# Bellman–Ford algorithm

The **Bellman-Ford algorithm** finds single-source shortest paths in a directed, weighted graph which contains no negative-weight cycles. That is, unlike Dijkstra's algorithm, it is guaranteed to correctly compute shortest paths even when some edge weights are negative. (Note however that it is still a requirement that no negative-weight *cycle* occurs; finding shortest paths in such a graph becomes either meaningless if non-simple paths are allowed, or computationally difficult when they are not.) When single-source shortest paths are all that which is needed, and not all-pairs shortest paths, The Bellman–Ford algorithm, with time complexity , outperforms the Floyd–Warshall algorithm at in sparse graphs. It may also be combined with Dijkstra's algorithm to yield Johnson's algorithm, which again outperforms Floyd–Warshall in sparse graphs.

## Contents

# The algorithm

The idea behind the algorithm is very easy to understand, and may be satisfactorily illustrated by the following pseudocode:

input G,v for each u ∈ V(G) let dist[u] = ∞ let dist[v] = 0 for each i ∈ [1..V-1] for each (u,w) ∈ E(G) dist[w] = min(dist[w],dist[u]+wt(u,w)) for each (u,w) ∈ E(G) if dist[w] > dist[u]+wt(u,w) error "Graph contains negative-weight cycles"

*G* is the directed, weighted graph in question, and *v* the source. The output is the array *dist*; at the completion of the algorithm, *dist[x]* contains the shortest-path distance from *v* to *x*. If the graph contains a cycle of negative weight, an error message is generated to that effect.

# Theory of the algorithm

## How the algorithm works

The algorithm works by performing a series of *relaxations*. A relaxation occurs whenever the current shortest distance from node *v* to node *w* is improved because, by travelling from *v* to some intermediate vertex *u*, and then from *u* to *w*, a shorter path is obtained. (Floyd–Warshall and Dijkstra's algorithms rely upon this same technique.) The key is that, after *N* passes of the main loop in Bellman–Ford have completed, at least *N*+1 of the shortest-path distances in *dist* are correct. (We consider all pairs of vertices to be connected, so that all "missing" edges are assigned a weight of positive infinity.)

## Proof of correctness for graphs containing no negative-weight cycles

We proceed by induction:

- When , there is at least 1 correct entry in
*dist*, the one stating that the distance from the source to itself is zero. - Now suppose that passes have occurred and that we know the shortest-path distances from the source to of the vertices. Now, either is equal to , and we are done, or the vertices may be partitioned into two sets: , which contains vertices for which we already know shortest-path distances (with any being chosen if there are more than this number), and , which contains the rest. Now, since a shortest-paths tree exists (it always does when there are no negative-weight cycles; the proof is in the Shortest path article), there must exist some vertex
*w*in whose parent*u*in the shortest-paths tree is in . Then, when the edge*(u,w)*is relaxed, the*dist*array will contain the correct shortest-path distance to*w*. Thus, after the next pass of the outer loop has occurred, passes will have occurred in total, and the shortest-path distances to at least vertices will be correctly known.

Thus, when a negative-weight cycle does not exist, after the main loop has finished, all distances in *dist* are correct. Now, if an edge *(u,w)* still exists such that `dist[w] > dist[u]+wt(u,w)`

, then the distances could not possibly have been correct, because relaxation of *(u,w)* would give a shorter path to *w*. Since this is a contradiction, the assumption of the non-existence of negative-weight cycles must be incorrect in this case. We see then that as long as there are no negative-weight cycles, the algorithm always computes all distances correctly and terminates successfully.

## Proof of detection of negative-weight cycles

Suppose vertices vertices form a negative-weight cycle, that at some point their entries in the *dist* array are , and that the numbers represent the weights of edges in the cycle, where is the weight of the edge from to . Now, because the cycle has negative weight, we know that . We show by contradiction that it is always possible to relax one of the edges.

Suppose we assume the opposite: that there exists no such that . Then, we have . Adding yields , which, after cancelling the 's, is a contradiction. So our assumption must be false, and it must always be possible to relax at least one of the edges in a negative-weight cycle. Thus, if there is a negative-weight cycle, it will always be detected at the end of the algorithm, because at least one edge on that cycle must be capable of relaxation.

# References

- Thomas H. Cormen, Charles E. Leiserson, Ronald L. Rivest, and Clifford Stein.
*Introduction to Algorithms*, Second Edition. MIT Press and McGraw-Hill, 2001. ISBN 0-262-03293-7. Section 24.1: The Bellman-Ford algorithm, pp.588–592. Problem 24-1, pp.614–615.