Bellman–Ford Algorithm

Imagine you have a map with different cities connected by roads, each road having a certain distance. The Bellman–Ford algorithm is like a guide that helps you find the shortest path from one city to all other cities, even if some roads have negative lengths. It’s like a GPS for computers, useful for figuring out the quickest way to get from one point to another in a network. In this article, we’ll take a closer look at how this algorithm works and why it’s so handy in solving everyday problems.

Bellman-Ford Algorithm

Bellman-Ford is a single source shortest path algorithm that determines the shortest path between a given source vertex and every other vertex in a graph. This algorithm can be used on both weighted and unweighted graphs.

A Bellman-Ford algorithm is also guaranteed to find the shortest path in a graph, similar to Dijkstra’s algorithm. Although Bellman-Ford is slower than Dijkstra’s algorithm, it is capable of handling graphs with negative edge weights, which makes it more versatile. The shortest path cannot be found if there exists a negative cycle in the graph. If we continue to go around the negative cycle an infinite number of times, then the cost of the path will continue to decrease (even though the length of the path is increasing). As a result, Bellman-Ford is also capable of detecting negative cycles, which is an important feature.

The idea behind Bellman Ford Algorithm:

The Bellman-Ford algorithm’s primary principle is that it starts with a single source and calculates the distance to each node. The distance is initially unknown and assumed to be infinite, but as time goes on, the algorithm relaxes those paths by identifying a few shorter paths. Hence it is said that Bellman-Ford is based on “Principle of Relaxation“.

Principle of Relaxation of Edges for Bellman-Ford:

• It states that for the graph having N vertices, all the edges should be relaxed N-1 times to compute the single source shortest path.
• In order to detect whether a negative cycle exists or not, relax all the edge one more time and if the shortest distance for any node reduces then we can say that a negative cycle exists. In short if we relax the edges N times, and there is any change in the shortest distance of any node between the N-1th and Nth relaxation than a negative cycle exists, otherwise not exist.

Why Relaxing Edges N-1 times, gives us Single Source Shortest Path?

In the worst-case scenario, a shortest path between two vertices can have at most N-1 edges, where N is the number of vertices. This is because a simple path in a graph with N vertices can have at most N-1 edges, as it’s impossible to form a closed loop without revisiting a vertex.

By relaxing edges N-1 times, the Bellman-Ford algorithm ensures that the distance estimates for all vertices have been updated to their optimal values, assuming the graph doesn’t contain any negative-weight cycles reachable from the source vertex. If a graph contains a negative-weight cycle reachable from the source vertex, the algorithm can detect it after N-1 iterations, since the negative cycle disrupts the shortest path lengths.

In summary, relaxing edges N-1 times in the Bellman-Ford algorithm guarantees that the algorithm has explored all possible paths of length up to N-1, which is the maximum possible length of a shortest path in a graph with N vertices. This allows the algorithm to correctly calculate the shortest paths from the source vertex to all other vertices, given that there are no negative-weight cycles.

Why Does the Reduction of Distance in the N’th Relaxation Indicates the Existence of a Negative Cycle?

As previously discussed, achieving the single source shortest paths to all other nodes takes N-1 relaxations. If the N’th relaxation further reduces the shortest distance for any node, it implies that a certain edge with negative weight has been traversed once more. It is important to note that during the N-1 relaxations, we presumed that each vertex is traversed only once. However, the reduction of distance during the N’th relaxation indicates revisiting a vertex.

Working of Bellman-Ford Algorithm to Detect the Negative cycle in the graph:

Let’s suppose we have a graph which is given below and we want to find whether there exists a negative cycle or not using Bellman-Ford.

Initial Graph

Step 1: Initialize a distance array Dist[] to store the shortest distance for each vertex from the source vertex. Initially distance of source will be 0 and Distance of other vertices will be INFINITY.

Initialize a distance array

Step 2: Start relaxing the edges, during 1st Relaxation:

• Current Distance of B > (Distance of A) + (Weight of A to B) i.e. Infinity > 0 + 5
  • Therefore, Dist[B] = 5

1st Relaxation

Step 3: During 2nd Relaxation:

• Current Distance of D > (Distance of B) + (Weight of B to D) i.e. Infinity > 5 + 2
  • Dist[D] = 7
• Current Distance of C > (Distance of B) + (Weight of B to C) i.e. Infinity > 5 + 1
  • Dist[C] = 6

2nd Relaxation

Step 4: During 3rd Relaxation:

• Current Distance of F > (Distance of D ) + (Weight of D to F) i.e. Infinity > 7 + 2
  • Dist[F] = 9
• Current Distance of E > (Distance of C ) + (Weight of C to E) i.e. Infinity > 6 + 1
  • Dist[E] = 7

3rd Relaxation

Step 5: During 4th Relaxation:

• Current Distance of D > (Distance of E) + (Weight of E to D) i.e. 7 > 7 + (-1)
  • Dist[D] = 6
• Current Distance of E > (Distance of F ) + (Weight of F to E) i.e. 7 > 9 + (-3)
  • Dist[E] = 6

4th Relaxation

Step 6: During 5th Relaxation:

• Current Distance of F > (Distance of D) + (Weight of D to F) i.e. 9 > 6 + 2
  • Dist[F] = 8
• Current Distance of D > (Distance of E ) + (Weight of E to D) i.e. 6 > 6 + (-1)
  • Dist[D] = 5
• Since the graph h 6 vertices, So during the 5th relaxation the shortest distance for all the vertices should have been calculated.

5th Relaxation

Step 7: Now the final relaxation i.e. the 6th relaxation should indicate the presence of negative cycle if there is any changes in the distance array of 5th relaxation.

During the 6th relaxation, following changes can be seen:

• Current Distance of E > (Distance of F) + (Weight of F to E) i.e. 6 > 8 + (-3)
  • Dist[E]=5
• Current Distance of F > (Distance of D ) + (Weight of D to F) i.e. 8 > 5 + 2
  • Dist[F]=7

Since, we observer changes in the Distance array Hence ,we can conclude the presence of a negative cycle in the graph.

6th Relaxation

Result: A negative cycle (D->F->E) exists in the graph.

Algorithm to Find Negative Cycle in a Directed Weighted Graph Using Bellman-Ford:

• Initialize distance array dist[] for each vertex ‘v‘ as dist[v] = INFINITY.
• Assume any vertex (let’s say ‘0’) as source and assign dist = 0.
• Relax all the edges(u,v,weight) N-1 times as per the below condition:
  • dist[v] = minimum(dist[v], distance[u] + weight)
• Now, Relax all the edges one more time i.e. the Nth time and based on the below two cases we can detect the negative cycle:
  • Case 1 (Negative cycle exists): For any edge(u, v, weight), if dist[u] + weight < dist[v]
  • Case 2 (No Negative cycle) : case 1 fails for all the edges.

Handling Disconnected Graphs in the Algorithm:

The above algorithm and program might not work if the given graph is disconnected. It works when all vertices are reachable from source vertex 0.
To handle disconnected graphs, we can repeat the above algorithm for vertices having distance = INFINITY, or simply for the vertices that are not visited.

Below is the implementation of the above approach: 

// A C++ program for Bellman-Ford's single source
// shortest path algorithm.
#include <bits/stdc++.h>
using namespace std;

// a structure to represent a weighted edge in graph
struct Edge {
    int src, dest, weight;
};

// a structure to represent a connected, directed and
// weighted graph
struct Graph {
    // V-> Number of vertices, E-> Number of edges
    int V, E;

    // graph is represented as an array of edges.
    struct Edge* edge;
};

// Creates a graph with V vertices and E edges
struct Graph* createGraph(int V, int E)
{
    struct Graph* graph = new Graph;
    graph->V = V;
    graph->E = E;
    graph->edge = new Edge[E];
    return graph;
}

// A utility function used to print the solution
void printArr(int dist[], int n)
{
    printf("Vertex   Distance from Source\n");
    for (int i = 0; i < n; ++i)
        printf("%d \t\t %d\n", i, dist[i]);
}

// The main function that finds shortest distances from src
// to all other vertices using Bellman-Ford algorithm.  The
// function also detects negative weight cycle
void BellmanFord(struct Graph* graph, int src)
{
    int V = graph->V;
    int E = graph->E;
    int dist[V];

    // Step 1: Initialize distances from src to all other
    // vertices as INFINITE
    for (int i = 0; i < V; i++)
        dist[i] = INT_MAX;
    dist[src] = 0;

    // Step 2: Relax all edges |V| - 1 times. A simple
    // shortest path from src to any other vertex can have
    // at-most |V| - 1 edges
    for (int i = 1; i <= V - 1; i++) {
        for (int j = 0; j < E; j++) {
            int u = graph->edge[j].src;
            int v = graph->edge[j].dest;
            int weight = graph->edge[j].weight;
            if (dist[u] != INT_MAX
                && dist[u] + weight < dist[v])
                dist[v] = dist[u] + weight;
        }
    }

    // Step 3: check for negative-weight cycles.  The above
    // step guarantees shortest distances if graph doesn't
    // contain negative weight cycle.  If we get a shorter
    // path, then there is a cycle.
    for (int i = 0; i < E; i++) {
        int u = graph->edge[i].src;
        int v = graph->edge[i].dest;
        int weight = graph->edge[i].weight;
        if (dist[u] != INT_MAX
            && dist[u] + weight < dist[v]) {
            printf("Graph contains negative weight cycle");
            return; // If negative cycle is detected, simply
                    // return
        }
    }

    printArr(dist, V);

    return;
}

// Driver's code
int main()
{
    /* Let us create the graph given in above example */
    int V = 5; // Number of vertices in graph
    int E = 8; // Number of edges in graph
    struct Graph* graph = createGraph(V, E);

    // add edge 0-1 (or A-B in above figure)
    graph->edge[0].src = 0;
    graph->edge[0].dest = 1;
    graph->edge[0].weight = -1;

    // add edge 0-2 (or A-C in above figure)
    graph->edge[1].src = 0;
    graph->edge[1].dest = 2;
    graph->edge[1].weight = 4;

    // add edge 1-2 (or B-C in above figure)
    graph->edge[2].src = 1;
    graph->edge[2].dest = 2;
    graph->edge[2].weight = 3;

    // add edge 1-3 (or B-D in above figure)
    graph->edge[3].src = 1;
    graph->edge[3].dest = 3;
    graph->edge[3].weight = 2;

    // add edge 1-4 (or B-E in above figure)
    graph->edge[4].src = 1;
    graph->edge[4].dest = 4;
    graph->edge[4].weight = 2;

    // add edge 3-2 (or D-C in above figure)
    graph->edge[5].src = 3;
    graph->edge[5].dest = 2;
    graph->edge[5].weight = 5;

    // add edge 3-1 (or D-B in above figure)
    graph->edge[6].src = 3;
    graph->edge[6].dest = 1;
    graph->edge[6].weight = 1;

    // add edge 4-3 (or E-D in above figure)
    graph->edge[7].src = 4;
    graph->edge[7].dest = 3;
    graph->edge[7].weight = -3;
    
      // Function call
    BellmanFord(graph, 0);

    return 0;
}

// A Java program for Bellman-Ford's single source shortest
// path algorithm.

import java.io.*;
import java.lang.*;
import java.util.*;

// A class to represent a connected, directed and weighted
// graph
class Graph {
  
    // A class to represent a weighted edge in graph
    class Edge {
        int src, dest, weight;
        Edge() { src = dest = weight = 0; }
    };

    int V, E;
    Edge edge[];

    // Creates a graph with V vertices and E edges
    Graph(int v, int e)
    {
        V = v;
        E = e;
        edge = new Edge[e];
        for (int i = 0; i < e; ++i)
            edge[i] = new Edge();
    }

    // The main function that finds shortest distances from
    // src to all other vertices using Bellman-Ford
    // algorithm. The function also detects negative weight
    // cycle
    void BellmanFord(Graph graph, int src)
    {
        int V = graph.V, E = graph.E;
        int dist[] = new int[V];

        // Step 1: Initialize distances from src to all
        // other vertices as INFINITE
        for (int i = 0; i < V; ++i)
            dist[i] = Integer.MAX_VALUE;
        dist[src] = 0;

        // Step 2: Relax all edges |V| - 1 times. A simple
        // shortest path from src to any other vertex can
        // have at-most |V| - 1 edges
        for (int i = 1; i < V; ++i) {
            for (int j = 0; j < E; ++j) {
                int u = graph.edge[j].src;
                int v = graph.edge[j].dest;
                int weight = graph.edge[j].weight;
                if (dist[u] != Integer.MAX_VALUE
                    && dist[u] + weight < dist[v])
                    dist[v] = dist[u] + weight;
            }
        }

        // Step 3: check for negative-weight cycles. The
        // above step guarantees shortest distances if graph
        // doesn't contain negative weight cycle. If we get
        // a shorter path, then there is a cycle.
        for (int j = 0; j < E; ++j) {
            int u = graph.edge[j].src;
            int v = graph.edge[j].dest;
            int weight = graph.edge[j].weight;
            if (dist[u] != Integer.MAX_VALUE
                && dist[u] + weight < dist[v]) {
                System.out.println(
                    "Graph contains negative weight cycle");
                return;
            }
        }
        printArr(dist, V);
    }

    // A utility function used to print the solution
    void printArr(int dist[], int V)
    {
        System.out.println("Vertex Distance from Source");
        for (int i = 0; i < V; ++i)
            System.out.println(i + "\t\t" + dist[i]);
    }

    // Driver's code
    public static void main(String[] args)
    {
        int V = 5; // Number of vertices in graph
        int E = 8; // Number of edges in graph

        Graph graph = new Graph(V, E);

        // add edge 0-1 (or A-B in above figure)
        graph.edge[0].src = 0;
        graph.edge[0].dest = 1;
        graph.edge[0].weight = -1;

        // add edge 0-2 (or A-C in above figure)
        graph.edge[1].src = 0;
        graph.edge[1].dest = 2;
        graph.edge[1].weight = 4;

        // add edge 1-2 (or B-C in above figure)
        graph.edge[2].src = 1;
        graph.edge[2].dest = 2;
        graph.edge[2].weight = 3;

        // add edge 1-3 (or B-D in above figure)
        graph.edge[3].src = 1;
        graph.edge[3].dest = 3;
        graph.edge[3].weight = 2;

        // add edge 1-4 (or B-E in above figure)
        graph.edge[4].src = 1;
        graph.edge[4].dest = 4;
        graph.edge[4].weight = 2;

        // add edge 3-2 (or D-C in above figure)
        graph.edge[5].src = 3;
        graph.edge[5].dest = 2;
        graph.edge[5].weight = 5;

        // add edge 3-1 (or D-B in above figure)
        graph.edge[6].src = 3;
        graph.edge[6].dest = 1;
        graph.edge[6].weight = 1;

        // add edge 4-3 (or E-D in above figure)
        graph.edge[7].src = 4;
        graph.edge[7].dest = 3;
        graph.edge[7].weight = -3;
        
          // Function call
        graph.BellmanFord(graph, 0);
    }
}
// Contributed by Aakash Hasija

# Python3 program for Bellman-Ford's single source
# shortest path algorithm.

# Class to represent a graph

class Graph:

    def __init__(self, vertices):
        self.V = vertices  # No. of vertices
        self.graph = []

    # function to add an edge to graph
    def addEdge(self, u, v, w):
        self.graph.append([u, v, w])

    # utility function used to print the solution
    def printArr(self, dist):
        print("Vertex Distance from Source")
        for i in range(self.V):
            print("{0}\t\t{1}".format(i, dist[i]))

    # The main function that finds shortest distances from src to
    # all other vertices using Bellman-Ford algorithm. The function
    # also detects negative weight cycle
    def BellmanFord(self, src):

        # Step 1: Initialize distances from src to all other vertices
        # as INFINITE
        dist = [float("Inf")] * self.V
        dist[src] = 0

        # Step 2: Relax all edges |V| - 1 times. A simple shortest
        # path from src to any other vertex can have at-most |V| - 1
        # edges
        for _ in range(self.V - 1):
            # Update dist value and parent index of the adjacent vertices of
            # the picked vertex. Consider only those vertices which are still in
            # queue
            for u, v, w in self.graph:
                if dist[u] != float("Inf") and dist[u] + w < dist[v]:
                    dist[v] = dist[u] + w

        # Step 3: check for negative-weight cycles. The above step
        # guarantees shortest distances if graph doesn't contain
        # negative weight cycle. If we get a shorter path, then there
        # is a cycle.

        for u, v, w in self.graph:
            if dist[u] != float("Inf") and dist[u] + w < dist[v]:
                print("Graph contains negative weight cycle")
                return

        # print all distance
        self.printArr(dist)


# Driver's code
if __name__ == '__main__':
    g = Graph(5)
    g.addEdge(0, 1, -1)
    g.addEdge(0, 2, 4)
    g.addEdge(1, 2, 3)
    g.addEdge(1, 3, 2)
    g.addEdge(1, 4, 2)
    g.addEdge(3, 2, 5)
    g.addEdge(3, 1, 1)
    g.addEdge(4, 3, -3)

    # function call
    g.BellmanFord(0)

# Initially, Contributed by Neelam Yadav
# Later On, Edited by Himanshu Garg

// C# program for Bellman-Ford's single source shortest
// path algorithm.

using System;

// A class to represent a connected, directed and weighted
// graph
class Graph {
    // A class to represent a weighted edge in graph
    class Edge {
        public int src, dest, weight;
        public Edge() { src = dest = weight = 0; }
    };

    int V, E;
    Edge[] edge;

    // Creates a graph with V vertices and E edges
    Graph(int v, int e)
    {
        V = v;
        E = e;
        edge = new Edge[e];
        for (int i = 0; i < e; ++i)
            edge[i] = new Edge();
    }

    // The main function that finds shortest distances from
    // src to all other vertices using Bellman-Ford
    // algorithm. The function also detects negative weight
    // cycle
    void BellmanFord(Graph graph, int src)
    {
        int V = graph.V, E = graph.E;
        int[] dist = new int[V];

        // Step 1: Initialize distances from src to all
        // other vertices as INFINITE
        for (int i = 0; i < V; ++i)
            dist[i] = int.MaxValue;
        dist[src] = 0;

        // Step 2: Relax all edges |V| - 1 times. A simple
        // shortest path from src to any other vertex can
        // have at-most |V| - 1 edges
        for (int i = 1; i < V; ++i) {
            for (int j = 0; j < E; ++j) {
                int u = graph.edge[j].src;
                int v = graph.edge[j].dest;
                int weight = graph.edge[j].weight;
                if (dist[u] != int.MaxValue
                    && dist[u] + weight < dist[v])
                    dist[v] = dist[u] + weight;
            }
        }

        // Step 3: check for negative-weight cycles. The
        // above step guarantees shortest distances if graph
        // doesn't contain negative weight cycle. If we get
        // a shorter path, then there is a cycle.
        for (int j = 0; j < E; ++j) {
            int u = graph.edge[j].src;
            int v = graph.edge[j].dest;
            int weight = graph.edge[j].weight;
            if (dist[u] != int.MaxValue
                && dist[u] + weight < dist[v]) {
                Console.WriteLine(
                    "Graph contains negative weight cycle");
                return;
            }
        }
        printArr(dist, V);
    }

    // A utility function used to print the solution
    void printArr(int[] dist, int V)
    {
        Console.WriteLine("Vertex Distance from Source");
        for (int i = 0; i < V; ++i)
            Console.WriteLine(i + "\t\t" + dist[i]);
    }

    // Driver's code
    public static void Main()
    {
        int V = 5; // Number of vertices in graph
        int E = 8; // Number of edges in graph

        Graph graph = new Graph(V, E);

        // add edge 0-1 (or A-B in above figure)
        graph.edge[0].src = 0;
        graph.edge[0].dest = 1;
        graph.edge[0].weight = -1;

        // add edge 0-2 (or A-C in above figure)
        graph.edge[1].src = 0;
        graph.edge[1].dest = 2;
        graph.edge[1].weight = 4;

        // add edge 1-2 (or B-C in above figure)
        graph.edge[2].src = 1;
        graph.edge[2].dest = 2;
        graph.edge[2].weight = 3;

        // add edge 1-3 (or B-D in above figure)
        graph.edge[3].src = 1;
        graph.edge[3].dest = 3;
        graph.edge[3].weight = 2;

        // add edge 1-4 (or B-E in above figure)
        graph.edge[4].src = 1;
        graph.edge[4].dest = 4;
        graph.edge[4].weight = 2;

        // add edge 3-2 (or D-C in above figure)
        graph.edge[5].src = 3;
        graph.edge[5].dest = 2;
        graph.edge[5].weight = 5;

        // add edge 3-1 (or D-B in above figure)
        graph.edge[6].src = 3;
        graph.edge[6].dest = 1;
        graph.edge[6].weight = 1;

        // add edge 4-3 (or E-D in above figure)
        graph.edge[7].src = 4;
        graph.edge[7].dest = 3;
        graph.edge[7].weight = -3;
        
          // Function call
        graph.BellmanFord(graph, 0);
    }
    // This code is contributed by Ryuga
}

// a structure to represent a connected, directed and
// weighted graph
class Edge {
  constructor(src, dest, weight) {
    this.src = src;
    this.dest = dest;
    this.weight = weight;
  }
}

class Graph {
  constructor(V, E) {
    this.V = V;
    this.E = E;
    this.edge = [];
  }
}

function createGraph(V, E) {
  const graph = new Graph(V, E);
  for (let i = 0; i < E; i++) {
    graph.edge[i] = new Edge();
  }
  return graph;
}

function printArr(dist, V) {
  console.log("Vertex Distance from Source");
  for (let i = 0; i < V; i++) {
    console.log(`${i} \t\t ${dist[i]}`);
  }
}

function BellmanFord(graph, src) {
  const V = graph.V;
  const E = graph.E;
  const dist = [];

  for (let i = 0; i < V; i++) {
    dist[i] = Number.MAX_SAFE_INTEGER;
  }
  dist[src] = 0;

  for (let i = 1; i <= V - 1; i++) {
    for (let j = 0; j < E; j++) {
      const u = graph.edge[j].src;
      const v = graph.edge[j].dest;
      const weight = graph.edge[j].weight;
      if (dist[u] !== Number.MAX_SAFE_INTEGER && dist[u] + weight < dist[v]) {
        dist[v] = dist[u] + weight;
      }
    }
  }

  for (let i = 0; i < E; i++) {
    const u = graph.edge[i].src;
    const v = graph.edge[i].dest;
    const weight = graph.edge[i].weight;
    if (dist[u] !== Number.MAX_SAFE_INTEGER && dist[u] + weight < dist[v]) {
      console.log("Graph contains negative weight cycle");
      return;
    }
  }

  printArr(dist, V);
}

// Driver program to test methods of graph class
     
// Create a graph given in the above diagram

const V = 5;
const E = 8;
const graph = createGraph(V, E);

graph.edge[0] = new Edge(0, 1, -1);
graph.edge[1] = new Edge(0, 2, 4);
graph.edge[2] = new Edge(1, 2, 3);
graph.edge[3] = new Edge(1, 3, 2);
graph.edge[4] = new Edge(1, 4, 2);
graph.edge[5] = new Edge(3, 2, 5);
graph.edge[6] = new Edge(3, 1, 1);
graph.edge[7] = new Edge(4, 3, -3);

BellmanFord(graph, 0);

Vertex   Distance from Source
0          0
1          -1
2          2
3          -2
4          1

Complexity Analysis of Bellman-Ford Algorithm:

• Time Complexity when graph is connected:
  • Best Case: O(E), when distance array after 1st and 2nd relaxation are same , we can simply stop further processing
  • Average Case: O(V*E)
  • Worst Case: O(V*E)
• Time Complexity when graph is disconnected:
  • All the cases: O(E*(V^2))
• Auxiliary Space: O(V), where V is the number of vertices in the graph.

Bellman Ford’s Algorithm Applications:

• Network Routing: Bellman-Ford is used in computer networking to find the shortest paths in routing tables, helping data packets navigate efficiently across networks.
• GPS Navigation: GPS devices use Bellman-Ford to calculate the shortest or fastest routes between locations, aiding navigation apps and devices.
• Transportation and Logistics: Bellman-Ford’s algorithm can be applied to determine the optimal paths for vehicles in transportation and logistics, minimizing fuel consumption and travel time.
• Game Development: Bellman-Ford can be used to model movement and navigation within virtual worlds in game development, where different paths may have varying costs or obstacles.
• Robotics and Autonomous Vehicles: The algorithm aids in path planning for robots or autonomous vehicles, considering obstacles, terrain, and energy consumption.

Drawback of Bellman Ford’s Algorithm:

• Bellman-Ford algorithm will fail if the graph contains any negative edge cycle.

Related articles on single source shortest path algorithms:

• Dijkstra’s Algorithm
• Floyd-Warshall Algorithm

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