Bellman–Ford Algorithm | DP-23

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

We have discussed Dijkstra’s algorithm for this problem. Dijkstra’s algorithm is a Greedy algorithm and the time complexity is O((V+E)LogV) (with the use of the Fibonacci heap). Dijkstra doesn’t work for Graphs with negative weights, Bellman-Ford works for such graphs. Bellman-Ford is also simpler than Dijkstra and suites well for distributed systems. But time complexity of Bellman-Ford is O(V * E), which is more than Dijkstra.

Steps for finding the shortest distance to all vertices from the source using the Bellman-Ford algorithm:

• This step initializes distances from the source to all vertices as infinite and distance to the source itself as 0. Create an array dist[] of size |V| with all values as infinite except dist[src] where src is source vertex.
• 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] to
  • dist[v] = dist[u] + weight of edge uv
• This step reports if there is a negative weight cycle in the graph. Again traverse every edge and 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 the shortest distances if the graph doesn’t contain a 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

How does this work?

Like other Dynamic Programming Problems, the algorithm calculates the shortest paths in a bottom-up manner. It first calculates the shortest distances which have at most one edge in the path. Then, it calculates the shortest paths with at-most 2 edges, and so on. After the i-th iteration of the outer loop, the shortest paths with at most i edges are calculated. There can be maximum |V| – 1 edges in any simple path, that is why the outer loop runs |v| – 1 times. The idea is, assuming that there is no negative weight cycle if we have calculated shortest paths with at most i edges, then an iteration over all edges guarantees to give the shortest path with at-most (i+1) edges

Below is the illustration of the above algorithm:

Step 1: Let the given source vertex be 0. Initialize all distances as infinite, except the distance to the source itself. Total number of vertices in the graph is 5, so all edges must be processed 4 times.

Step 2: Let all edges are processed in the following order: (B, E), (D, B), (B, D), (A, B), (A, C), (D, C), (B, C), (E, D). We get the following distances when all edges are processed the first time. The first row shows initial distances. The second row shows distances when edges (B, E), (D, B), (B, D) and (A, B) are processed. The third row shows distances when (A, C) is processed. The fourth row shows when (D, C), (B, C) and (E, D) are processed. 

Step 3: The first iteration guarantees to give all shortest paths which are at most 1 edge long. We get the following distances when all edges are processed second time (The last row shows final values). 

Step 4: The second iteration guarantees to give all shortest paths which are at most 2 edges long. The algorithm processes all edges 2 more times. The distances are minimized after the second iteration, so third and fourth iterations don’t update the distances.

Below is the implementation of the above approach: 

C++

// 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;
}

Java

// 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

# 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#

// 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
}

Javascript

// 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);

Output

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

Time Complexity:  O(V * E), where V is the number of vertices in the graph and E is the number of edges in the graph
Auxiliary Space: O(E)

Notes:

• Negative weights are found in various applications of graphs. For example, instead of paying the cost for a path, we may get some advantage if we follow the path.
• Bellman-Ford works better (better than Dijkstra’s) for distributed systems. Unlike Dijkstra’s where we need to find the minimum value of all vertices, in Bellman-Ford, edges are considered one by one.                                                                  
• Bellman-Ford does not work with an undirected graph with negative edges as it will be declared as a negative cycle.

Exercise:

• The standard Bellman-Ford algorithm reports the shortest path only if there are no negative weight cycles. Modify it so that it reports minimum distances even if there is a negative weight cycle.
• Can we use Dijkstra’s algorithm for shortest paths for graphs with negative weights – one idea can be, to calculate the minimum weight value, add a positive value (equal to the absolute value of minimum weight value) to all weights and run the Dijkstra’s algorithm for the modified graph. Will this algorithm work?

Bellman Ford Algorithm (Simple Implementation)