Bellman–Ford single source shortest path algorithm

Bellman–Ford single source shortest path algorithm

Given a graph and a source vertex src in graph, find shortest paths from src to all vertices in the given graph. The graph may contain negative weight edges.

Following are the detailed steps.

Input: Graph and a source vertex src

Output: Shortest distance to all vertices from src. If there is a negative weight cycle, then shortest 

distances are not calculated, negative weight cycle is reported.

1) This step initializes distances from source to all vertices as infinite and distance to source itself as 0. Create an array dist[] of size |V| with all values as infinite except dist[src] where src is source vertex.

2) This step calculates shortest distances. Do following |V|-1 times where |V| is the number of vertices in given graph.

Do following for each edge u-v
             If dist[v] > dist[u] + weight of edge uv, then update dist[v]
             dist[v] = dist[u] + weight of edge uv

3) This step reports if there is a negative weight cycle in graph. Do following for each edge u-v
          If dist[v] > dist[u] + weight of edge uv, then “Graph contains negative weight cycle”

The idea of step 3 is, step 2 guarantees shortest distances if graph doesn’t contain negative weight cycle. If we iterate through all edges one more time and get a shorter path for any vertex, then there is a negative weight cycle

Example:

In this particular example, we have 5 vertices so there will be |V-1| that is 5-1= 4 passes. Each pass relaxes the edges in the order

Is the given graph with source node ‘s’ and initialized to 0 and all remaining distances initialized to infinity.

Start with all the ordered edges and start finding and updating distances

After Pass 4: there is no updating of distances after pass 4. It is the last pass and the shortest path is