红黑树介绍

红黑树，是一种二叉搜索树，但在每个结点上增加一个存储位表示结点的颜色，可以是Red或Black。

通过对任何一条从根到叶子的路径上各个结点着色方式的限制，红黑树确保没有一条路径会比其他路径长出俩倍，因而是接近平衡的。

最长路径不超过最短路径的二倍

红黑树的性质：
1. 每个结点不是红色就是黑色
2. 根节点是黑色的
3. 如果一个节点是红色的，则它的两个孩子结点是黑色的（树中没有连续的红色节点）
4. 对于每个结点，从该结点到其所有后代叶结点的简单路径上，均包含相同数目的黑色结点。（每条路径的黑色节点数量相等）
5. 每个叶子结点都是黑色的(此处的叶子结点指的是空结点

红黑树实现

节点的插入

当有新节点要插入时，我们尽量遵守规则4，因此插入新节点时，我们插入红色节点。

情况一: cur为红，p为红，g为黑，u存在且为红

abcde是符合红黑树的子树。

如果u存在且为红，p和u变黑，g变红，继续往上处理或者g变黑

新增可以在ab的任意孩子位置

情况二: cur为红，p为红，g为黑，u不存在/u存在且为黑。

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
p、g变色--p变黑，g变红

情况三: cur为红，p为红，g为黑，u不存在/u存在且为黑。

p为g的左孩子，cur为p的右孩子，则针对p做左单旋转；相反，
p为g的右孩子，cur为p的左孩子，则针对p做右单旋转
则转换成了情况2

完整代码

#include
#include 
#include 
#include 
using namespace std;#include
#include enum Colour
{RED,BLACK
};template
struct RBTreeNode
{RBTreeNode* _left;RBTreeNode* _right;RBTreeNode* _parent;pair _kv;Colour _col;RBTreeNode(const pair& kv):_left(nullptr), _right(nullptr), _parent(nullptr), _kv(kv){}
};template
struct RBTree
{typedef RBTreeNode Node;
public:bool Insert(const pair& kv){if (_root == nullptr){_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;}}cur = new Node(kv);cur->_col = RED;if (parent->_kv.first < kv.first){parent->_right = cur;}else{parent->_left = cur;}cur->_parent = parent;while (parent && parent->_col == RED){Node* grandfater = parent->_parent;assert(grandfater);assert(grandfater->_col == BLACK);// 关键看叔叔if (parent == grandfater->_left){Node* uncle = grandfater->_right;// 情况一 : uncle存在且为红，变色+继续往上处理if (uncle && uncle->_col == RED){parent->_col = uncle->_col = BLACK;grandfater->_col = RED;// 继续往上处理cur = grandfater;parent = cur->_parent;}// 情况二+三：uncle不存在 + 存在且为黑else{// 情况二：右单旋+变色//     g //   p   u// cif (cur == parent->_left){RotateR(grandfater);parent->_col = BLACK;grandfater->_col = RED;}else{// 情况三：左右单旋+变色//     g //   p   u//     cRotateL(parent);RotateR(grandfater);cur->_col = BLACK;grandfater->_col = RED;}break;}}else // (parent == grandfater->_right){Node* uncle = grandfater->_left;// 情况一if (uncle && uncle->_col == RED){parent->_col = uncle->_col = BLACK;grandfater->_col = RED;// 继续往上处理cur = grandfater;parent = cur->_parent;}else{// 情况二：左单旋+变色//     g //   u   p//         cif (cur == parent->_right){RotateL(grandfater);parent->_col = BLACK;grandfater->_col = RED;}else{// 情况三：右左单旋+变色//     g //   u   p//     cRotateR(parent);RotateL(grandfater);cur->_col = BLACK;grandfater->_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){//cout << blackNum << endl;//return;if (benchmark == 0)//黑色节点数量的基准值{benchmark = blackNum;return true;}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)//如果有连续红色节点，则不是红黑树{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;parent->_right = subRL;if (subRL)subRL->_parent = parent;Node* ppNode = parent->_parent;subR->_left = parent;parent->_parent = subR;if (_root == parent){_root = subR;subR->_parent = nullptr;}else{if (ppNode->_left == parent){ppNode->_left = subR;}else{ppNode->_right = subR;}subR->_parent = ppNode;}}void RotateR(Node* parent){Node* subL = parent->_left;Node* subLR = subL->_right;parent->_left = subLR;if (subLR){subLR->_parent = parent;}Node* ppNode = parent->_parent;subL->_right = parent;parent->_parent = subL;if (_root == parent){_root = subL;subL->_parent = nullptr;}else{if (ppNode->_left == parent){ppNode->_left = subL;}else{ppNode->_right = subL;}subL->_parent = ppNode;}}private:Node* _root = nullptr;
};
void TestRBTree1()
{size_t N = 1000;srand(time(0));RBTree 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;
}
int main()
{TestRBTree1();return 0;
}

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C++——红黑树

红黑树介绍

红黑树实现

节点的插入

完整代码

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