Bellman ford algorithm
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Bellman Ford's algorithm finds the shortest paths from a source vertex to all other vertices in a weighted graph, even if edge weights are negative. It works by repeatedly relaxing all edges to update the distance and previous vertex for each vertex. After iterating through all edges |V|-1 times, if an edge can still be relaxed, then a negative cycle exists in the graph.
![3rd Iteration
Visited
Vertex
A B C D E F
A 0 1 3 5 1 4
B 0 1 3 5 1 4
C 0 1 3 5 0 4
D 0 1 3 5 0 4
E 0 1 3 5 0 4
F 0 1 3 5 0 3
4th Iteration
Visited
Vertex
A B C D E F
A 0 1 3 5 0 3
B 0 1 3 5 0 3
C 0 1 3 5 0 3
D 0 1 3 5 0 3
E 0 1 3 5 0 3
F 0 1 3 5 0 3
Bellman Ford’s Algorithm Pseudocode
function bellmanFord(G, S)
for each vertex V in G
distance[V] <- infinite
previous[V] <- NULL
distance[S] <- 0
for each vertex V in G
for each edge (U,V) in G
tempDistance <- distance[U] + edge_weight(U, V)
if tempDistance < distance[V]
distance[V] <- tempDistance](https://image.slidesharecdn.com/bellmanfordalgorithm-210706143844/85/Bellman-ford-algorithm-2-320.jpg)
![previous[V] <- U
for each edge (U,V) in G
If distance[U] + edge_weight(U, V) < distance[V}
Error: Negative Cycle Exists
return distance[], previous[]](https://image.slidesharecdn.com/bellmanfordalgorithm-210706143844/85/Bellman-ford-algorithm-3-320.jpg)
Bellman ford algorithm
- 1. Bellman Ford’s Algorithm Bellman Ford algorithm helps us find the shortest path from a vertex to all other vertices of a weighted graph. It is similar to Dijkstra's algorithm but it can work with graphs in which edges can have negative weights. Example: Solution if(d(u) + c(u,v) < d(v)) then d(v) = d(u) + c(u,v) 1st Iteration Visited Vertex A B C D E F A 0 ∞ ∞ ∞ ∞ ∞ B 0 6 4 5 ∞ ∞ C 0 6 4 5 5 ∞ D 0 2 4 5 5 ∞ E 0 2 3 5 5 4 F 0 2 3 5 5 4 2nd Iteration Visited Vertex A B C D E F A 0 2 3 5 5 4 B 0 2 3 5 5 4 C 0 2 3 5 1 4 D 0 1 3 5 1 4 E 0 1 3 5 1 4 F 0 1 3 5 1 4 A B E C D F 6 -1 3 -1 5 4 -2 3 -2
- 2. 3rd Iteration Visited Vertex A B C D E F A 0 1 3 5 1 4 B 0 1 3 5 1 4 C 0 1 3 5 0 4 D 0 1 3 5 0 4 E 0 1 3 5 0 4 F 0 1 3 5 0 3 4th Iteration Visited Vertex A B C D E F A 0 1 3 5 0 3 B 0 1 3 5 0 3 C 0 1 3 5 0 3 D 0 1 3 5 0 3 E 0 1 3 5 0 3 F 0 1 3 5 0 3 Bellman Ford’s Algorithm Pseudocode function bellmanFord(G, S) for each vertex V in G distance[V] <- infinite previous[V] <- NULL distance[S] <- 0 for each vertex V in G for each edge (U,V) in G tempDistance <- distance[U] + edge_weight(U, V) if tempDistance < distance[V] distance[V] <- tempDistance
- 3. previous[V] <- U for each edge (U,V) in G If distance[U] + edge_weight(U, V) < distance[V} Error: Negative Cycle Exists return distance[], previous[]