Breadth First Search or BFS for a Graph

Breadth-First Traversal (or Search) for a graph is similar to Breadth-First Traversal of a tree (See method 2 of this post). 

The only catch here is, that, unlike trees, graphs may contain cycles, so we may come to the same node again. To avoid processing a node more than once, we divide the vertices into two categories:

• Visited and
• Not visited.

A boolean visited array is used to mark the visited vertices. For simplicity, it is assumed that all vertices are reachable from the starting vertex. BFS uses a queue data structure for traversal.

Example:

In the following graph, we start traversal from vertex 2.

When we come to vertex 0, we look for all adjacent vertices of it. 

• 2 is also an adjacent vertex of 0. 
• If we don’t mark visited vertices, then 2 will be processed again and it will become a non-terminating process.

There can be multiple BFS traversals for a graph. Different BFS traversals for the above graph :
2, 3, 0, 1
2, 0, 3, 1

Implementation of BFS traversal:

Follow the below method to implement BFS traversal.

• Declare a queue and insert the starting vertex.
• Initialize a visited array and mark the starting vertex as visited.
• Follow the below process till the queue becomes empty:
  • Remove the first vertex of the queue.
  • Mark that vertex as visited.
  • Insert all the unvisited neighbours of the vertex into the queue.

Illustration:

IMAGES TO BE ADDED

The implementation uses an adjacency list representation of graphs. STL‘s list container stores lists of adjacent nodes and the queue of nodes needed for BFS traversal.

C++

// Program to print BFS traversal from a given
// source vertex. BFS(int s) traverses vertices
// reachable from s.
#include<bits/stdc++.h>
using namespace std;

// This class represents a directed graph using
// adjacency list representation
class Graph
{
    int V;    // No. of vertices

    // Pointer to an array containing adjacency
    // lists
    vector<list<int>> adj;
public:
    Graph(int V);  // Constructor

    // function to add an edge to graph
    void addEdge(int v, int w);

    // prints BFS traversal from a given source s
    void BFS(int s);
};

Graph::Graph(int V)
{
    this->V = V;
    adj.resize(V);
}

void Graph::addEdge(int v, int w)
{
    adj[v].push_back(w); // Add w to v’s list.
}

void Graph::BFS(int s)
{
    // Mark all the vertices as not visited
    vector<bool> visited;
    visited.resize(V,false);

    // Create a queue for BFS
    list<int> queue;

    // Mark the current node as visited and enqueue it
    visited[s] = true;
    queue.push_back(s);

    while(!queue.empty())
    {
        // Dequeue a vertex from queue and print it
        s = queue.front();
        cout << s << " ";
        queue.pop_front();

        // Get all adjacent vertices of the dequeued
        // vertex s. If a adjacent has not been visited,
        // then mark it visited and enqueue it
        for (auto adjecent: adj[s])
        {
            if (!visited[adjecent])
            {
                visited[adjecent] = true;
                queue.push_back(adjecent);
            }
        }
    }
}

// Driver program to test methods of graph class
int main()
{
    // Create a graph given in the above diagram
    Graph g(4);
    g.addEdge(0, 1);
    g.addEdge(0, 2);
    g.addEdge(1, 2);
    g.addEdge(2, 0);
    g.addEdge(2, 3);
    g.addEdge(3, 3);

    cout << "Following is Breadth First Traversal "
         << "(starting from vertex 2) \n";
    g.BFS(2);

    return 0;
}

Java

// Java program to print BFS traversal from a given source vertex.
// BFS(int s) traverses vertices reachable from s.
import java.io.*;
import java.util.*;

// This class represents a directed graph using adjacency list
// representation
class Graph
{
    private int V;   // No. of vertices
    private LinkedList<Integer> adj[]; //Adjacency Lists

    // Constructor
    Graph(int v)
    {
        V = v;
        adj = new LinkedList[v];
        for (int i=0; i<v; ++i)
            adj[i] = new LinkedList();
    }

    // Function to add an edge into the graph
    void addEdge(int v,int w)
    {
        adj[v].add(w);
    }

    // prints BFS traversal from a given source s
    void BFS(int s)
    {
        // Mark all the vertices as not visited(By default
        // set as false)
        boolean visited[] = new boolean[V];

        // Create a queue for BFS
        LinkedList<Integer> queue = new LinkedList<Integer>();

        // Mark the current node as visited and enqueue it
        visited[s]=true;
        queue.add(s);

        while (queue.size() != 0)
        {
            // Dequeue a vertex from queue and print it
            s = queue.poll();
            System.out.print(s+" ");

            // Get all adjacent vertices of the dequeued vertex s
            // If a adjacent has not been visited, then mark it
            // visited and enqueue it
            Iterator<Integer> i = adj[s].listIterator();
            while (i.hasNext())
            {
                int n = i.next();
                if (!visited[n])
                {
                    visited[n] = true;
                    queue.add(n);
                }
            }
        }
    }

    // Driver method to
    public static void main(String args[])
    {
        Graph g = new Graph(4);

        g.addEdge(0, 1);
        g.addEdge(0, 2);
        g.addEdge(1, 2);
        g.addEdge(2, 0);
        g.addEdge(2, 3);
        g.addEdge(3, 3);

        System.out.println("Following is Breadth First Traversal "+
                           "(starting from vertex 2)");

        g.BFS(2);
    }
}
// This code is contributed by Aakash Hasija

Python3

# Python3 Program to print BFS traversal
# from a given source vertex. BFS(int s)
# traverses vertices reachable from s.
from collections import defaultdict

# This class represents a directed graph
# using adjacency list representation
class Graph:

    # Constructor
    def __init__(self):

        # default dictionary to store graph
        self.graph = defaultdict(list)

    # function to add an edge to graph
    def addEdge(self,u,v):
        self.graph[u].append(v)

    # Function to print a BFS of graph
    def BFS(self, s):

        # Mark all the vertices as not visited
        visited = [False] * (max(self.graph) + 1)

        # Create a queue for BFS
        queue = []

        # Mark the source node as
        # visited and enqueue it
        queue.append(s)
        visited[s] = True

        while queue:

            # Dequeue a vertex from
            # queue and print it
            s = queue.pop(0)
            print (s, end = " ")

            # Get all adjacent vertices of the
            # dequeued vertex s. If a adjacent
            # has not been visited, then mark it
            # visited and enqueue it
            for i in self.graph[s]:
                if visited[i] == False:
                    queue.append(i)
                    visited[i] = True

# Driver code

# Create a graph given in
# the above diagram
g = Graph()
g.addEdge(0, 1)
g.addEdge(0, 2)
g.addEdge(1, 2)
g.addEdge(2, 0)
g.addEdge(2, 3)
g.addEdge(3, 3)

print ("Following is Breadth First Traversal"
                  " (starting from vertex 2)")
g.BFS(2)

# This code is contributed by Neelam Yadav

C#

// C# program to print BFS traversal
// from a given source vertex.
// BFS(int s) traverses vertices
// reachable from s.
using System;
using System.Collections.Generic;
using System.Linq;
using System.Text;

// This class represents a directed
// graph using adjacency list
// representation
class Graph{

// No. of vertices
private int _V;

//Adjacency Lists
LinkedList<int>[] _adj;

public Graph(int V)
{
    _adj = new LinkedList<int>[V];
    for(int i = 0; i < _adj.Length; i++)
    {
        _adj[i] = new LinkedList<int>();
    }
    _V = V;
}

// Function to add an edge into the graph
public void AddEdge(int v, int w)
{
    _adj[v].AddLast(w);

}

// Prints BFS traversal from a given source s
public void BFS(int s)
{

    // Mark all the vertices as not
    // visited(By default set as false)
    bool[] visited = new bool[_V];
    for(int i = 0; i < _V; i++)
        visited[i] = false;

    // Create a queue for BFS
    LinkedList<int> queue = new LinkedList<int>();

    // Mark the current node as
    // visited and enqueue it
    visited[s] = true;
    queue.AddLast(s);

    while(queue.Any())
    {

        // Dequeue a vertex from queue
        // and print it
        s = queue.First();
        Console.Write(s + " " );
        queue.RemoveFirst();

        // Get all adjacent vertices of the
        // dequeued vertex s. If a adjacent
        // has not been visited, then mark it
        // visited and enqueue it
        LinkedList<int> list = _adj[s];

        foreach (var val in list)
        {
            if (!visited[val])
            {
                visited[val] = true;
                queue.AddLast(val);
            }
        }
    }
}

// Driver code
static void Main(string[] args)
{
    Graph g = new Graph(4);

    g.AddEdge(0, 1);
    g.AddEdge(0, 2);
    g.AddEdge(1, 2);
    g.AddEdge(2, 0);
    g.AddEdge(2, 3);
    g.AddEdge(3, 3);

    Console.Write("Following is Breadth First " +
                  "Traversal(starting from " +
                  "vertex 2)\n");
    g.BFS(2);
}
}

// This code is contributed by anv89

Output

Following is Breadth First Traversal (starting from vertex 2) 
2 0 3 1 

Time Complexity: O(V+E), where V is the number of nodes and E is the number of edges.
Auxiliary Space: O(V)

BFS for Disconnected Graph:

Note that the above code traverses only the vertices reachable from a given source vertex. In every situation, all the vertices may not be reachable from a given vertex (i.e. for a disconnected graph). 

To print all the vertices, we can modify the BFS function to do traversal starting from all nodes one by one (Like the DFS modified version). 

Below is the implementation for BFS traversal for the entire graph (valid for directed as well as undirected graphs) with possible multiple disconnected components:

C++

/*
-> Generic Function for BFS traversal of a Graph
 (valid for directed as well as undirected graphs
 which can have multiple disconnected components)
-- Inputs --
-> V - represents number of vertices in the Graph
-> adj[] - represents adjacency list for the Graph
-- Output --
-> bfs_traversal - a vector containing bfs traversal
for entire graph
*/

vector<int> bfsOfGraph(int V, vector<int> adj[])
{
    vector<int> bfs_traversal;
    vector<bool> vis(V, false);
    for (int i = 0; i < V; ++i) {

        // To check if already visited
        if (!vis[i]) {
            queue<int> q;
            vis[i] = true;
            q.push(i);

            // BFS starting from ith node
            while (!q.empty()) {
                int g_node = q.front();
                q.pop();
                bfs_traversal.push_back(g_node);
                for (auto it : adj[g_node]) {
                    if (!vis[it]) {
                        vis[it] = true;
                        q.push(it);
                    }
                }
            }
        }
    }
    return bfs_traversal;
}

Python3

'''
Generic Function for BFS traversal of a Graph
 (valid for directed as well as undirected graphs
 which can have multiple disconnected components)
-- Inputs --
-> V - represents number of vertices in the Graph
-> adj[] - represents adjacency list for the Graph
-- Output --
-> bfs_traversal - a vector containing bfs traversal
for entire graph
'''
def bfsOfGraph(V, adj):

    bfs_traversal = []
    vis = [False]*V
    for i in range(V):

        # To check if already visited
        if (vis[i] == False):
            q = []
            vis[i] = True
            q.append(i)

            # BFS starting from ith node
            while (len(q) > 0):
                g_node = q.pop(0)

                bfs_traversal.append(g_node)
                for it in adj[g_node]:
                    if (vis[it] == False):
                        vis[it] = True
                        q.append(it)

    return bfs_traversal

  # This code is contributed by Abhijeet Kumar(abhijeet19403)

You may like to see below also :

• Recent Articles on BFS
• Depth First Traversal
• Applications of Breadth First Traversal
• Applications of Depth First Search

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