[1\,5\,6\right]\,3\right]\right]$

$\mathrm{BellmanFordAlgorithm}\left(\mathrm{C6}\,2\right)$

$\left[\left[\left[\right]\,\mathrm{\infty }\right]\,\left[\left[2\right]\,0\right]\,\left[\left[2\,3\right]\,\mathrm{-3}\right]\,\left[\left[2\,3\,4\right]\,4\right]\,\left[\left[\right]\,\mathrm{\infty }\right]\,\left[\left[\right]\,\mathrm{\infty }\right]\right]$