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The '''Bellman-Ford algorithm''' finds [[Shortest_path#Single-source_shortest_paths|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 [[Shortest path#All-pairs_shortest_paths|all-pairs shortest paths]], The Bellman–Ford algorithm, with time complexity <math>\mathcal{O}(VE)</math>, outperforms the [[Floyd–Warshall algorithm]] at <math>\mathcal{O}(V^3)</math> 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.
 
The '''Bellman-Ford algorithm''' finds [[Shortest_path#Single-source_shortest_paths|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 [[Shortest path#All-pairs_shortest_paths|all-pairs shortest paths]], The Bellman–Ford algorithm, with time complexity <math>\mathcal{O}(VE)</math>, outperforms the [[Floyd–Warshall algorithm]] at <math>\mathcal{O}(V^3)</math> 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.
  
 
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==The algorithm==
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The idea behind the algorithm is very easy to understand, and may be satisfactorily illustrated by the following pseudocode:
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<pre>
 
input G,v
 
input G,v
 
for each u ∈ V(G)
 
for each u ∈ V(G)

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