RB-tree(红黑树)

红黑树的规则如下：

1.每个节点不是红色就是黑色

2.根节点为黑色

3.如果节点为红色，那么它的子节点必须为黑色

4.任何一个节点到NULL（树的尾端）的任何路径所包含的黑节点个数相同

简而言之就是每个路径的黑色节点数量相同

新增的节点必须为红，因为增加红色节点所要付出的代价会比黑色的要小很多，如果新增节点为黑色，那么就会打破第四条规则，整棵树都要调整，代价是相当之大的

红黑树节点的定义

enum Colour
{Red,Black
};template<class K, class V>
struct RBTreeNode
{RBTreeNode<K, V>* _left;RBTreeNode<K, V>* _right;RBTreeNode<K, V>* _parent;pair<K, V> _kv;Colour _col;RBTreeNode():_left(nullptr),_right(nullptr),_parent(nullptr),_kv(kv){}
};

红黑树的插入操作

红黑树的插入操作分为几种情况

情况一：

当cur为红、parent为红、grandparent为黑、uncle存在并且为红

操作过程：

将parent和uncle全部染成黑色，grandparent染成红色，然后把cur给grandparent继续向上判断，如果cur的parent依旧为红色，那么就要继续染色，如果cur为_root了，将根节点染成黑色。

具象图：

抽象图：

在抽象图当中，在a或者b子树当中触发了染色机制，然后一路染色上来，一直到cur也染成了红色，cur和p就造成冲突，需要继续染色。

情况二：

当cur为红，p为红，g黑，u不存在/u为黑

操作：

p为g的左孩子，cur为p的左孩子，则进行右单旋转；

相反，p为g的右孩子，cur为p的右孩子，则进行左单旋转

最后p变为黑色，cur和g变为红色

具象图：

当uncle为空的时候

抽象图：

情况三：

当cur和p为红，g为黑，u为黑/不存在

如果parent为grandparent的左儿子，对parent左旋，再对grandparent右旋

如果parent为grandparent的右儿子，对parent右旋，再对grandparent左旋

cur染为黑色，cur和grandparent染为红色。

具象图：

抽象图：

红黑树+测试代码

#pragma once
#include"AVLTree.h"enum Colour
{Red,Black
};template<class K, class V>
struct RBTreeNode
{RBTreeNode<K, V>* _left;RBTreeNode<K, V>* _right;RBTreeNode<K, V>* _parent;pair<K, V> _kv;Colour _col;RBTreeNode(const pair<K, V>& kv):_left(nullptr), _right(nullptr), _parent(nullptr), _kv(kv){}
};template<class K,class V>
class RBTree
{typedef RBTreeNode<K,V> Node;
public:bool Insert(const pair<K, V>& kv){if (_root == nullptr){//创建根节点_root = new Node(kv);_root->_col = Black;return true;}Node* parent = nullptr;Node* cur = _root;while (cur){if (cur->_kv.first<kv.first){parent = cur;cur = cur->_right;}else if (cur->_kv.first > kv.first){parent = cur;cur = cur->_left;}else{return false;}}//找到插入点cur = new Node(kv);cur->_col = Red;//链接节点if (parent->_kv.first < kv.first){parent->_right = cur;}else{parent->_left = cur;}cur->_parent = parent;//检查是否满足红黑树的要求while (parent && parent->_col == Red){//如果父亲节点的颜色为红色，那么就有可能需要改变颜色或者旋转Node* grandparent = parent->_parent;assert(grandparent);assert(grandparent->_col == Black);//父亲为红色，那么祖父一定为黑色，不然之前的插入就存在问题if (parent == grandparent->_left){Node* uncle = grandparent->_right;//情况一，uncle存在并且uncle为红色，就要继续往上染色if (uncle && uncle->_col == Red){parent->_col = uncle->_col = Black;grandparent->_col = Red;cur = grandparent;parent = cur->_parent;}else{//uncle不存在或者uncle为黑色就会走到这里//情况二+情况三if (cur == parent->_left){// 情况二：右单旋+变色//     g //   p   u// c//右旋+染色RotateR(grandparent);parent->_col = Black;grandparent->_col = Red;}else{// 情况三：左右单旋+变色//     g //   p   u//     cRotateL(parent);RotateR(grandparent);cur->_col = Black;parent->_col = grandparent->_col = Red;}break;}}else{Node* uncle = grandparent->_left;if (uncle && uncle->_col == Red){parent->_col = uncle->_col = Black;grandparent->_col = Red;cur = grandparent;parent = cur->_parent;}else{if (cur == parent->_right){RotateL(grandparent);parent->_col = Black;grandparent->_col = Red;}else{RotateR(parent);RotateL(grandparent);cur->_col = Black;grandparent->_col = Red;}break;}}}_root->_col = Black;return true;}void InOrder(){_InOrder(_root);cout << endl;}bool IsBalance(){if (_root == nullptr){return true;}if (_root->_col == Red){cout << "根节点不是黑色" << endl;return false;}// 黑色节点数量基准值int benchmark = 0;Node* cur = _root;while (cur){if (cur->_col == Black)++benchmark;cur = cur->_left;}return PrevCheck(_root, 0, benchmark);}private:bool PrevCheck(Node* root, int blackNum, int& benchmark){if (root == nullptr){if (blackNum != benchmark){cout << "某条黑色节点的数量不相等" << endl;return false;}else{return true;}}if (root->_col == Black){++blackNum;}if (root->_col == Red && root->_parent->_col == Red){cout << "存在连续的红色节点" << endl;return false;}return PrevCheck(root->_left, blackNum, benchmark)&& PrevCheck(root->_right, blackNum, benchmark);}void _InOrder(Node* root){if (root == nullptr){return;}_InOrder(root->_left);cout << root->_kv.first << ":" << root->_kv.second << endl;_InOrder(root->_right);}void RotateL(Node* parent){Node* subR = parent->_right;Node* subRL = subR->_left;Node* pparent = parent->_parent;parent->_right = subRL;if (subRL){subRL->_parent = parent;}subR->_left = parent;parent->_parent = subR;if (parent == _root){_root = subR;subR->_parent = nullptr;}else{if (pparent->_left == parent){pparent->_left = subR;}else{pparent->_right = subR;;}subR->_parent = pparent;}}void RotateR(Node* parent){Node* subL = parent->_left;Node* subLR = subL->_right;parent->_left = subLR;Node* pparent = parent->_parent;//防止空指针if (subLR){subLR->_parent = parent;}subL->_right = parent;parent->_parent = subL;if (parent == _root){//如果parent就是根节点_root = subL;subL->_parent = nullptr;}else{//如果parent只是一颗子树的根节点，就还需要连接好parent//判断是左还是右节点if (pparent->_left == parent){pparent->_left = subL;}else{pparent->_right = subL;}subL->_parent = pparent;}}Node* _root = nullptr;
};void TestRBTree2()
{size_t N = 1000;srand(time(0));RBTree<int, int> t1;for (size_t i = 0; i < N; ++i){int x = rand();cout << "Insert:" << x << ":" << i << endl;t1.Insert(make_pair(x, i));}cout << "IsBalance:" << t1.IsBalance() << endl;
}

运行结果：