Bellman-Ford Shortest Path Algorithm

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 ) then
         distance-from-source-to [ node_v ] = distance-from-source-to [ node_u ] + weight-of-path-from-node-u-to-v )

  • With 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
3.       For every edge E in the EdgeList
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.



Bellman-Ford’s shortest implementation

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 BellmanFord (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)+") : Length => "+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.BellmanFord(source_node)

if __name__ == "__main__":
    main()

Output

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

#include<iostream>
#include<vector>
#include<list>

using namespace std;

class Edge {

    public :

    Edge(int arg_src, int arg_dst, int arg_weight) : src(arg_src), dst(arg_dst), weight(arg_weight)
    {}
    int src;
    int dst;
    int weight;
};

class Graph {

    private :

    int node_count;
    list<Edge> edge_list;

    public:

    Graph (int arg_node_count, list<Edge> arg_edge_list) : node_count(arg_node_count), edge_list(arg_edge_list)
    {}

    void BellmanFord (int src) {

        // Initialize the distance / cost from the source node to all other nodes to some max value.
        vector<int> distance(node_count, 999999999);

        // Distance/cost from the source node to itself is 0.
        distance[src] = 0;

        for (int i=0; i<node_count; i++) {
            for (auto& it : edge_list) {
                if (distance[it.dst] > distance[it.src] + it.weight) {
                    distance[it.dst] = distance[it.src] + it.weight;
                }
            }
        }

        for (auto& it : edge_list) {
            if (distance[it.dst] > distance[it.src] + it.weight) {
               cout << "Negative weight cycle exist in the graph !!!" << endl;
            }
        }

        for (int i=0; i<node_count; i++)
            cout << "Source Node(" << src << ") -> Destination Node(" << i << ") : Length => " << distance[i] << endl;
    }
};

int main(){

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

    int node_count = 6;
    int source_node = 0;
    list<Edge> edge_list = { e01, e05, e12, e15, e23, e24, e43, e45, e51, e52 };

    Graph g(node_count, edge_list);
    g.BellmanFord(source_node);

    return 0;
}

Output

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

import java.util.ArrayList;
import java.util.List;
import java.util.Collections;

class Edge {

    Edge(int arg_src, int arg_dst, int arg_weight) {
         this.src = arg_src;
         this.dst = arg_dst;
         this.weight = arg_weight;
    }
    int src;
    int dst;
    int weight;
}

class Graph {

    int node_count;
    List<Edge> edge_list;

    Graph (int arg_node_count, List<Edge> arg_edge_list) {
        this.node_count = arg_node_count;
        this.edge_list  = arg_edge_list;
    }

    void BellmanFord (int src) {

        // Initialize the distance/cost from the source node to all other nodes to some max value.
        Long INF = (long) 999999999;
        List<Long> distance = new ArrayList<Long>(Collections.nCopies(node_count, INF));

        // Distance/cost from the source node to itself is 0.
        distance.set(src, (long) 0);

        for (int i=0; i<node_count; i++) {
            for (Edge it : edge_list) {
                if (distance.get(it.dst) > distance.get(it.src) + it.weight) {
                    distance.set(it.dst, distance.get(it.src) + it.weight);
                }
            }
        }

        for (Edge it : edge_list) {
            if (distance.get(it.dst) > distance.get(it.src) + it.weight) {
               System.out.println("Negative weight cycle exist in the graph !!!");
            }
        }

        for (int i=0; i<node_count; i++)
            System.out.println("Source Node(" + src + ") -> Destination Node(" + i + ") : Length => " + distance.get(i));
    }

    public static void main (String[] args) {

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

        int node_count = 6;
        int source_node = 0;
        List<Edge> edge_list = new ArrayList<>();
        Collections.addAll(edge_list, e01, e05, e12, e15, e23, e24, e43, e45, e51, e52);

        Graph g = new Graph(node_count, edge_list);
        g.BellmanFord(source_node);
    }
}

Output

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


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