# Bellman–Ford algorithm ¶

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This algorithm is similar to Dijkstra's algorithm, but the weight can be either positive or negative (complexity $O(n*m)$).

The algorithm will fail if there is a negative-weight cycle because we can always make another iteration and reduce the shortest path value.

You will have up to $|V|-1$ iterations, but if you have an iteration with no changes, then the algorithm is done.

The differences with Dijkstra are for $i+1$

• you start at a vertex
• you will replace the distance if the old one is bigger
• but you can also use the distances calculated in the current iteration

It can be summarized as "at each iteration, you will try to find if you can have a better predecessor for each vertex".

## Example ¶

We are starting from A

Step A B C D E F
$0$ $0$ $+\infty$ $+\infty$ $+\infty$ $+\infty$ $+\infty$
$1$ $A(6)$ $A(4)$ $A(5)$ $+\infty$
$B(5)$
$\xcancel{C(7)}$
$+\infty$
$D(4)$
$\xcancel{E(8)}$
$2$ $C(2)$ $D(3)$ $B(1)$
$3$ $C(1)$ $B(0)$ $E(3)$
$4$

The interpretation is the same as for Dijkstra.