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### Bellman –Ford

The bellman-Ford algorithm solves the single source shortest path problems even in the cases in which edge weights are negative. This algorithm returns a Boolean value indicating whether or not there is a negative weight cycle that is reachable from the source. If there is such a cycle, the algorithm indicates that no solution exists and it there is no such cycle, it produces the shortest path and their weights.

### Complexity

The Bellman-Ford algorithm runs in time O(VE), since the initialization takes Θ(V) time, each of the |V| - 1 passes over the edges takes O(E) time and calculating the distance takes O(E) times.

Complete Tutorial with Example :

## Bellman Ford Algorithm - C Program source code

`#include<stdio.h>#include<assert.h>/* maxVertices represents maximum number of vertices that can be present in the graph. */#define maxVertices   100#define INF 123456789/* Input Format: Graph is directed and weighted. First two integers must be number of vertces and edges    which must be followed by pairs of vertices which has an edge between them.  */void BellmanFord(int graph[][maxVertices],int cost[][maxVertices],int size[],int source,int vertices){        int distance[vertices];        int iter,jter,from,to;        for(iter=0;iter<vertices;iter++)        {                distance[iter] = INF;        }        distance[source] = 0;        /* We have to repeatedly update the distance |V|-1 times where |V| represents           number of vertices */        for(iter=0;iter<vertices-1;iter++)        {                for(from=0;from<vertices;from++)                {                        for(jter=0;jter<size[from];jter++)                        {                                to = graph[from][jter];                                if(distance[from] + cost[from][jter] < distance[to])                                {                                        distance[to] = distance[from] + cost[from][jter];                                }                        }                }        }        for(iter=0;iter<vertices;iter++)        {                printf("The shortest distance to %d is %d\n",iter,distance[iter]);        }}int main(){        int graph[maxVertices][maxVertices],size[maxVertices]={0},visited[maxVertices]={0};        int cost[maxVertices][maxVertices];        int vertices,edges,iter,jter;        /* vertices represent number of vertices and edges represent number of edges in the graph. */        scanf("%d%d",&vertices,&edges);        int vertex1,vertex2,weight;        /* Here graph[i][j] represent the weight of edge joining i and j */        for(iter=0;iter<edges;iter++)        {                scanf("%d%d%d",&vertex1,&vertex2,&weight);                assert(vertex1>=0 && vertex1<vertices);                assert(vertex2>=0 && vertex2<vertices);                graph[vertex1][size[vertex1]] = vertex2;                cost[vertex1][size[vertex1]] = weight;                size[vertex1]++;        }        int source;        scanf("%d",&source);        BellmanFord(graph,cost,size,source,vertices);        return 0;}`
```Rough notes about the Algorithm and how it is implemented in the code above:

Input Format: Graph is directed and weighted. First two integers must be number of vertices and
edges which must be followed by pairs of vertices which has an edge between them.

maxVertices represents maximum number of vertices that can be present in the graph.
vertices represent number of vertices and edges represent number of edges in the graph.
graph[i][j] represent the weight of edge joining i and j.
size[maxVertices] is initialed to{0}, represents the size of every vertex i.e. the number of
edges corresponding to the vertex.
cost[maxVertices][maxVertices] represents the cost of going from one vertex to another.
visited[maxVertices]={0} represents the vertex that have been visited.
Initialize the graph and input the source vertex.
BellmanFord function is called to get the shortest path.

BellmanFord function: This function takes the graph obtained (graph[ ][ maxVertices]), cost
(cost[ ][maxVertices]) of going from one vertex to other, size (size[maxVertices]) of vertices,
source vertex and the total number of vertices (vertices) as arguments.

Initialize an array distance[vertices] to store the shortest distance travelled to reach vertex i. ```
`Initially distance[source] = 0 i.e. distance to reach the source vertex is zero and distance to reach every other vertex is initialed to infinity (distance[i] = INF, where INF is a very large integer value).`
```We have to repeatedly update the distance |V|-1 times where |V| represents number of vertices as we visit the vertices.
for iter=0 to vertices-2
for from=0 to vertices-1
for jter=0 to size[from]
to = graph[from][jter]
if(distance[from] + cost[from][jter] < distance[to])
distance[to] = distance[from] + cost[from][jter]
jter + 1
from + 1
iter + 1
Lastly for every vertex print the shortest distance traveled to reach that vertex.```

Related Tutorials (basic Graph Algorithms) :

 Depth First Search Traversing through a graph using Depth First Search in which unvisited neighbors of the current vertex are pushed into a stack and visited in that order. Breadth First Search Traversing through a graph using Breadth First Search in which unvisited neighbors of the current vertex are pushed into a queue and then visited in that order. Minimum Spanning Trees: Kruskal Algorithm Finding the Minimum Spanning Tree using the Kruskal Algorithm which is a greedy technique. Introducing the concept of Union Find. Minumum Spanning Trees: Prim's Algorithm Finding the Minimum Spanning Tree using the Prim's Algorithm. Dijkstra Algorithm for Shortest Paths Popular algorithm for finding shortest paths : Dijkstra Algorithm. Floyd Warshall Algorithm for Shortest Paths All the all shortest path algorithm: Floyd Warshall Algorithm Bellman Ford Algorithm Another common shortest path algorithm : Bellman Ford Algorithm.

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