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### To go through the C program / source-code, scroll down to the end of this page

Breadth-first search is one of the simplest algorithms for searching a graph. Given a graph and
a distinguished source vertex, breadth-first search explores the edges of the graph to find every
vertex reachable from source. It computes the distance (fewest number of edges) from source
to all reachable vertices and produces a “breadth-first tree” with source vertex as root, which
contains all such reachable vertices. It works on both directed and undirected graphs. This
algorithm uses a first-in, first-out Queue Q to manage the vertices.

### Analysis

Initializing takes O(V) time The queue operations take time of O(V) as enqueuing and dequeuing takes O(1) time. In scanning the adjacency list at most O(E) time is spent. Thus, the total running time of BFS is O(V + E).

Complete Tutorial with Examples :

## Breadth First Search - C Program Source Code

`#include<stdio.h>#include<stdlib.h>#include<assert.h>/* maxVertices represents maximum number of vertices that can be present in the graph. */#define maxVertices   100/*Queue has five properties. capacity stands for the maximum number of elements Queue can hold.  Size stands for the current size of the Queue and elements is the array of elements. front is the index of first element (the index at which we remove the element) and rear is the index of last element (the index at which we insert the element) */typedef struct Queue{        int capacity;        int size;        int front;        int rear;        int *elements;}Queue;/* crateQueue function takes argument the maximum number of elements the Queue can hold, creates   a Queue according to it and returns a pointer to the Queue. */Queue * CreateQueue(int maxElements){        /* Create a Queue */        Queue *Q;        Q = (Queue *)malloc(sizeof(Queue));        /* Initialise its properties */        Q->elements = (int *)malloc(sizeof(int)*maxElements);        Q->size = 0;        Q->capacity = maxElements;        Q->front = 0;        Q->rear = -1;        /* Return the pointer */        return Q;}void Dequeue(Queue *Q){        /* If Queue size is zero then it is empty. So we cannot pop */        if(Q->size==0)        {                printf("Queue is Empty\n");                return;        }        /* Removing an element is equivalent to incrementing index of front by one */        else        {                Q->size--;                Q->front++;                /* As we fill elements in circular fashion */                if(Q->front==Q->capacity)                {                        Q->front=0;                }        }        return;}int Front(Queue *Q){        if(Q->size==0)        {                printf("Queue is Empty\n");                exit(0);        }        /* Return the element which is at the front*/        return Q->elements[Q->front];}void Enqueue(Queue *Q,int element){        /* If the Queue is full, we cannot push an element into it as there is no space for it.*/        if(Q->size == Q->capacity)        {                printf("Queue is Full\n");        }        else        {                Q->size++;                Q->rear = Q->rear + 1;                /* As we fill the queue in circular fashion */                if(Q->rear == Q->capacity)                {                        Q->rear = 0;                }                /* Insert the element in its rear side */                 Q->elements[Q->rear] = element;        }        return;}void Bfs(int graph[][maxVertices], int *size, int presentVertex,int *visited){        visited[presentVertex] = 1;        /* Iterate through all the vertices connected to the presentVertex and perform bfs on those           vertices if they are not visited before */        Queue *Q = CreateQueue(maxVertices);        Enqueue(Q,presentVertex);        while(Q->size)        {                presentVertex = Front(Q);                printf("Now visiting vertex %d\n",presentVertex);                Dequeue(Q);                int iter;                for(iter=0;iter<size[presentVertex];iter++)                {                        if(!visited[graph[presentVertex][iter]])                        {                                visited[graph[presentVertex][iter]] = 1;                                Enqueue(Q,graph[presentVertex][iter]);                        }                }        }        return;        }/* Input Format: Graph is directed and unweighted. First two integers must be number of vertces and edges    which must be followed by pairs of vertices which has an edge between them. */int main(){        int graph[maxVertices][maxVertices],size[maxVertices]={0},visited[maxVertices]={0};        int vertices,edges,iter;        /* vertices represent number of vertices and edges represent number of edges in the graph. */        scanf("%d%d",&vertices,&edges);        int vertex1,vertex2;        for(iter=0;iter<edges;iter++)        {                scanf("%d%d",&vertex1,&vertex2);                assert(vertex1>=0 && vertex1<vertices);                assert(vertex2>=0 && vertex2<vertices);                graph[vertex1][size[vertex1]++] = vertex2;        }        int presentVertex;        for(presentVertex=0;presentVertex<vertices;presentVertex++)        {                if(!visited[presentVertex])                {                        Bfs(graph,size,presentVertex,visited);                }        }        return 0;}`

`Rough Notes about the Algorithm : `
```
Input Format: Graph is directed and unweighted. 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.
visited[maxVertices]={0} represents the vertex that have been visited.```
```
Initialize the graph.
For presentVertex = 0 to vertices if visited[presentVertex] is 0, i.e. if the vertex has not been visited then call Bfs function. presentVertex represents the vertex that is being tackled.
Bfs function is called to get the shortest path.

Bfs function: This function takes the graph obtained (graph[ ][ maxVertices]), pointer to the array size and visited, and the presentValue as arguments.

visited[presentVertex] = 1 as the vertex has now been visited.
Iterate through all the vertices connected to the presentVertex and perform bfs on those vertices if they are not visited before.
Create a queue Q using createQueue function and enqueue the presentVertex using Enqueue function.
Until the Q is not empty i.e. Q->size ≠ 0
• Store the front element of Q using Front function in presentVertex
• Print the vertex that is being visited now, which is presentVertex
• Remove the element from the front of Q using Dequeue function
• For iter=0 to size[presentVertex] – 1

If (!visited[graph[presentVertex][iter]])

visited[graph[presentVertex][iter]] = 1

Enqueue(Q,graph[presentVertex][iter])

Iter + 1

This for loop visits every vertex that is adjacent to the presentVertex and has not
been visited yet. These vertices are then inserted in the Q using Enqueue function
and their visited status is updated to 1.

The Queue has five properties - capacity stands for the maximum number of elements Queue can
hold, Size stands for the current size of the Queue, elements is the array of elements, front is the
index of first element (the index at which we remove the element) and rear is the index of last
element (the index at which we insert the element). Functions on Queue

1. createQueue function takes argument the maximum number of elements the Queue can
hold, creates a Queue according to it and returns a pointer to the Queue. It initializes Q-
>size to 0, Q->capacity to maxElements, Q->front to 0 and Q->rear to -1.
2. enqueue function - This function takes the pointer to the top of the queue Q and the item
(element) to be inserted as arguments. Check for the emptiness of queue
a. If Q->size is equal to Q->capacity, we cannot push an element into Q as there is
no space for it.
b. Else, enqueue an element at the end of Q, increase its size by one. Increase the
value of Q->rear to Q->rear + 1. As we fill the queue in circular fashion, if
Q->rear is equal to Q->capacity make Q->rear = 0. Now, Insert the element in its
rear side

Q->elements[Q->rear] = element

3. dequeue function - This function takes the pointer to the top of the stack S as an
argument.
a. If Q->size is equal to zero, then it is empty. So, we cannot dequeue.
b. Else, remove an element which is equivalent to incrementing index of front by
one. Decrease the size by 1. As we fill elements in circular fashion, if Q->front is
equal to Q->capacity make Q->front=0.
4. front function – This function takes the pointer to the top of the queue Q as an argument
and returns the front element of the queue Q. It first checks if the queue is empty
(Q->size is equal to zero). If it’s not it returns the element which is at the front of the
queue.

Q->elements[Q->front]```

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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 Arrays : Popular Sorting and Searching Algorithms Bubble Sort Insertion Sort Selection Sort Shell Sort Merge Sort Quick Sort Heap Sort Binary Search Algorithm Basic Data Structures  and Operations on them Stacks Queues Single Linked List Double Linked List Circular Linked List
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