红黑树

 1、 红黑树（Red Black Tree） 是一种自平衡二叉查找树，是在计算机科学中用到的一种数据结构，典型的用途是实现关联数组。 红黑树是在1972年由Rudolf Bayer发明的，当时被称为平衡二叉B树（symmetric binary B-trees）。后来，在1978年被 Leo J. Guibas 和 Robert Sedgewick 修改为如今的“红黑树”。 红黑树是一种特化的AVL树（平衡二叉树），都是在进行插入和删除操作时通过特定操作保持二叉查找树的平衡，从而获得较高的查找性能。 它虽然是复杂的，但它的最坏情况运行时间也是非常良好的，并且在实践中是高效的： 它可以在O(log n)时间内做查找，插入和删除，这里的n 是树中元素的数目。

2、特性：

3、插入（对于红黑树其比avl树多了颜色，在插入方面差异大）

  对于插入的新增节点而言其可能为cur/在cur的左右插入，对于cur而言其有父亲节点parent（p），祖父grandfather（g），叔叔（u），叔叔的变化是造成插入差异的原因：

（1）u存在且为红，cur为新增节点：

（2）cur为红，p为红，g为黑，u不存在/u存在且为黑（单旋）：

      注：cur为g左边的p的左边时用右单旋，反之左单旋

（3）cur为红，p为红，g为黑，u不存在/u存在且为黑（双旋）：

注：cur为g左边的p的右边时先用左单旋再用右单旋，反之先右后左单旋

//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);
	// 新增节点。颜色红色给红色，黑色的话会破坏规则
	cur->_col = RED;
	if (parent->_kv.first < kv.first)
	{
		parent->_right = cur;
	}
	else
	{
		parent->_left = cur;
	}
	cur->_parent = parent;

	//调整树维持红黑树的规则
	//当树的父亲节点存在或者父亲节点为红继续调整
	while (parent && parent->_col == RED)
	{
		Node* grandfather = parent->_parent;
		//    g
		//  p   u
		if (parent == grandfather->_left)
		{
			Node* uncle = grandfather->_right;
			// u存在且为红，变色再继续往上处理
			if (uncle && uncle->_col == RED)
			{
				//将p与u变黑g变红
				parent->_col = uncle->_col = BLACK;
				grandfather->_col = RED;
				//g变为新的cur继续调整
				cur = grandfather;
				parent = cur->_parent;
			}
			else
			{
				// u存在且为黑或不存在，旋转+变色
				if (cur == parent->_left)
				{
					//    g
					//  p   u
					//c
					//单旋
					RotateR(grandfather);
					parent->_col = BLACK;
					grandfather->_col = RED;
				}
				else
				{
					//    g
					//  p   u
					//    c
					//双旋
					RotateL(parent);
					RotateR(grandfather);

					cur->_col = BLACK;
					grandfather->_col = RED;
				}

				break;
			}
		}
		//p和u反向
		else
		{
			//    g
			//  u   p
			Node* uncle = grandfather->_left;
			// 叔叔存在且为红，-》变色即可
			if (uncle && uncle->_col == RED)
			{
				parent->_col = uncle->_col = BLACK;
				grandfather->_col = RED;

				// 继续往上处理
				cur = grandfather;
				parent = cur->_parent;
			}
			else // 叔叔不存在，或者存在且为黑
			{
				// 情况二：叔叔不存在或者存在且为黑
				// 旋转+变色
				//      g
				//   u     p
				//            c
				if (cur == parent->_right)
				{
					RotateL(grandfather);
					parent->_col = BLACK;
					grandfather->_col = RED;
				}
				else
				{
					//		g
					//   u     p
					//      c
					RotateR(parent);
					RotateL(grandfather);
					cur->_col = BLACK;
					grandfather->_col = RED;
				}
				break;
			}
		}
	}
	//保持根节点为黑
	_root->_col = BLACK;



	return true;
}

4、判断红黑树的合理性：

//判断平衡
bool IsBalance()
{
	if (_root == nullptr)
		return true;

	if (_root->_col == RED)
	{
		return false;
	}

	// 参考值
	int refNum = 0;
	Node* cur = _root;
	while (cur)
	{
		if (cur->_col == BLACK)
		{
			++refNum;
		}

		cur = cur->_left;
	}

	return Check(_root, 0, refNum);
}
////////////////////////////////////////////////////////////////
bool Check(Node* root, int blackNum, const int refNum)
{
	if (root == nullptr)
	{
		//cout << blackNum << endl;
		if (refNum != blackNum)
		{
			cout << "存在黑色节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 << "存在连续的红色节点" << endl;
		return false;
	}

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

	return Check(root->_left, blackNum, refNum)
		&& Check(root->_right, blackNum, refNum);
}

5、与anl的区别：

6、完整代码：

#pragma once
#pragma once
#include<iostream>
#include<vector>
#include<assert.h>
using namespace std;

//利用枚举控制节点颜色
enum Colour
{
	RED,
	BLACK
};

//控制树的节点
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)
	{}
};

//控制树的行为
template<class K, class V>
class RBTree
{
	typedef RBTreeNode<K, V> Node;
public:
	//强制编译器生成默认构造
	RBTree() = default;

	//利用copy完成深层拷贝
	RBTree(const RBTree<K, V>& t)
	{
		_root = Copy(t._root);
	}

	//赋值
	RBTree<K, V>& operator=(RBTree<K, V> t)
	{
		swap(_root, t._root);
		return *this;
	}
	//析构
	~RBTree()
	{
		Destroy(_root);
		_root = nullptr;
	}
	//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);
		// 新增节点。颜色红色给红色，黑色的话会破坏规则
		cur->_col = RED;
		if (parent->_kv.first < kv.first)
		{
			parent->_right = cur;
		}
		else
		{
			parent->_left = cur;
		}
		cur->_parent = parent;

		//调整树维持红黑树的规则
		//当树的父亲节点存在或者父亲节点为红继续调整
		while (parent && parent->_col == RED)
		{
			Node* grandfather = parent->_parent;
			//    g
			//  p   u
			if (parent == grandfather->_left)
			{
				Node* uncle = grandfather->_right;
				// u存在且为红，变色再继续往上处理
				if (uncle && uncle->_col == RED)
				{
					//将p与u变黑g变红
					parent->_col = uncle->_col = BLACK;
					grandfather->_col = RED;
					//g变为新的cur继续调整
					cur = grandfather;
					parent = cur->_parent;
				}
				else
				{
					// u存在且为黑或不存在，旋转+变色
					if (cur == parent->_left)
					{
						//    g
						//  p   u
						//c
						//单旋
						RotateR(grandfather);
						parent->_col = BLACK;
						grandfather->_col = RED;
					}
					else
					{
						//    g
						//  p   u
						//    c
						//双旋
						RotateL(parent);
						RotateR(grandfather);

						cur->_col = BLACK;
						grandfather->_col = RED;
					}

					break;
				}
			}
			//p和u反向
			else
			{
				//    g
				//  u   p
				Node* uncle = grandfather->_left;
				// 叔叔存在且为红，-》变色即可
				if (uncle && uncle->_col == RED)
				{
					parent->_col = uncle->_col = BLACK;
					grandfather->_col = RED;

					// 继续往上处理
					cur = grandfather;
					parent = cur->_parent;
				}
				else // 叔叔不存在，或者存在且为黑
				{
					// 情况二：叔叔不存在或者存在且为黑
					// 旋转+变色
					//      g
					//   u     p
					//            c
					if (cur == parent->_right)
					{
						RotateL(grandfather);
						parent->_col = BLACK;
						grandfather->_col = RED;
					}
					else
					{
						//		g
						//   u     p
						//      c
						RotateR(parent);
						RotateL(grandfather);
						cur->_col = BLACK;
						grandfather->_col = RED;
					}
					break;
				}
			}
		}
		//保持根节点为黑
		_root->_col = BLACK;



		return true;
	}

	void InOrder()
	{
		_InOrder(_root);
		cout << endl;
	}

	int Height()
	{
		return _Height(_root);
	}

	int Size()
	{
		return _Size(_root);
	}
	//判断平衡
	bool IsBalance()
	{
		if (_root == nullptr)
			return true;

		if (_root->_col == RED)
		{
			return false;
		}

		// 参考值
		int refNum = 0;
		Node* cur = _root;
		while (cur)
		{
			if (cur->_col == BLACK)
			{
				++refNum;
			}

			cur = cur->_left;
		}

		return Check(_root, 0, refNum);
	}

	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;
	}
private:
	bool Check(Node* root, int blackNum, const int refNum)
	{
		if (root == nullptr)
		{
			//cout << blackNum << endl;
			if (refNum != blackNum)
			{
				cout << "存在黑色节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 << "存在连续的红色节点" << endl;
			return false;
		}

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

		return Check(root->_left, blackNum, refNum)
			&& Check(root->_right, blackNum, refNum);
	}
	int _Size(Node* root)
	{
		return root == nullptr ? 0 : _Size(root->_left) + _Size(root->_right) + 1;//左右加自己
	}

	int _Height(Node* root)
	{
		if (root == nullptr)
			return 0;

		int leftHeight = _Height(root->_left);
		int rightHeight = _Height(root->_right);
		//取大的相加
		return leftHeight > rightHeight ? leftHeight + 1 : rightHeight + 1;
	}

	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* 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  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 (parentParent == nullptr)
		{
			_root = subL;
			subL->_parent = nullptr;
		}
		else
		{
			if (parent == parentParent->_left)
			{
				parentParent->_left = subL;
			}
			else
			{
				parentParent->_right = subL;
			}

			subL->_parent = parentParent;
		}

		
	}

	void RotateRL(Node* parent)
	{
		Node* subR = parent->_right;
		Node* subRL = subR->_left;
		int bf = subRL->_bf;

		RotateR(parent->_right);
		RotateL(parent);

		if (bf == 0)
		{
			subR->_bf = 0;
			subRL->_bf = 0;
			parent->_bf = 0;
		}
		else if (bf == 1)
		{
			subR->_bf = 0;
			subRL->_bf = 0;
			parent->_bf = -1;
		}
		else if (bf == -1)
		{
			subR->_bf = 1;
			subRL->_bf = 0;
			parent->_bf = 0;
		}
		else
		{
			assert(false);
		}
	}

	void RotateLR(Node* parent)
	{
		Node* subL = parent->_left;
		Node* subLR = subL->_right;
		int bf = subLR->_bf;

		RotateL(parent->_left);
		RotateR(parent);

		if (bf == 0)
		{
			subL->_bf = 0;
			subLR->_bf = 0;
			parent->_bf = 0;
		}
		else if (bf == -1)
		{
			subL->_bf = 0;
			subLR->_bf = 0;
			parent->_bf = 1;
		}
		else if (bf == 1)
		{
			subL->_bf = -1;
			subLR->_bf = 0;
			parent->_bf = 0;
		}
		else
		{
			assert(false);
		}

	}
	//中序完成删除
	void Destroy(Node* root)
	{
		if (root == nullptr)
			return;

		Destroy(root->_left);
		Destroy(root->_right);
		delete root;
	}
	//先序遍历完成拷贝
	Node* Copy(Node* root)
	{
		if (root == nullptr)
			return nullptr;

		Node* newRoot = new Node(root->_kv);
		newRoot->_left = Copy(root->_left);
		newRoot->_right = Copy(root->_right);

		return newRoot;
	}

private:
	Node* _root = nullptr;
};
void TestRBTree1()
{
	RBTree<int, int> t;
	//int a[] = { 16, 3, 7, 11, 9, 26, 18, 14, 15 };
	int a[] = { 4, 2, 6, 1, 3, 5, 15, 7, 16, 14 };
	for (auto e : a)
	{
		/*if (e == 9)
		{
			int i = 0;
		}*/

		t.Insert({ e, e });

		//cout << e << "->" << t.IsBalanceTree() << endl;
	}

	t.InOrder();
	cout << t.IsBalance() << endl;
}