并查集

我们可以在一开始把每个元素都看成一棵树，并且根结点的父结点指向自己，在此基础上完成查找、合并操作。

两种优化方法：第一种按秩合并，即每次都把较矮的树并到较高的树中。第二种路径压缩，即让所有的并入的结点都指向整棵树的根结点。

具体代码如下：

//并查集
class DisjointSet {
private:
    //rank is the height of tree
    int *father, *rank;
public:
    DisjointSet(int size) {
        father = new int[size];
        rank = new int[size];
        for (int i = 0; i < size; ++i) {
            //initially every node is a tree so father points to itself
            father[i] = i;
            rank[i] = 0;
        }
    }
    ~DisjointSet() {
        delete[] father;
        delete[] rank;
    }
    int find_set(int node) {
        if (father[node] != node) {
            //path compression
            father[node]=find_set(father[node]);
        }
        return father[node];
    }
    bool merge(int node1, int node2) {
        int ancestor1 = find_set(node1);
        int ancestor2 = find_set(node2);
        if (ancestor1 != ancestor2) {
            //rank merge
            if (rank[ancestor1] > rank[ancestor2]) {
                swap(ancestor1, ancestor2);
            }
            //shorter tree points to higher tree
            father[ancestor1] = ancestor2;
            rank[ancestor2] = max(rank[ancestor2], rank[ancestor1] + 1);
            return true;
        }
        return false;
    }
};