波奇学C++：红黑树

红黑树性质：

1.每个结点不是红色就是黑色。

2.根节点是黑色

3.如果一个节点是红色，它的两个孩子必须是黑色

4.对于每个节点，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

5.每个叶结点都是黑色的（此处的叶子结点指的是空结点）

6.红黑路径的计算从根结点到NIL结点（红黑树特殊的黑色结点）

7.红黑树路径节点个数计算，不包括叶节点

8.最长路径的节点个数不会超过最短路径节点个数的两倍

证明满足上述条件，最长路径的节点个数不会超过最短路径节点个数的两倍。

最长路径*1/2<=黑节点个数，最短路径的黑结点个数=最长路径黑结点个数，

最小路径>=黑节点个数>=最长路径*1/2。

 cur,parent,uncle为红，

grandparent再进行判断

cur,parent为红，uncle为黑或者uncle不存在

parent为黑，grandparent为红，三角型表示存在至少一个黑色节点来维持左右黑色节点数目平衡。

双旋，parent先左旋转，cur再右旋。

uncle不存在时同理

总结：同理左右单旋取决于cur在parent的哪一边 ，unle颜色决定是否旋转。

补充：uncle为黑时，cur必然不是新插入节点，原因为了维持左右黑节点数目平衡，cur下面必然还有其他节点。

分析：红黑树最重要的两个性质，红红不能相连，左右黑数目相同。当红红相连时，必然有其一变变黑，新增的黑色节点只能在根或者子树的根处来保持平衡，旋转的目的就是如此平衡左右黑节点。

代码如下：

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)
		,_col(BLACK)
	{}
};
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* grandfather = parent->_parent;
			if (parent==grandfather->_left)
			{
				Node* uncle = grandfather->_right;
				if (uncle && uncle->_col == RED)
				{
					parent->_col = uncle->_col = BLACK;
					grandfather->_col = RED;

					cur = grandfather;
					parent = cur->_parent;
				}
				else //u不存在或者存在且为黑
				{
					if (cur==parent->_left)
					{
						RotateR(grandfather);
						parent->_col = BLACK;
						grandfather->_col = RED;
					}
					else
					{
						RotateL(parent);
						RotateR(grandfather);
						cur->_col = BLACK;
						grandfather->_col = RED;
					}
					break;
				}
			}
			else
			{
				Node* uncle = grandfather->_right;
				if (uncle && uncle->_col == RED)
				{
					parent->_col = uncle->_col = BLACK;
					grandfather->_col = RED;

					cur = grandfather;
					parent = cur->_parent;
				}
				else //u不存在或者存在且为黑
				{
					if (cur == parent->_right)
					{
						RotateL(grandfather);
						parent->_col = BLACK;
						grandfather->_col = RED;
					}
					else
					{
						RotateR(parent);
						RotateL(grandfather);
						cur->_col = BLACK;
						grandfather->_col = RED;
					}
					break;
				}
				
			}
		}
		_root->_col = BLACK;
		return true;
	}void RotateL(Node* parent)
	{
		Node* cur = parent->_right;
		Node* curleft = parent->_left;
		parent->_left = curleft;
		if (curleft)
		{
			curleft->_parent = parent;
		}
		cur->_left = parent;
		Node* ppnode = parent->_parent;
		parent->_parent = cur;
		if (parent == _root)
		{
			_root = cur;
			cur->_parent = nullptr;
		}
		else
		{
			if (ppnode->_left == parent)
			{
				ppnode->_left = cur;
			}
			else
			{
				ppnode->_right = cur;
			}
			cur->_parent = ppnode;
		}
		
	}	void RotateR(Node* parent)
	{
		Node* cur = parent->_left;
		Node* curright = cur->_right;
		parent->_left = curright;
		cur->_right = parent;

		Node* ppnode = parent->_parent;
		if (curright)
		{
			curright->_parent = parent;
		}
		parent->_parent = cur;
		if (ppnode == _root)
		{
			_root = cur;
			cur->_parent = nullptr;
		}
		else
		{
			if (ppnode->_left == parent)
			{
				ppnode->_left = cur;
			}
			else
			{
				ppnode->_right = cur;
			}
		}
		
		
	}
	bool CheckColour(Node* root,int blacknum,int benchmark)
	{
		if (root == nullptr)
		{ 
			if (blacknum > 2 * benchmark)
			{
				return false;
			}
			return true;
		}

		if (root->_col == BLACK)
		{
			++blacknum;

		}
		if (root->_col == RED && root->_parent && root->_parent->_col == RED)
		{
			cout << root->_kv.first << "出现连续红色节点" << endl;
			return false;
		}
		return CheckColour(root->_left,blacknum,benchmark)
			&& CheckColour(root->_right,blacknum,benchmark);
	}
	bool IaBalance()
	{
		IsBalance(_root);
	}
	bool IsBalance(Node* root)
	{
		if (root == nullptr)
			return true;
		if (root->_col != BLACK)
		{
			return false;
		}
		//基准值
		int benchmark = 0;
		Node* cur = _root;
		while (cur)
		{
			if (cur->_col == BLACK)
			{
				++benchmark;
			}
			cur = cur->_left;
		}

		return CheckColour(root,0,benchmark);
	}