红黑树的概念
红黑树也是一颗搜索二叉树，它每个节点的颜色不是红色就是黑色，所以叫做红黑树
C++：二叉搜索树-CSDN博客
它可以确保没有一条路径比其他路径长出2倍，因此它接近平衡
红黑树的规则
- 每个节点不是红色就是黑色
- 根节点是黑色
- 如果⼀个结点是红色的，则它的两个孩⼦结点必须是⿊色的，也就是说任意⼀条路径不会有连续的红色结点
- 对于任意一个节点，该节点到NULL节点的路径上均包含相同数量的黑色节点
根据红黑树的规则，它之所以能做到没有一条路径比其他路径长出2倍是因为规则3和4
最短的节点路径上的颜色就是全黑，而相对最长的路径就是全部一黑一红，所以最长只能是两倍
红黑树和AVL树相比，红黑树的整体高度会偏高一些，但是由于它的特性，它旋转的次数会相较于AVL树的旋转次数少，并且它们的高度差也只差很小的常数，这高度差对于CPU的运行速度来讲都没有区别，所以红黑树的效率也是极高的
红黑树的实现
红黑树的结构
enum Colour
{RED,BLACK
};template<class K, class V>
struct RBTreeNode
{pair<K, V> _kv;RBTreeNode<K, V>* _left;RBTreeNode<K, V>* _right;RBTreeNode<K, V>* _parent;Colour _col;RBTreeNode(const pair<K, V>& kv):_kv(kv), _left(nullptr), _right(nullptr), _parent(nullptr){}
};template<class K, class V>
class RBTree
{typedef RBTreeNode<K, V> Node;
public:
private:Node* _root = nullptr;
};首先用了一个枚举的类型枚举了红色和黑色两种颜色，当然也可以使用数字来表示
这种枚举好处在于在调试的时候是可以直观的看见RED和BLACK的，而不是数字等
红黑树的节点和AVL树一样都需要三叉链，值用pair来存储
红黑树类里面只需要一个root节点的指针即可
红黑树的插入
bool Insert(const pair<K, V>& 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* grandfather = parent->_parent;if (grandfather->_left == parent){Node* uncle = grandfather->_right;if (uncle && uncle->_col == RED){// 变色parent->_col = uncle->_col = BLACK;grandfather->_col = RED;// 继续向上处理cur = grandfather;parent = cur->_parent;}else{if (cur == parent->_left){RotateR(grandfather);parent->_col = BLACK;grandfather->_col = RED;}else{RotateL(parent);RotateR(grandfather);cur->_col = BLACK;parent->_col = grandfather->_col = RED; // 1}break;}}else{Node* uncle = grandfather->_left;if (uncle && uncle->_col == RED){// 变色parent->_col = uncle->_col = BLACK;grandfather->_col = RED;// 继续向上处理cur = grandfather;parent = cur->_parent;}else{if (cur == parent->_right){RotateL(grandfather);parent->_col = BLACK;grandfather->_col = RED;}else{RotateR(parent);RotateL(grandfather);cur->_col = BLACK;parent->_col = grandfather->_col = RED; // 1}break;}}}// 确保根节点颜色是黑色_root->_col = BLACK;return true;
}前面的插入逻辑和搜索二叉树、AVL树的插入逻辑都一样，毕竟它们都是同一个规则
只需要注意的是插入的如果是根节点，则需要插入的是黑色的节点（红黑树的规则）
其余的只需要插入红色的节点即可
为什么插入的新节点是红色节点而不是黑色节点
根据红黑树的四个规则
- 每个节点不是红色就是黑色
- 根节点是黑色
下面大多的代码就是维护红黑树的结构
因为我们插入的是红色节点，所以我们只需要判断它的parent节点的颜色是否为红色
当parent节点在grandparent节点的左边时
若uncle节点也为红色，我们可以把grandparent节点变红，parent和uncle节点变黑，这样我们的规则3的问题就解决了，规则4也没有触犯到
我们这个语句是需要循环往上执行的，因为当我们把grandfather变红后，保不齐它的father就是红色，这样又会违反规则3，所以我们需要循环处理
若uncle不如我们所料，那我们就需要旋转处理了
若是cur在parent的左边上图所示的情况，无论我们怎么改变颜色都解决不了问题，这时候我们就需要引入AVL中写过的旋转来解决
只需要对grandfather进行右旋+变色即可
旋转完后需要注意颜色不能违反规则，让parent变成黑色（它有可能是根节点），grandfather和cur变成红色即可
若是cur在parent的右边如上图所示，这样我们只进行单旋是解决不了问题的，很明显p和c偏向右，需要左单选，而从整体的g看来是需要右单旋的，这时候就只需要双旋+变色即可
先对p左单旋，解决它偏向右的问题，然后再对g右单旋，这时候整体就平衡了
这时候只需要和上面一样调整变色即可
剩下下面的代码的逻辑完全和上面的一样
和上面的区别就是上面的parent是在grandparent的左边，而下面的parent是在grandparent的右边，只有方向上面的区别
最后需要确保根节点的颜色是黑色，因为我们在调整的过程中可能会无意中把根节点的颜色变成红色，在各个 if 语句中解决起来相对麻烦一些，所以我们在最后修改root的颜色为黑色就解决了
红黑树的查找
Node* Find(const K& key)
{Node* cur = _root;while (cur){if (cur->_kv.first < key){cur = cur->_right;}else if (cur->_kv.first > key){cur = cur->_left;}else{return cur;}}return nullptr;
}查找逻辑和搜索二叉树一致，搜索效率为O(logN)
红黑树的验证
bool Check(Node* root, int blackNum, const int refNum)
{if (root == nullptr){if (blackNum != refNum){cout << "存在黑色结点的数量不相等的路径" << endl;return false;}return true;}if (root->_col == RED && root->_parent->_col == RED){cout << root->_kv.first << "存在连续的红色结点" << endlC;return false;}if (root->_col == BLACK){blackNum++;}return Check(root->_left, blackNum, refNum) && Check(root->_right, blackNum, refNum);
}bool IsBalance()
{if (_root == nullptr)return true;if (_root->_col == RED)return false;int refNum = 0;Node* cur = _root;while (cur){if (cur->_col == BLACK)refNum++;cur = cur->_left;}return Check(_root, 0, refNum);
}Check函数的第一个参数为根节点，第二个参数需要记录黑色节点的数量，第三个参数为黑色节点的其中一个参考值
这里的参考值是在IsBalance中传递的，这个参考值是随机选取一条路径记录黑色节点的数量，接着用这个参考值在Check函数中检查每一条路径的黑色节点与之对比，若不相等则说明存在黑色节点的数量不相等的路径
判断连续相同的红色节点
如果我们从parent来判断它的左右孩子是否为红色就比较麻烦
我们可以直接判断若当前节点颜色为红色，并且它的parent也为红色，则说明存在连续的红色节点
完整代码
#pragma onceenum Colour
{RED,BLACK
};template<class K, class V>
struct RBTreeNode
{pair<K, V> _kv;RBTreeNode<K, V>* _left;RBTreeNode<K, V>* _right;RBTreeNode<K, V>* _parent;Colour _col;RBTreeNode(const pair<K, V>& kv):_kv(kv), _left(nullptr), _right(nullptr), _parent(nullptr){}
};template<class K, class V>
class RBTree
{typedef RBTreeNode<K, V> Node;
public:bool Insert(const pair<K, V>& 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* grandfather = parent->_parent;if (grandfather->_left == parent){Node* uncle = grandfather->_right;if (uncle && uncle->_col == RED){// 变色parent->_col = uncle->_col = BLACK;grandfather->_col = RED;// 继续向上处理cur = grandfather;parent = cur->_parent;}else{if (cur == parent->_left){RotateR(grandfather);parent->_col = BLACK;grandfather->_col = RED;}else{RotateL(parent);RotateR(grandfather);cur->_col = BLACK;parent->_col = grandfather->_col = RED;}break;}}else{Node* uncle = grandfather->_left;if (uncle && uncle->_col == RED){// 变色parent->_col = uncle->_col = BLACK;grandfather->_col = RED;// 继续向上处理cur = grandfather;parent = cur->_parent;}else{if (cur == parent->_right){RotateL(grandfather);parent->_col = BLACK;grandfather->_col = RED;}else{RotateR(parent);RotateL(grandfather);cur->_col = BLACK;parent->_col = grandfather->_col = RED;}break;}}}// 确保根节点颜色是黑色_root->_col = BLACK;return true;}void RotateR(Node* parent){Node* subL = parent->_left;Node* subLR = subL->_right;parent->_left = subLR;if (subLR)subLR->_parent = parent;Node* parentParent = parent->_parent;subL->_right = parent;parent->_parent = subL;if (parent == _root){_root = subL;subL->_parent = nullptr;}else{if (parentParent->_left == parent){parentParent->_left = subL;}else{parentParent->_right = subL;}subL->_parent = parentParent;}}void RotateL(Node* parent){Node* subR = parent->_right;Node* subRL = subR->_left;parent->_right = subRL;if (subRL)subRL->_parent = parent;Node* parentParent = parent->_parent;subR->_left = parent;parent->_parent = subR;if (parentParent == nullptr){_root = subR;subR->_parent = nullptr;}else{if (parent == parentParent->_left){parentParent->_left = subR;}else{parentParent->_right = subR;}subR->_parent = parentParent;}}void InOrder(){_InOrder(_root);cout << endl;}int Height(){return _Height(_root);}int Size(){return _Size(_root);}Node* Find(const K& key){Node* cur = _root;while (cur){if (cur->_kv.first < key){cur = cur->_right;}else if (cur->_kv.first > key){cur = cur->_left;}else{return cur;}}return nullptr;}bool IsBalance(){if (_root == nullptr)return true;if (_root->_col == RED)return false;int refNum = 0;Node* cur = _root;while (cur){if (cur->_col == BLACK)refNum++;cur = cur->_left;}return Check(_root, 0, refNum);}
private:bool Check(Node* root, int blackNum, const int refNum){if (root == nullptr){if (blackNum != refNum){cout << "存在黑色结点的数量不相等的路径" << endl;return false;}return true;}if (root->_col == RED && root->_parent->_col == RED){cout << root->_kv.first << "存在连续的红色结点" << endlC;return false;}if (root->_col == BLACK){blackNum++;}return Check(root->_left, blackNum, refNum) && Check(root->_right, blackNum, refNum);}void _InOrder(Node* root){if (root == nullptr){return;}_InOrder(root->_left);cout << root->_kv.first << ":" << root->_kv.second << endl;_InOrder(root->_right);}int _Height(Node* root){if (root == nullptr)return 0;int leftHeight = _Height(root->_left);int rightHeight = _Height(root->_right);return leftHeight > rightHeight ? leftHeight + 1 : rightHeight + 1;}int _Size(Node* root){if (root == nullptr)return 0;return _Size(root->_left) + _Size(root->_right) + 1;}private:Node* _root = nullptr;
};完