无向图中的连通分量
给定一个无向图，逐行打印所有连通分量。例如，考虑下图。
我们在后续文章中讨论了在有向图中查找强连通分量的算法。
Kosaraju 的强连通分量算法。
Tarjan 算法寻找强连通分量
为无向图查找连通分量是一项更容易的任务。我们只需要从每个未访问的顶点开始执行 BFS 或 DFS，并且我们得到所有强连接的组件。以下是基于 DFS 的步骤。
1) Initialize all vertices as not visited.
2) Do following for every vertex 'v'.
(a) If 'v' is not visited before, call DFSUtil(v)
(b) Print new line character
DFSUtil(v)
1) Mark 'v' as visited.
2) Print 'v'
3) Do following for every adjacent 'u' of 'v'.
If 'u' is not visited, then recursively call DFSUtil(u)下面是上述算法的实现。
C++
// C++ program to print connected components in
// an undirected graph
#include
#include
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* 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::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[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, 3);
g.addEdge(3, 4);
cout << "Following are connected components \n";
g.connectedComponents();
return 0;
}
Java
// 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 > 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); // 5 vertices numbered from 0 to 4
g.addEdge(1, 0);
g.addEdge(2, 3);
g.addEdge(3, 4);
System.out.println(
"Following are connected components");
g.connectedComponents();
}
} Python3
# 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, 3)
g.addEdge(3, 4)
cc = g.connectedComponents()
print("Following are connected components")
print(cc)
# This code is contributed by Abhishek ValsanC#
// C++ program to print connected components in
// an undirected graph
#include
#include
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* 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::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[ 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, 3);
g.addEdge(3, 4);
cout << "Following are connected components \n";
g.connectedComponents();
return 0;
}
Javascript
C++
#include
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 >& 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 > m;
for (int i = 0; i < n; i++) {
m[parent[i]].push_back(i);
}
for (auto it = m.begin(); it != m.end(); it++) {
list l = it->second;
for (auto x : l) {
cout << x << " ";
}
cout << endl;
}
return ans;
}
int main()
{
int n = 5;
vector e1 = { 0, 1 };
vector e2 = { 2, 3 };
vector e3 = { 3, 4 };
vector > e;
e.push_back(e1);
e.push_back(e2);
e.push_back(e3);
int a = connectedcomponents(n, e);
cout << "total no. of connected components are: " << a
<< endl;
return 0;
} 输出：
0 1
2 3 4上述解决方案的时间复杂度为 O(V + E)，因为它对给定图执行简单的 DFS。
这可以使用时间复杂度为 O(N) 的不相交集联合来解决。 DSU 解决方案更易于理解和实施。
算法 ：
!.声明一个大小为 n 的数组 arr，其中 n 是节点的总数。
2. 对于数组 arr 的每个索引 i，该值表示第 i 个顶点的父节点是谁。比如 arr[0]=3，那么我们可以说顶点 0 的父节点是 3。
3. 将每个节点初始化为自身的父节点，然后在将它们相加的同时，相应地更改它们的父节点。
C++
#include
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 >& 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 > m;
for (int i = 0; i < n; i++) {
m[parent[i]].push_back(i);
}
for (auto it = m.begin(); it != m.end(); it++) {
list l = it->second;
for (auto x : l) {
cout << x << " ";
}
cout << endl;
}
return ans;
}
int main()
{
int n = 5;
vector e1 = { 0, 1 };
vector e2 = { 2, 3 };
vector e3 = { 3, 4 };
vector > e;
e.push_back(e1);
e.push_back(e2);
e.push_back(e3);
int a = connectedcomponents(n, e);
cout << "total no. of connected components are: " << a
<< endl;
return 0;
}
输出
0 1
2 3 4
total no. of connected components are: 2