Breadth First Search or BFS for a Graph

27 Jun, 2022
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 use a boolean visited array. For simplicity, it is assumed that all vertices are reachable from the starting vertex. BFS uses a queue data structure for traversal.
For 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
Following are the implementations of simple Breadth-First Traversal from a given source. 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++
Java
Python3
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 representationclass 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 classint 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 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// representationclass 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 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 representationclass 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 diagramg = 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# 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// representationclass Graph{     // No. of vertices    private int _V; //Adjacency ListsLinkedList<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 graphpublic void AddEdge(int v, int w){            _adj[v].AddLast(w); } // Prints BFS traversal from a given source spublic 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 codestatic 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
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)
Illustration : 
 
 
 
 
Note that the above code traverses only the vertices reachable from a given source vertex. The vertices may not be reachable from a given vertex (for example 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). 
The C++ code for BFS traversal for the entire graph (valid for directed as well as undirected graphs) with possible multiple disconnected components is as follows:
C++
/******************************************************* * Generic Function for BFS traversal of a Graph * (valid for directed as well as undirected graphs *  which can have multiple disconnected commponents) * ********** 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) {        if (!vis[i]) {            queue<int> q;            vis[i] = true;            q.push(i);            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;}
 
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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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