Given an undirected graph, the task is to print all the connected components line by line.
Examples:
Input: Consider the following graph
Output:
0 1 2
3 4
Explanation: There are 2 different connected components.
They are {0, 1, 2} and {3, 4}.
We have discussed algorithms for finding strongly connected components in directed graphs in following posts.
Kosaraju’s algorithm for strongly connected components.
Tarjan’s Algorithm to find Strongly Connected Components
Connected Components for undirected graph using DFS:
Finding connected components for an undirected graph is an easier task. The idea is to
Do either BFS or DFS starting from every unvisited vertex, and we get all strongly connected components.
Follow the steps mentioned below to implement the idea using DFS:
- Initialize all vertices as not visited.
- Do the following for every vertex v:
- If v is not visited before, call the DFS. and print the newline character to print each component in a new line
- Mark v as visited and print v.
- For every adjacent u of v, If u is not visited, then recursively call the DFS.
- If v is not visited before, call the DFS. and print the newline character to print each component in a new line
Below is the implementation of above algorithm.
// C++ program to print connected components in // an undirected graph #include <bits/stdc++.h> using namespace std;
// Graph class represents a undirected graph // using adjacency list representation class Graph {
int V; // No. of vertices
// Pointer to an array containing adjacency lists
list< int >* adj;
// A function used by DFS
void DFSUtil( int v, bool visited[]);
public :
Graph( int V); // Constructor
~Graph();
void addEdge( int v, int w);
void connectedComponents();
}; // Method to print connected components in an // undirected graph void Graph::connectedComponents()
{ // Mark all the vertices as not visited
bool * visited = new bool [V];
for ( int v = 0; v < V; v++)
visited[v] = false ;
for ( int v = 0; v < V; v++) {
if (visited[v] == false ) {
// print all reachable vertices
// from v
DFSUtil(v, visited);
cout << "\n" ;
}
}
delete [] visited;
} void Graph::DFSUtil( int v, bool visited[])
{ // Mark the current node as visited and print it
visited[v] = true ;
cout << v << " " ;
// Recur for all the vertices
// adjacent to this vertex
list< int >::iterator i;
for (i = adj[v].begin(); i != adj[v].end(); ++i)
if (!visited[*i])
DFSUtil(*i, visited);
} Graph::Graph( int V)
{ this ->V = V;
adj = new list< int >[V];
} Graph::~Graph() { delete [] adj; }
// method to add an undirected edge void Graph::addEdge( int v, int w)
{ adj[v].push_back(w);
adj[w].push_back(v);
} // Driver code int main()
{ // Create a graph given in the above diagram
Graph g(5); // 5 vertices numbered from 0 to 4
g.addEdge(1, 0);
g.addEdge(2, 1);
g.addEdge(3, 4);
cout << "Following are connected components \n" ;
g.connectedComponents();
return 0;
} |
// Java program to print connected components in // an undirected graph import java.util.ArrayList;
class Graph {
// A user define class to represent a graph.
// A graph is an array of adjacency lists.
// Size of array will be V (number of vertices
// in graph)
int V;
ArrayList<ArrayList<Integer> > adjListArray;
// constructor
Graph( int V)
{
this .V = V;
// define the size of array as
// number of vertices
adjListArray = new ArrayList<>();
// Create a new list for each vertex
// such that adjacent nodes can be stored
for ( int i = 0 ; i < V; i++) {
adjListArray.add(i, new ArrayList<>());
}
}
// Adds an edge to an undirected graph
void addEdge( int src, int dest)
{
// Add an edge from src to dest.
adjListArray.get(src).add(dest);
// Since graph is undirected, add an edge from dest
// to src also
adjListArray.get(dest).add(src);
}
void DFSUtil( int v, boolean [] visited)
{
// Mark the current node as visited and print it
visited[v] = true ;
System.out.print(v + " " );
// Recur for all the vertices
// adjacent to this vertex
for ( int x : adjListArray.get(v)) {
if (!visited[x])
DFSUtil(x, visited);
}
}
void connectedComponents()
{
// Mark all the vertices as not visited
boolean [] visited = new boolean [V];
for ( int v = 0 ; v < V; ++v) {
if (!visited[v]) {
// print all reachable vertices
// from v
DFSUtil(v, visited);
System.out.println();
}
}
}
// Driver code
public static void main(String[] args)
{
// Create a graph given in the above diagram
Graph g = new Graph( 5 );
g.addEdge( 1 , 0 );
g.addEdge( 2 , 1 );
g.addEdge( 3 , 4 );
System.out.println(
"Following are connected components" );
g.connectedComponents();
}
} |
# Python program to print connected # components in an undirected graph class Graph:
# init function to declare class variables
def __init__( self , V):
self .V = V
self .adj = [[] for i in range (V)]
def DFSUtil( self , temp, v, visited):
# Mark the current vertex as visited
visited[v] = True
# Store the vertex to list
temp.append(v)
# Repeat for all vertices adjacent
# to this vertex v
for i in self .adj[v]:
if visited[i] = = False :
# Update the list
temp = self .DFSUtil(temp, i, visited)
return temp
# method to add an undirected edge
def addEdge( self , v, w):
self .adj[v].append(w)
self .adj[w].append(v)
# Method to retrieve connected components
# in an undirected graph
def connectedComponents( self ):
visited = []
cc = []
for i in range ( self .V):
visited.append( False )
for v in range ( self .V):
if visited[v] = = False :
temp = []
cc.append( self .DFSUtil(temp, v, visited))
return cc
# Driver Code if __name__ = = "__main__" :
# Create a graph given in the above diagram
# 5 vertices numbered from 0 to 4
g = Graph( 5 )
g.addEdge( 1 , 0 )
g.addEdge( 2 , 1 )
g.addEdge( 3 , 4 )
cc = g.connectedComponents()
print ( "Following are connected components" )
print (cc)
# This code is contributed by Abhishek Valsan |
using System;
using System.Collections.Generic;
class Graph
{ // A user defined class to represent a graph.
// A graph is an array of adjacency lists.
// Size of array will be V (number of vertices
// in graph)
int V;
List<List< int >> adjListArray;
// constructor
public Graph( int V)
{
this .V = V;
// define the size of array as
// number of vertices
adjListArray = new List<List< int >>();
// Create a new list for each vertex
// such that adjacent nodes can be stored
for ( int i = 0; i < V; i++)
{
adjListArray.Add( new List< int >());
}
}
// Adds an edge to an undirected graph
public void addEdge( int src, int dest)
{
// Add an edge from src to dest.
adjListArray[src].Add(dest);
// Since graph is undirected, add an edge from dest
// to src also
adjListArray[dest].Add(src);
}
void DFSUtil( int v, bool [] visited)
{
// Mark the current node as visited and print it
visited[v] = true ;
Console.Write(v + " " );
// Recur for all the vertices
// adjacent to this vertex
foreach ( int x in adjListArray[v])
{
if (!visited[x])
DFSUtil(x, visited);
}
}
public void connectedComponents()
{
// Mark all the vertices as not visited
bool [] visited = new bool [V];
for ( int v = 0; v < V; ++v)
{
if (!visited[v])
{
// print all reachable vertices
// from v
DFSUtil(v, visited);
Console.WriteLine();
}
}
}
// Driver code
public static void Main( string [] args)
{
// Create a graph given in the above diagram
Graph g = new Graph(5);
g.addEdge(1, 0);
g.addEdge(2, 1);
g.addEdge(3, 4);
Console.WriteLine( "Following are connected components:" );
g.connectedComponents();
}
} // This code is contributed by lokeshpotta20. |
<script> // Javascript program to print connected components in // an undirected graph // A user define class to represent a graph. // A graph is an array of adjacency lists.
// Size of array will be V (number of vertices
// in graph)
let V; let adjListArray=[]; // constructor function Graph(v)
{ V = v
// define the size of array as
// number of vertices
// Create a new list for each vertex
// such that adjacent nodes can be stored
for (let i = 0; i < V; i++) {
adjListArray.push([]);
}
} // Adds an edge to an undirected graph function addEdge(src,dest)
{ // Add an edge from src to dest.
adjListArray[src].push(dest);
// Since graph is undirected, add an edge from dest
// to src also
adjListArray[dest].push(src);
} function DFSUtil(v,visited)
{ // Mark the current node as visited and print it
visited[v] = true ;
document.write(v + " " );
// Recur for all the vertices
// adjacent to this vertex
for (let x = 0; x < adjListArray[v].length; x++)
{
if (!visited[adjListArray[v][x]])
DFSUtil(adjListArray[v][x], visited);
}
} function connectedComponents()
{ // Mark all the vertices as not visited
let visited = new Array(V);
for (let i = 0; i < V; i++)
{
visited[i] = false ;
}
for (let v = 0; v < V; ++v)
{
if (!visited[v])
{
// print all reachable vertices
// from v
DFSUtil(v, visited);
document.write( "<br>" );
}
}
} // Driver code Graph(5); addEdge(1, 0); addEdge(2, 1); addEdge(3, 4); document.write( "Following are connected components<br>" );
connectedComponents(); // This code is contributed by rag2127 </script> |
Following are connected components 0 1 2 3 4
Time Complexity: O(V + E) where V is the number of vertices and E is the number of edges.
Auxiliary Space: O(V)
Connected Component for undirected graph using Disjoint Set Union:
The idea to solve the problem using DSU (Disjoint Set Union) is
Initially declare all the nodes as individual subsets and then visit them. When a new unvisited node is encountered, unite it with the under. In this manner, a single component will be visited in each traversal.
Follow the below steps to implement the idea:
- Declare an array arr[] of size V where V is the total number of nodes.
- For every index i of array arr[], the value denotes who the parent of ith vertex is.
- Initialise every node as the parent of itself and then while adding them together, change their parents accordingly.
- Traverse the nodes from 0 to V:
- For each node that is the parent of itself start the DSU.
- Print the nodes of that disjoint set as they belong to one component.
Below is the implementation of the above approach.
#include <bits/stdc++.h> using namespace std;
int merge( int * parent, int x)
{ if (parent[x] == x)
return x;
return merge(parent, parent[x]);
} int connectedcomponents( int n, vector<vector< int > >& edges)
{ int parent[n];
for ( int i = 0; i < n; i++) {
parent[i] = i;
}
for ( auto x : edges) {
parent[merge(parent, x[0])] = merge(parent, x[1]);
}
int ans = 0;
for ( int i = 0; i < n; i++) {
ans += (parent[i] == i);
}
for ( int i = 0; i < n; i++) {
parent[i] = merge(parent, parent[i]);
}
map< int , list< int > > m;
for ( int i = 0; i < n; i++) {
m[parent[i]].push_back(i);
}
for ( auto it = m.begin(); it != m.end(); it++) {
list< int > l = it->second;
for ( auto x : l) {
cout << x << " " ;
}
cout << endl;
}
return ans;
} int main()
{ int n = 5;
vector< int > e1 = { 0, 1 };
vector< int > e2 = { 2, 1 };
vector< int > e3 = { 3, 4 };
vector<vector< int > > e;
e.push_back(e1);
e.push_back(e2);
e.push_back(e3);
cout << "Following are connected components:\n" ;
int a = connectedcomponents(n, e);
return 0;
} |
import java.util.*;
class ConnectedComponents {
public static int merge( int [] parent, int x) {
if (parent[x] == x)
return x;
return merge(parent, parent[x]);
}
public static int connectedComponents( int n, List<List<Integer>> edges) {
int [] parent = new int [n];
for ( int i = 0 ; i < n; i++) {
parent[i] = i;
}
for (List<Integer> x : edges) {
parent[merge(parent, x.get( 0 ))] = merge(parent, x.get( 1 ));
}
int ans = 0 ;
for ( int i = 0 ; i < n; i++) {
if (parent[i] == i) ans++;
}
for ( int i = 0 ; i < n; i++) {
parent[i] = merge(parent, parent[i]);
}
Map<Integer, List<Integer>> m = new HashMap<>();
for ( int i = 0 ; i < n; i++) {
m.computeIfAbsent(parent[i], k -> new ArrayList<>()).add(i);
}
for (Map.Entry<Integer, List<Integer>> it : m.entrySet()) {
List<Integer> l = it.getValue();
for ( int x : l) {
System.out.print(x + " " );
}
System.out.println();
}
return ans;
}
public static void main(String[] args) {
int n = 5 ;
List<List<Integer>> edges = new ArrayList<>();
edges.add(Arrays.asList( 0 , 1 ));
edges.add(Arrays.asList( 2 , 1 ));
edges.add(Arrays.asList( 3 , 4 ));
System.out.println( "Following are connected components:" );
int ans = connectedComponents(n, edges);
}
} |
using System;
using System.Collections.Generic;
class ConnectedComponents {
public static int Merge( int [] parent, int x) {
if (parent[x] == x)
return x;
return Merge(parent, parent[x]);
} public static int CountConnectedComponents( int n, List<List< int >> edges) {
int [] parent = new int [n];
for ( int i = 0; i < n; i++) {
parent[i] = i;
}
foreach (List< int > x in edges) {
parent[Merge(parent, x[0])] = Merge(parent, x[1]);
}
int ans = 0;
for ( int i = 0; i < n; i++) {
if (parent[i] == i) ans++;
}
for ( int i = 0; i < n; i++) {
parent[i] = Merge(parent, parent[i]);
}
Dictionary< int , List< int >> m = new Dictionary< int , List< int >>();
for ( int i = 0; i < n; i++) {
if (!m.ContainsKey(parent[i])) {
m[parent[i]] = new List< int >();
}
m[parent[i]].Add(i);
}
foreach (KeyValuePair< int , List< int >> it in m) {
List< int > l = it.Value;
foreach ( int x in l) {
Console.Write(x + " " );
}
Console.WriteLine();
}
return ans;
} public static void Main() {
int n = 5;
List<List< int >> edges = new List<List< int >>();
edges.Add( new List< int > { 0, 1 });
edges.Add( new List< int > { 2, 1 });
edges.Add( new List< int > { 3, 4 });
Console.WriteLine( "Following are connected components:" );
int ans = CountConnectedComponents(n, edges);
} } |
# Python equivalent of above Java code # importing utilities import collections
# class to find connected components class ConnectedComponents:
# function to merge two components
def merge( self , parent, x):
if parent[x] = = x:
return x
return self .merge(parent, parent[x])
# function to find connected components
def connectedComponents( self , n, edges):
# list to store parents of each node
parent = [i for i in range (n)]
# loop to set parent of each node
for x in edges:
parent[ self .merge(parent, x[ 0 ])] = self .merge(parent, x[ 1 ])
# count to store number of connected components
ans = 0
# loop to count number of connected components
for i in range (n):
if parent[i] = = i:
ans + = 1
# loop to merge all components
for i in range (n):
parent[i] = self .merge(parent, parent[i])
# map to store parent and its connected components
m = collections.defaultdict( list )
for i in range (n):
m[parent[i]].append(i)
# loop to print connected components
for it in m.items():
l = it[ 1 ]
print ( " " .join([ str (x) for x in l]))
return ans
# driver code if __name__ = = "__main__" :
n = 5
edges = [[ 0 , 1 ], [ 2 , 1 ], [ 3 , 4 ]]
# print connected components
print ( "Following are connected components:" )
ans = ConnectedComponents().connectedComponents(n, edges)
|
// JavaScript program to find connected components in a graph // Function to find the parent of a node using path compression function merge(parent, x) {
if (parent[x] === x) {
return x;
}
return merge(parent, parent[x]);
} // Function to find the number of connected components in the graph function connectedcomponents(n, edges) {
// Initialize parent array for each node
let parent = [];
for (let i = 0; i < n; i++) {
parent[i] = i;
} // Union operation for all edges
for (let x of edges) {
parent[merge(parent, x[0])] = merge(parent, x[1]);
}
// Count the number of nodes with self as parent, which are the roots of connected components
let ans = 0;
for (let i = 0; i < n; i++) {
ans += parent[i] === i;
}
// Find the parent of each node again, and group nodes with the same parent
for (let i = 0; i < n; i++) {
parent[i] = merge(parent, parent[i]);
}
let m = new Map();
for (let i = 0; i < n; i++) {
if (!m.has(parent[i])) {
m.set(parent[i], []);
}
m.get(parent[i]).push(i);
}
// Print the nodes in each connected component
console.log( "Following are connected components:" );
for (let [key, value] of m) {
console.log(value.join( " " ));
}
// Return the number of connected components
return ans;
} // Sample input let n = 5; let e1 = [0, 1]; let e2 = [2, 1]; let e3 = [3, 4]; let e = [e1, e2, e3]; // Find connected components and print them let a = connectedcomponents(n, e); |
Following are connected components: 0 1 2 3 4
Time Complexity: O(V)
Auxiliary Space: O(V)