1. 连通分量是什么

无向图G的极大连通子图称为G的连通分量( Connected Component)。任何连通图的连通分量只有一个，即是其自身，非连通的无向图有多个连通分量。

2. 案例

2.1.图极其数据结构初始化

2.2.求连通分量的方法

从每个顶点出发，判断是否有连通分量
BFS[BFS](https://blog.csdn.net/qq_44423388/article/details/127591933?spm=1001.2014.3001.5501)
DFS[DFS](https://blog.csdn.net/qq_44423388/article/details/127583096?spm=1001.2014.3001.5501)
并查集（本篇主讲,实现步骤见下）

2.3 具体实现

/*
测试用例：
1 2
1 4
2 4
*/


#include <vector>
#include <iostream>
#include <queue>
#include <unordered_map>

using namespace std;

/*
如果节点是相互连通的（从一个节点可以到达另一个节点），那么他们在同一棵树里，或者说在同一个集合里，或者说他们的祖先是相同的。
*/
//并查集的数据结构
class UnionFind {

private:
    // 记录每一个节点的父节点father<当前节点下标,父节点下标>
    unordered_map<int, int> father;
    // 记录集合数量
    int num_of_sets = 0;

public:

    //找节点x的父节点
    int find(int x) 
    {
        int root = x;

        while (father[root] != -1) 
        {
            root = father[root];
        }
        //优化的点：如果我们树很深，那么每次查询的效率都会非常低。这一步把树的深度固定为二。
        while (x != root) 
        {
            int original_father = father[x];
            father[x] = root;
            x = original_father;
        }

        return root;
    }

    bool is_connected(int x, int y) 
    {
        return find(x) == find(y);
    }

    //将连通的两个节点合并为同一个祖先，同时并查集的数目--
    void merge(int x, int y) 
    {
        int root_x = find(x);
        int root_y = find(y);

        if (root_x != root_y)
        {
            father[root_y] = root_x;
            num_of_sets--;
        }
    }
    //将新节点添加到并查集中
    void add(int x) 
    {
        if (!father.count(x))
        {
            father[x] = -1;
            num_of_sets++;
        }
    }
    //返回并查集个数
    int get_num_of_sets()
    {
        auto it = father.begin();
        while (it != father.end())
        {
            cout << it->first<<" ->"<<it->second << endl;
            it++;
        }
        
        return num_of_sets;
    }
};

class Connectedcomponent:protected UnionFind
{
private:
    int vertice = 0;//顶点数
    int edge = 0;//边数
    vector<vector<int>> e;
    //因为dfs和bfs都会对其进行改变，所有设置两个book
    vector<bool> book;//判断顶点j是否扩展过
    vector<bool> book1;//判断顶点j是否扩展过
    queue<int> qu;

    //DFS求连通分量个数
    void DFS_Alg(int current, int sum)//current当前所在的节点编号
    {
        sum++;
        if (sum == vertice)//所有的节点均已被访问
        {
            cout << current << endl;
            return;
        }
        else
        {
            cout << current << " ->";
        }
        for (int k = 1; k <= vertice; k++)
        {
            if (e[current][k] != 0 && book[k] == 0)
            {
                book[k] = 1;
                DFS_Alg(k, sum);
            }
        }
    }
    
public:
    Connectedcomponent(int x, int y) :vertice(x), edge(y)
    {
        //图的初始化从下标1开始
        e.resize(vertice + 1);//初始化二维数组的行
        for (int i = 0; i <= vertice; i++)
        {
            e[i].resize(vertice + 1,0);//初始化二维数组的列
        }
        
        book.resize(vertice + 1);
        book1.resize(vertice + 1);
    }
    //图的初始化
    void Init_tu()
    {
        for (int i = 0; i <= vertice; i++)
        {
            for (int j = 0; j <= vertice; j++)
            {
                if (i == 0 || j == 0)
                {
                    e[i][j] = 0;
                }
                if (i == j)
                {
                    e[i][j] = 0;
                }
                else
                {
                    e[i][j] = INT_MAX;
                }
            }
        }
    }
    //读入图的边，并且根据边的信息初始化数组dis，数组book
    void GetEdgeInfo()
    {
        cout << "输入边的信息（节点1，节点2）：" << endl;
        int e1 = 0, e2 = 0, weigth = 0;
        for (int i = 1; i <= edge; i++)//无向图
        {
            cin >> e1 >> e2;
            e[e1][e2] = 1;
            e[e2][e1] = 1;
        }        
    }

    //打印
    void Print()
    {
        for (int i = 1; i <= vertice; i++)
        {
            for (int j = 1; j <= vertice; j++)
            {
                cout << e[i][j] << "    ";
            }
            cout << endl;
        }
        cout << endl;
    }
    
    int DFS_Num()
    {
        int num = 0;
        for (int i = 1; i <= vertice; i++)
        {
            if (book[i] == false)
            {
                DFS_Alg(i,0);
                cout <<"end" <<endl;
                num++;
            }               
        }

        return num;
    }
    //BFS求连通分量个数
    int BFS_Num()
    {     
        int num = 0;
        for (int i = 1; i <= vertice; i++)//遍历每个节点，查看是否从该节点出发是否有连通分量
        {
            if (book1[i] == false)
            {
                qu.push(i);
                while (!qu.empty())
                {
                    int v = qu.front();
                    qu.pop();
                    book1[v] = true;
                    cout << v << "->";
                    for (int i = 1; i <= vertice; i++)//循坏找节点v的相邻节点
                    {
                        if (e[v][i] != 0 && book1[i] == false)
                        {
                            qu.push(i);
                            book1[i] = true;
                        }
                    }
                }
                num++;
            }            
            
            cout << "end" << endl;
        }
        return num;       

    }
    //并查集求连通分量的个数
    /*
    每个节点会记录它的父节点。
    */
    int UnionFindSet()
    {
        UnionFind uf;
        for (int i = 1; i <= vertice; i++)
        {
            uf.add(i);
            for (int j = 1; j < i; j++)
            {
                if (e[i][j] == 1)
                {
                    uf.merge(i, j);
                }
            }
        }

        return uf.get_num_of_sets();
    }
};

int main()
{
    int num1 = 0, num2 = 0,num3 = 0;
    Connectedcomponent Conn(5, 3);
    Conn.GetEdgeInfo();

    cout << "初始信息：" << endl;
    Conn.Print();
    
    cout << "DFS:::" << endl;
    num1 = Conn.DFS_Num();
    cout << "BFS:::" << endl;
    num2 = Conn.BFS_Num();


    cout << "Union Find Set:::" << endl;
    num3 = Conn.UnionFindSet();
    cout << num1 << "  " << num2 <<"   "<<num3<< endl;

    return 0;
}