Detect a negative cycle in a Graph | (Bellman Ford)

We are given a directed graph. We need to compute whether the graph has a negative cycle or not. A negative cycle is one in which the overall sum of the cycle becomes negative.

Negative weights are found in various applications of graphs. For example, instead of paying cost for a path, we may get some advantage if we follow the path.

Examples: 

Input : 4 4
        0 1 1
        1 2 -1
        2 3 -1
        3 0 -1

Output : Yes
The graph contains a negative cycle.

The idea is to use Bellman-Ford Algorithm. 

Below is an algorithm to find if there is a negative weight cycle reachable from the given source.

1. Initialize 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 the source vertex.
2. This step calculates the shortest distances. Do the following |V|-1 times where |V| is the number of vertices in the given graph. 
   1. Do the following for each edge u-v.
   2. If dist[v] > dist[u] + weight of edge uv, then update dist[v]. 
   3. dist[v] = dist[u] + weight of edge uv.
3. This step reports if there is a negative weight cycle in the graph. Do the following for each edge u-v  
   1. If dist[v] > dist[u] + weight of edge uv, then the “Graph has a 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.

Implementation:

C++

// A C++ program to check if a graph contains negative
// weight cycle using Bellman-Ford algorithm. This program
// works only if all vertices are reachable from a source
// vertex 0.
#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[graph->E];
    return graph;
}

// The main function that finds shortest distances
// from src to all other vertices using Bellman-
// Ford algorithm.  The function also detects
// negative weight cycle
bool isNegCycleBellmanFord(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])
            return true;
    }

    return false;
}

// Driver program to test above functions
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 A-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;

    if (isNegCycleBellmanFord(graph, 0))
        cout << "Yes";
    else
        cout << "No";

    return 0;
}

Java

// Java program to check if a graph contains negative
// weight cycle using Bellman-Ford algorithm. This program
// works only if all vertices are reachable from a source
// vertex 0.
import java.util.*;

class GFG {

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

    // a structure to represent a connected, directed and
    // weighted graph
    static class Graph {

        // V-> Number of vertices, E-> Number of edges
        int V, E;

        // graph is represented as an array of edges.
        Edge edge[];

    }

    // Creates a graph with V vertices and E edges
    static Graph createGraph(int V, int E) {
        Graph graph = new Graph();
        graph.V = V;
        graph.E = E;
        graph.edge = new Edge[graph.E];

        for (int i = 0; i < graph.E; i++) {
            graph.edge[i] = new Edge();
        }

        return graph;
    }

    // The main function that finds shortest distances
    // from src to all other vertices using Bellman-
    // Ford algorithm. The function also detects
    // negative weight cycle
    static boolean isNegCycleBellmanFord(Graph graph, int src) {
        int V = graph.V;
        int 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 - 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] != 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 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] != Integer.MAX_VALUE && dist[u] + weight < dist[v])
                return true;
        }

        return false;
    }

    // Driver Code
    public static void main(String[] args) {
        int V = 5, E = 8;
        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 A-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;

        if (isNegCycleBellmanFord(graph, 0))
            System.out.println("Yes");
        else
            System.out.println("No");
    }
}

// This code is contributed by
// sanjeev2552

Python3

# A Python3 program to check if a graph contains negative
# weight cycle using Bellman-Ford algorithm. This program
# works only if all vertices are reachable from a source
# vertex 0.

# a structure to represent a weighted edge in graph
class Edge:

    def __init__(self):
        self.src = 0
        self.dest = 0
        self.weight = 0

# a structure to represent a connected, directed and
# weighted graph
class Graph:

    def __init__(self):

        # V. Number of vertices, E. Number of edges
        self.V = 0
        self.E = 0

        # graph is represented as an array of edges.
        self.edge = None

# Creates a graph with V vertices and E edges
def createGraph(V, E):

    graph = Graph()
    graph.V = V;
    graph.E = E;
    graph.edge =[Edge() for i in range(graph.E)]
    return graph;

# 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 isNegCycleBellmanFord(graph, src):

    V = graph.V;
    E = graph.E;
    dist = [1000000 for i in range(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 i in range(1, V):
        for j in range(E):

            u = graph.edge[j].src;
            v = graph.edge[j].dest;
            weight = graph.edge[j].weight;
            if (dist[u] != 1000000 and 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 i in range(E):

        u = graph.edge[i].src;
        v = graph.edge[i].dest;
        weight = graph.edge[i].weight;
        if (dist[u] != 1000000 and dist[u] + weight < dist[v]):
            return True;

    return False;

# Driver program to test above functions
if __name__=='__main__':

    # Let us create the graph given in above example
    V = 5; # Number of vertices in graph
    E = 8; # Number of edges in 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 A-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;

    if (isNegCycleBellmanFord(graph, 0)):
        print("Yes")
    else:
        print("No")

        # This code is contributed by pratham76

C#

// C# program to check if a graph contains negative
// weight cycle using Bellman-Ford algorithm. This program
// works only if all vertices are reachable from a source
// vertex 0.
using System;
using System.Collections;
using System.Collections.Generic;

class GFG {

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

    // a structure to represent a connected, directed and
    // weighted graph
    class Graph {

        // V-> Number of vertices, E-> Number of edges
        public int V, E;

        // graph is represented as an array of edges.
        public Edge []edge;

    }

    // Creates a graph with V vertices and E edges
    static Graph createGraph(int V, int E) {
        Graph graph = new Graph();
        graph.V = V;
        graph.E = E;
        graph.edge = new Edge[graph.E];

        for (int i = 0; i < graph.E; i++) {
            graph.edge[i] = new Edge();
        }

        return graph;
    }

    // The main function that finds shortest distances
    // from src to all other vertices using Bellman-
    // Ford algorithm. The function also detects
    // negative weight cycle
    static bool isNegCycleBellmanFord(Graph graph, int src) {
        int V = graph.V;
        int 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] = 1000000;
        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] != 1000000 && 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] != 1000000 && dist[u] + weight < dist[v])
                return true;
        }

        return false;
    }

    // Driver Code
    public static void Main(string[] args) {
        int V = 5, E = 8;
        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 A-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;

        if (isNegCycleBellmanFord(graph, 0))
            Console.Write("Yes");
        else
            Console.Write("No");
    }
}

// This code is contributed by rutvik_56

Javascript

<script>

// Javascript program to check if a graph contains negative
// weight cycle using Bellman-Ford algorithm. This program
// works only if all vertices are reachable from a source
// vertex 0.

// A structure to represent a weighted edge in graph
class Edge
{
    constructor()
    {
        let src, dest, weight;
    }
}

// A structure to represent a connected, directed and
// weighted graph
class Graph
{
    constructor()
    {

        // V-> Number of vertices, E-> Number of edges
        let V, E;

        // graph is represented as an array of edges.
        let edge = [];
    }
}

// Creates a graph with V vertices and E edges
function createGraph(V,E)
{
    let graph = new Graph();
    graph.V = V;
    graph.E = E;
    graph.edge = new Array(graph.E);

    for(let i = 0; i < graph.E; i++)
    {
        graph.edge[i] = new Edge();
    }
    return graph;
}

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

    // Step 1: Initialize distances from src
    // to all other vertices as INFINITE
    for(let i = 0; i < V; i++)
        dist[i] = Number.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(let i = 1; i <= V - 1; i++)
    {
        for(let j = 0; j < E; j++)
        {
            let u = graph.edge[j].src;
            let v = graph.edge[j].dest;
            let weight = graph.edge[j].weight;

            if (dist[u] != Number.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(let i = 0; i < E; i++)
    {
        let u = graph.edge[i].src;
        let v = graph.edge[i].dest;
        let weight = graph.edge[i].weight;

        if (dist[u] != Number.MAX_VALUE &&
            dist[u] + weight < dist[v])
            return true;
    }
    return false;
}

// Driver Code
let V = 5, E = 8;
let 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 A-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;

if (isNegCycleBellmanFord(graph, 0))
    document.write("Yes");
else
    document.write("No");

// This code is contributed by unknown2108

</script>

Output

No

Time Complexity: O(VE), where V is the number of vertices and E is the number of edges in the graph.

Space Complexity: O(V), where V is the number of vertices in the graph.

How does it work?
As discussed, the Bellman-Ford algorithm, for a given source, 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 a maximum |V| – 1 edge on any simple path, that is why the outer loop runs |v| – 1 time. If there is a negative weight cycle, then one more iteration would give a short route.

How to handle a disconnected graph (If the cycle is not reachable from the source)?
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 process for vertices for which distance is infinite.

Implementation:

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[graph->E];
    return graph;
}

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

    // 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])
            return true;
    }

    return false;
}

// Returns true if given graph has negative weight
// cycle.
bool isNegCycleDisconnected(struct Graph* graph)
{

    int V = graph->V;
    // To keep track of visited vertices to avoid
    // recomputations.
    bool visited[V];
    memset(visited, 0, sizeof(visited));

    // This array is filled by Bellman-Ford
    int dist[V];

    // Call Bellman-Ford for all those vertices
    // that are not visited
    for (int i = 0; i < V; i++) {
        if (visited[i] == false) {
            // If cycle found
            if (isNegCycleBellmanFord(graph, i, dist))
                return true;

            // Mark all vertices that are visited
            // in above call.
            for (int i = 0; i < V; i++)
                if (dist[i] != INT_MAX)
                    visited[i] = true;
        }
    }

    return false;
}

// Driver program to test above functions
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 A-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;

    if (isNegCycleDisconnected(graph))
        cout << "Yes";
    else
        cout << "No";

    return 0;
}

Java

// A Java program for Bellman-Ford's single source
// shortest path algorithm.
import java.util.*;

class GFG{

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

// A structure to represent a connected,
// directed and weighted graph
static class Graph
{

    // V-> Number of vertices,
    // E-> Number of edges
    int V, E;

    // Graph is represented as
    // an array of edges.
    Edge edge[];
}

// Creates a graph with V vertices and E edges
static Graph createGraph(int V, int E)
{
    Graph graph = new Graph();
    graph.V = V;
    graph.E = E;
    graph.edge = new Edge[graph.E];

    for(int i = 0; i < graph.E; i++)
    {
        graph.edge[i] = new Edge();
    }

    return graph;
}

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

    // 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 - 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] != 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 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] != Integer.MAX_VALUE &&
            dist[u] + weight < dist[v])
            return true;
    }

    return false;
}

// Returns true if given graph has negative weight
// cycle.
static boolean isNegCycleDisconnected(Graph graph)
{
    int V = graph.V;

    // To keep track of visited vertices
    // to avoid recomputations.
    boolean visited[] = new boolean[V];
    Arrays.fill(visited, false);

    // This array is filled by Bellman-Ford
    int dist[] = new int[V];

    // Call Bellman-Ford for all those vertices
    // that are not visited
    for(int i = 0; i < V; i++)
    {
        if (visited[i] == false)
        {

            // If cycle found
            if (isNegCycleBellmanFord(graph, i, dist))
                return true;

            // Mark all vertices that are visited
            // in above call.
            for(int j = 0; j < V; j++)
                if (dist[j] != Integer.MAX_VALUE)
                    visited[j] = true;
        }
    }
    return false;
}

// Driver Code
public static void main(String[] args)
{
    int V = 5, E = 8;
    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 A-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;

    if (isNegCycleDisconnected(graph))
        System.out.println("Yes");
    else
        System.out.println("No");
}
}

// This code is contributed by adityapande88

Python3

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

# 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 isNegCycleBellmanFord(src, dist):
    global graph, V, E

    # Step 1: Initialize distances from src
    # to all other vertices as INFINITE
    for i in range(V):
        dist[i] = 10**18
    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 i in range(1,V):
        for j in range(E):
            u = graph[j][0]
            v = graph[j][1]
            weight = graph[j][2]
            if (dist[u] != 10**18 and 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 i in range(E):
        u = graph[i][0]
        v = graph[i][1]
        weight = graph[i][2]
        if (dist[u] != 10**18 and dist[u] + weight < dist[v]):
            return True

    return False
# Returns true if given graph has negative weight
# cycle.
def isNegCycleDisconnected():
    global V, E, graph

    # To keep track of visited vertices to avoid
    # recomputations.
    visited = [0]*V
    # memset(visited, 0, sizeof(visited))

    # This array is filled by Bellman-Ford
    dist = [0]*V

    # Call Bellman-Ford for all those vertices
    # that are not visited
    for i in range(V):
        if (visited[i] == 0):

            # If cycle found
            if (isNegCycleBellmanFord(i, dist)):
                return True

            # Mark all vertices that are visited
            # in above call.
            for i in range(V):
                if (dist[i] != 10**18):
                    visited[i] = True
    return False

# Driver code
if __name__ == '__main__':

    # /* Let us create the graph given in above example */
    V = 5 # Number of vertices in graph
    E = 8 # Number of edges in graph
    graph = [[0, 0, 0] for i in range(8)]

    # add edge 0-1 (or A-B in above figure)
    graph[0][0] = 0
    graph[0][1] = 1
    graph[0][2] = -1

    # add edge 0-2 (or A-C in above figure)
    graph[1][0] = 0
    graph[1][1] = 2
    graph[1][2] = 4

    # add edge 1-2 (or B-C in above figure)
    graph[2][0] = 1
    graph[2][1] = 2
    graph[2][2] = 3

    # add edge 1-3 (or B-D in above figure)
    graph[3][0] = 1
    graph[3][1] = 3
    graph[3][2] = 2

    # add edge 1-4 (or A-E in above figure)
    graph[4][0] = 1
    graph[4][1] = 4
    graph[4][2] = 2

    # add edge 3-2 (or D-C in above figure)
    graph[5][0] = 3
    graph[5][1] = 2
    graph[5][2] = 5

    # add edge 3-1 (or D-B in above figure)
    graph[6][0] = 3
    graph[6][1] = 1
    graph[6][2] = 1

    # add edge 4-3 (or E-D in above figure)
    graph[7][0] = 4
    graph[7][1] = 3
    graph[7][2] = -3

    if (isNegCycleDisconnected()):
        print("Yes")
    else:
        print("No")

# This code is contributed by mohit kumar 29

C#

// A C# program for Bellman-Ford's single source
// shortest path algorithm.
using System;
using System.Collections.Generic;
public class GFG
{

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

  // A structure to represent a connected,
  // directed and weighted graph
  public
    class Graph
    {

      // V-> Number of vertices,
      // E-> Number of edges
      public
        int V, E;

      // Graph is represented as
      // an array of edges.
      public
        Edge []edge;
    }

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

    return graph;
  }

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

    // 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 - 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.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 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.MaxValue &&
          dist[u] + weight < dist[v])
        return true;
    }

    return false;
  }

  // Returns true if given graph has negative weight
  // cycle.
  static bool isNegCycleDisconnected(Graph graph)
  {
    int V = graph.V;

    // To keep track of visited vertices
    // to avoid recomputations.
    bool []visited = new bool[V];


    // This array is filled by Bellman-Ford
    int []dist = new int[V];

    // Call Bellman-Ford for all those vertices
    // that are not visited
    for(int i = 0; i < V; i++)
    {
      if (visited[i] == false)
      {

        // If cycle found
        if (isNegCycleBellmanFord(graph, i, dist))
          return true;

        // Mark all vertices that are visited
        // in above call.
        for(int j = 0; j < V; j++)
          if (dist[j] != int.MaxValue)
            visited[j] = true;
      }
    }
    return false;
  }

  // Driver Code
  public static void Main(String[] args)
  {
    int V = 5, E = 8;
    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 A-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;

    if (isNegCycleDisconnected(graph))
      Console.WriteLine("Yes");
    else
      Console.WriteLine("No");
  }
}

// This code is contributed by aashish1995

Javascript

<script>
// A Javascript program for Bellman-Ford's single source
// shortest path algorithm.

    // A structure to represent a weighted
    // edge in graph
    class Edge
    {
        constructor()
        {
            let src, dest, weight;
        }
    }

    // A structure to represent a connected,
    // directed and weighted graph
    class Graph
    {
        constructor()
        {
            // V-> Number of vertices,
            // E-> Number of edges
            let V, E;

            // Graph is represented as
            // an array of edges.
            let edge=[];
        }
    }

    // Creates a graph with V vertices and E edges
    function createGraph(V,E)
    {
        let graph = new Graph();
        graph.V = V;
           graph.E = E;
        graph.edge = new Array(graph.E);

    for(let i = 0; i < graph.E; i++)
    {
        graph.edge[i] = new Edge();
    }

    return graph;
    }

// The main function that finds shortest distances
// from src to all other vertices using Bellman-
// Ford algorithm. The function also detects
// negative weight cycle
function isNegCycleBellmanFord(graph,src,dist)
{
    let V = graph.V;
    let E = graph.E;

    // Step 1: Initialize distances from src
    // to all other vertices as INFINITE
    for(let i = 0; i < V; i++)
        dist[i] = Number.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(let i = 1; i <= V - 1; i++)
    {
        for(let j = 0; j < E; j++)
        {
            let u = graph.edge[j].src;
            let v = graph.edge[j].dest;
            let weight = graph.edge[j].weight;

            if (dist[u] != Number.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(let i = 0; i < E; i++)
    {
        let u = graph.edge[i].src;
        let v = graph.edge[i].dest;
        let weight = graph.edge[i].weight;

        if (dist[u] != Number.MAX_VALUE &&
            dist[u] + weight < dist[v])
            return true;
    }

    return false;
}

// Returns true if given graph has negative weight
// cycle.
function isNegCycleDisconnected(graph)
{
    let V = graph.V;

    // To keep track of visited vertices
    // to avoid recomputations.
    let visited = new Array(V);
    for(let i=0;i<V;i++)
    {
        visited[i]=false;
    }

    // This array is filled by Bellman-Ford
    let dist = new Array(V);

    // Call Bellman-Ford for all those vertices
    // that are not visited
    for(let i = 0; i < V; i++)
    {
        if (visited[i] == false)
        {

            // If cycle found
            if (isNegCycleBellmanFord(graph, i, dist))
                return true;

            // Mark all vertices that are visited
            // in above call.
            for(let j = 0; j < V; j++)
                if (dist[j] != Number.MAX_VALUE)
                    visited[j] = true;
        }
    }
    return false;
}

// Driver Code

let V = 5, E = 8;
let 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 A-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;

if (isNegCycleDisconnected(graph))
    document.write("Yes");
else
    document.write("No");


// This code is contributed by patel2127
</script>

Output

No

 Time complexity : O(E*V2), where V is the number of vertices and E is the number of edges.

Space Complexity : O(V + E) which is the space required to store the graph. 

Detecting negative cycle using Floyd Warshall

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