**Basic Theory**

Breadth – first searches are performed by exploring all nodes at a given depth before proceeding to the next level. This means that all immediate children of nodes are explored before any of the children’s children are considered. It has obvious advantage of always finding a minimal path length solution when one exists. However, a great many nodes may need to be explored before a solution is found, especially if the tree is very full.

**Algorithm**

BFS uses a queue structure to hold all generate but still unexplored nodes. The order in which nodes are placed on the queue for removal and exploration determines the type of search. The BFS algorithm proceeds as follows.

- Place the starting node
*s*on the queue. - If the queue is empty, return failure and stop.
- If the first element on the queue is a goal node
*g*, return success and stop Otherwise, - Remove and expand the first element from the queue and place all the children at the end of the queue in any order.
- Return to step 2.

**Example**

Consider a graph

Applying above algorithm, the BFS of the graph starting from node 1 is : 1, 2, 3, 4, 6, 7, 5**Source Code**

/**************************************************************** BFS.cpp Written by Bibek Subedi ****************************************************************/ #include <iostream> #include <ctime> using namespace std; /**************************************************************** Performs the Breadth-First Graph search for both directed and undirected graphs. This algorithm explores all the findable nodes in "layers". @author Bibek Subedi *****************************************************************/ /**************************************************************** Class Queue represent a Queue data structure which is First In First Out [FIFO] structured. It has operations like Enqueue which adds an element at the rear side and Dequeue which removes the element from front. *****************************************************************/ struct node { int info; node *next; }; class Queue { private: node *front; node *rear; public: Queue(); ~Queue(); bool isEmpty(); void enqueue(int); int dequeue(); void display(); }; void Queue::display(){ node *p = new node; p = front; if(front == NULL){ cout<<"\nNothing to Display\n"; }else{ while(p!=NULL){ cout<<endl<<p->info; p = p->next; } } } Queue::Queue() { front = NULL; rear = NULL; } Queue::~Queue() { delete front; } void Queue::enqueue(int data) { node *temp = new node(); temp->info = data; temp->next = NULL; if(front == NULL){ front = temp; }else{ rear->next = temp; } rear = temp; } int Queue::dequeue() { node *temp = new node(); int value; if(front == NULL){ cout<<"\nQueue is Emtpty\n"; }else{ temp = front; value = temp->info; front = front->next; delete temp; } return value; } bool Queue::isEmpty() { return (front == NULL); } /************************************************************ Class Graph represents a Graph [V,E] having vertices V and edges E. ************************************************************/ class Graph { private: int n; /// n is the number of vertices in the graph int **A; /// A stores the edges between two vertices public: Graph(int size = 2); ~Graph(); bool isConnected(int, int); void addEdge(int u, int v); void BFS(int ); }; Graph::Graph(int size) { int i, j; if (size < 2) n = 2; else n = size; A = new int*[n]; for (i = 0; i < n; ++i) A[i] = new int[n]; for (i = 0; i < n; ++i) for (j = 0; j < n; ++j) A[i][j] = 0; } Graph::~Graph() { for (int i = 0; i < n; ++i) delete [] A[i]; delete [] A; } /****************************************************** Checks if two given vertices are connected by an edge @param u vertex @param v vertex @return true if connected false if not connected ******************************************************/ bool Graph::isConnected(int u, int v) { return (A[u-1][v-1] == 1); } /***************************************************** adds an edge E to the graph G. @param u vertex @param v vertex *****************************************************/ void Graph::addEdge(int u, int v) { A[u-1][v-1] = A[v-1][u-1] = 1; } /***************************************************** performs Breadth First Search @param s initial vertex *****************************************************/ void Graph::BFS(int s) { Queue Q; /** Keeps track of explored vertices */ bool *explored = new bool[n+1]; /** Initailized all vertices as unexplored */ for (int i = 1; i <= n; ++i) explored[i] = false; /** Push initial vertex to the queue */ Q.enqueue(s); explored[s] = true; /** mark it as explored */ cout << "Breadth first Search starting from vertex "; cout << s << " : " << endl; /** Unless the queue is empty */ while (!Q.isEmpty()) { /** Pop the vertex from the queue */ int v = Q.dequeue(); /** display the explored vertices */ cout << v << " "; /** From the explored vertex v try to explore all the connected vertices */ for (int w = 1; w <= n; ++w) /** Explores the vertex w if it is connected to v and and if it is unexplored */ if (isConnected(v, w) && !explored[w]) { /** adds the vertex w to the queue */ Q.enqueue(w); /** marks the vertex w as visited */ explored[w] = true; } } cout << endl; delete [] explored; } int main() { /** Creates a graph with 12 vertices */ Graph g(12); /** Adds edges to the graph */ g.addEdge(1, 2); g.addEdge(1, 3); g.addEdge(2, 4); g.addEdge(3, 4); g.addEdge(3, 6); g.addEdge(4 ,7); g.addEdge(5, 6); g.addEdge(5, 7); clock_t t1; t1 = clock(); /** Explores all vertices findable from vertex 1 */ g.BFS(1); float diff = (double)(clock() - t1)/CLOCKS_PER_SEC ; cout <<endl<< "The time taken for Breadth first search: "<< diff << endl; }

**Complexity**

The time complexity of the breadth-first search is O(

*b*). This can be seen by noting that all nodes up to the goal depth d are generated. Therefore, the number generated is^{d}*b + b*which is O(^{2}+ . . . + b^{d}*b*). The space complexity is also O(^{d}*b*) since all nodes at a given depth must be stored in order to generate the nodes at the next depth, that is,^{d}*b*nodes must be stored at depth^{d-1}*d – 1*to generate nodes at depth*d*, which gives space complexity of O(*b*).^{d}
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In your BFS function, I think you have to remove the 'continue' in the if-statement. Otherwise it's not doing the right thing. Just use for example g.BFS(1, 3); and you will see that the output after the 3rd value is different.

ReplyDeleteIn your bfs program remove that time thing from the void main part..

ReplyDeletemain function should return an integer value ... :)

ReplyDeleteOf course, but my compiler does it for me :) ... any way thank you for pointing.

Deleteaccording to the c++ standard, it is not necessary have a return statement in main()

Deletewhat does this algorithm do exactly?

ReplyDeleteGot a small memory leak in dequeue. You are leaking temp when you assign front to it.

ReplyDeleteI think there are memory leak in the queue destructor

ReplyDeleteWhy did you implement your own queue anyway, when there's std::queue?

ReplyDeleteuse of std:queue is recommended.

DeleteI sense a memory leak .. no need to allocate memory for the temp variables.

ReplyDeleteWhen does the display() function ever get called?

ReplyDeletepost Output of the program i.e screenshot of ouput

ReplyDeleteline 156 is added because of the graph is undirected, sure?

ReplyDeleteThank a lot.

ReplyDeleteThe style you code is neat and perspicuous.

It helps me.