LintCode 591. Connecting Graph III (并查集经典题！！！)

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本文介绍了一种使用并查集数据结构解决图连接问题的方法。在一个初始无边的图中，通过connect操作连接节点，并用query操作返回当前图的连通组件数量。示例代码展示了如何实现并查集，包括路径压缩优化，以高效地处理节点连接和查询。

Connecting Graph III
Given n nodes in a graph labeled from 1 to n. There is no edges in the graph at beginning.

You need to support the following method:

connect(a, b), an edge to connect node a and node b
query(), Returns the number of connected component in the graph
Example
5 // n = 5
query() return 5
connect(1, 2)
query() return 4
connect(2, 4)
query() return 3
connect(1, 4)
query() return 3

思路：并查集经典题。
代码如下：

class ConnectingGraph3 {
public:
    /**
     * @param a: An integer
     * @param b: An integer
     * @return: nothing
     */
    ConnectingGraph3(int n) {
        father.resize(n + 1);
        for (int i = 1; i <= n; ++i) {
            father[i] = i;
        }
        count = n; //total # of connected components
    }
     
    void connect(int a, int b) {
        int fatherA = find(a);
        int fatherB = find(b);
        if (fatherA != fatherB) {
            father[fatherA] = fatherB;    //not father[a] = father[b], not father[a] = fatherB
            count--;
        }
    }

    /**
     * @return: An integer
     */
    int query() {
        return count;
    }

private:
    vector<int> father;
    int count;
    int find(int x) {
        if (father[x] == x) return x;
        int x2 = x;
                
        while(x != father[x]) {
            x = father[x];
        }
        
        //path compression!!!
        while(x2 != x) {    
            int temp = father[x2];
            father[x2] = x;
            x2 = temp;
        }
        
        return x;
    }
    
};