Bellman-Ford Shortest Path Algorithm in Python and C++

Bellman-Ford Single Source Shortest Path

The gist of Bellman-Ford single source shortest path algorithm is a below :

  • Bellman-Ford algorithm finds the shortest path (in terms of distance / cost ) from a single source in a directed, weighted graph containing positive and negative edge weights.
  • Bellman-Ford algorithm performs edge relaxation of all the edges for every node.
    Bellman-Ford Edge Relaxation
    if (distance-from-source-to [node_v] > distance-from-source-to [node_u] + weight-of-path-from-node-u-to-v) {
        distance-from-source-to [node_v] = distance-from-source-to [node_u] + weight-of-path-from-node-u-to-v)
    }
  • In presence of negative edge weights in a graph Bellman-Ford algorithm is preferred over Dijkstra’s algorithm as Dijkstra’s algorithm cannot handle negative edge weights in a graph.

Below data structures are used for storing the graph before running Bellman-Ford algorithm

  • EdgeList : List of all the edges in the graph.
  • EdgeWeight : Map of edges and their corresponding weights.

Algorithm : Bellman-Ford Single Source Shortest Path ( EdgeList, EdgeWeight )

1.    Initialize the distance from the source node S to all other nodes as infinite (999999999) and to itself as 0.
      Distance [ AllNodes ] = 999999999, Distance [ S ] = 0.
2.    For every node in the graph do
3.       For every edge E in the EdgeList do
4.           Node_u = E.first, Node_v = E.second
5.           Weight_u_v = EdgeWeight ( Node_u, Node_v )
6.           If ( Distance [v] > Distance [u] + Weight_u_v )
7.                Distance [v] = Distance [u] + Weight_u_v

Example of single source shortest path from source node 0 using Bellman-Ford algorithm

Note: Weight of the path corresponds to the distance / cost between the nodes.

Bellman-Ford iteration 1
Dist [ 0 ] = 0. Distance from source node 0 to itself is 0.
Dist [ 1 ] = Dist [ 2 ] = Dist [ 3 ] = Dist [ 4 ] = Dist [ 5 ] = 999999999.
Initialize the distance from the source node 0 to all other nodes to a max value (ex:999999999).

Node Count
Iteration		 Edge (U-V) Weight (Cost/Distance)
of Edge (U-V)		 Distance from source
to node ‘U’		 Distance from source
to node ‘V’		 Update
1 ( 0 - 1 ) -1 Dist [ 0 ] = 0 Dist [ 1 ] = 999999999 If : Dist [ 1 ] > Dist [ 0 ] + ( -1 )
Yes
Update : Dist [ 1 ] = 0 + -1 = -1
1 ( 0 - 5 ) 2 Dist [ 0 ] = 0 Dist [ 5 ] = 999999999 If : Dist [ 5 ] > Dist [ 0 ] + ( 2 )
Yes
Update : Dist [ 5 ] = 0 + 2 = 2
1 ( 1 - 2 ) 2 Dist [ 1 ] = -1 Dist [ 2 ] = 999999999 If : Dist [ 2 ] > Dist [ 1 ] + ( 2 )
Yes
Update : Dist [ 2 ] = -1 + 2 = 1
1 ( 1 - 5 ) -2 Dist [ 1 ] = -1 Dist [ 5 ] = 2 If : Dist [ 5 ] > Dist [ 1 ] + ( -2 )
Yes
Update : Dist [ 5 ] = -1 + -2 = -3
1 ( 2 - 3 ) 5 Dist [ 2 ] = 1 Dist [ 3 ] = 999999999 If : Dist [ 3 ] > Dist [ 2 ] + ( 5 )
Yes
Update : Dist [ 3 ] = 1 + 5 = 6
1 ( 2 - 4 ) 1 Dist [ 2 ] = 1 Dist [ 4 ] = 999999999 If : Dist [ 4 ] > Dist [ 2 ] + ( 1 )
Yes
Update : Dist [ 4 ] = 1 + 1 = 2
1 ( 4 - 3 ) -4 Dist [ 4 ] = 2 Dist [ 3 ] = 6 If : Dist [ 3 ] > Dist [ 4 ] + ( -4 )
Yes
Update : Dist [ 3 ] = 6 + (-4) = 2
1 ( 4 - 5 ) 3 Dist [ 4 ] = 2 Dist [ 5 ] = -3 If : Dist [ 5 ] > Dist [ 4 ] + ( 3 )
No
No change : Dist [ 5 ] = -3
1 ( 5 - 1 ) 2 Dist [ 5 ] = -3 Dist [ 1 ] = -1 If : Dist [ 1 ] > Dist [ 5 ] + ( 2 )
No
No change : Dist [ 1 ] = -1
1 ( 5 - 2 ) 3 Dist [ 5 ] = -3 Dist [ 2 ] = 1 If : Dist [ 2 ] > Dist [ 5 ] + ( 3 )
Yes
Update Dist [ 2 ] = -3 + 3 = 0

…
the algorithm continues with iterations 2, 3, 4, 5 and 6 (number of nodes). During each iteration the shortest path from source node to other nodes is updated.

Bellman_Ford_Shortest_Path

Graph type : Designed for directed graph containing positive and negative edge weights.
Time complexity of Bellman-Ford’s algorithm : O(E.V). V is the number of vertices and E is the number of edges in a graph.

C++

Bellman-Ford's shortest path algorithm in C++14

Python : Bellman-Ford’s shortest path algorithm in Python 3

class Edge :

    def __init__(self, src, dst, weight) :
         self.src = src 
         self.dst = dst 
         self.weight = weight

class Graph :

    def __init__(self, edge_list, node_cnt) :
         self.edge_list = edge_list
         self.node_cnt  = node_cnt

    def BellmanFordShortestPath(self, src) :

        # Initialize the distance from the source node S to all other nodes as infinite (999999999) and to itself as 0.
        distance = [999999999999] * self.node_cnt
        distance[src] = 0 

        for node in range(self.node_cnt) :
            for edge in self.edge_list :

                if (distance[edge.dst] > distance[edge.src] + edge.weight) :
                    distance[edge.dst] = distance[edge.src] + edge.weight

        for edge in self.edge_list :

            if (distance[edge.dst] > distance[edge.src] + edge.weight) :
                print("Negative weight cycle exist in the graph")

        for node in range(self.node_cnt) : 
            print("Source Node("+str(src)+") -> Destination Node("+str(node)+") : "+str(distance[node]))

def main() :

    e01 = Edge(0, 1, -1) 
    e05 = Edge(0, 5, 2)
    e12 = Edge(1, 2, 2)
    e15 = Edge(1, 5, -2) 
    e23 = Edge(2, 3, 5)
    e24 = Edge(2, 4, 1)
    e43 = Edge(4, 3, -4) 
    e45 = Edge(4, 5, 3)
    e51 = Edge(5, 1, 2)
    e52 = Edge(5, 2, 3)

    edge_list = [e01, e05, e12, e15, e23, e24, e43, e45, e51, e52]
    node_cnt = 6 
    source_node = 0 
 
    g = Graph(edge_list, node_cnt)
    g.BellmanFordShortestPath(source_node)

Output of Bellman-Ford’s shortest path algorithm implemented in Python 3

Source Node(0) -> Destination Node(0) : 0
Source Node(0) -> Destination Node(1) : -1
Source Node(0) -> Destination Node(2) : 0
Source Node(0) -> Destination Node(3) : -3
Source Node(0) -> Destination Node(4) : 1
Source Node(0) -> Destination Node(5) : -3
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