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What is Bellman Ford Algorithm?

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



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