Bellman-Ford Algorithm

This algorithm iterates on the no. of edges in a path to get the shortest path. Since the no. of  hops  possible  is  limited (cycles  are  implicitly  not  allowed), the  algorithm terminates giving the direct path.

Notation:

d i,j      =  Length of path between nodes i and j,  indicating the cost of the link. h     =   Number of hops.

D[ i,h]   =  Shortest path length from node i to node 1, with upto 'h' hops. D[ 1,h] =   0 for all h .

Algorithm :

Initial condition    :                 D[ i, 0]  = infinity,  for all  i ( i != 1 )

Iteration             :                 D[i, h+1]  = min { di,j + D[j,h] }      over all values of j .

Termination         :                 The algorithm terminates when D[i, h]  = D [ i,  h+1] for all  i .

Principle:

For 0 hops, the minimum length path has length of time without end, for every node.  For one hop the shortest-path length related with a node is equal to the length of the edge among that node and node 1. after this, we increment the no. of hops allowed, (from h to h+1 ) and find out whether a shorter path exists through every of the  other nodes.  If  it exists, say through node 'j',  then its length must be the sum of the lengths among these 2 nodes (i.e. di,j ) and the shortest path between j and 1 obtainable in up to h paths. If such a path does not exist, then the path length remains the same. The algorithm is confident to terminate, since there are highest N nodes, and so N-1 paths. It has time complexity of O ( N3 ) .

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