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Bellman ford algorithm returns the least-cost paths between any pair of nodes.

 Bellman_Ford (B, x, c)single source Initialize (B, c)for(j=0 to V-1) For each_edge (v1, v2) in E1   relax(v1, v2);for each_edge (v1, v2) in E1  if d[v2] > d[v1] + w(v1, v2)  return weight cycle -ve

RUNNING_TIME

 for (j=1 to V-1) for each_edge (v1, v2) in E1   relax (v1, v2);above time taken (V-1)O(E1) =   O(VE1)for each_edge (v1, v2) in E1  if d[v2] > d[v1] + w[v1, v2]  return weight cycle -ve above time taken O(E1)

Properties of Algorithms:

• Specified Input
• Specified Output
• Definiteness
• Effectiveness

Bellman  ford algorithm example:

1. Let W want to find shortest path to each destination
2. first, calculate shortest one-link paths from each node: easy, X(i, j)[1]=x(i, j)

 X( W, V   )[ 1 ], X( W, X )[ 1 ], andX( V, U   )[ 1 ], X( V, Y )[ 1 ], X( V, W )[ 1 ], andX( U, V   )[ 1 ], X( U, Y )[ 1 ], X( U, Z )[ 1 ], andX( Y, U   )[ 1 ], X( Y, V )[ 1 ], X( Y, X )[ 1 ], X( Y, Z )[ 1 ], andX( Z, U   )[ 1 ], X( Z, Y )[ 1 ]