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GPT-4o mini 来袭:开发者如何驾驭新一代AI模型?

 🌈个人主页:秦jh_-CSDN博客
🔥 系列专栏:https://blog.csdn.net/qinjh_/category_12575764.html?spm=1001.2014.3001.5482

 9efbcbc3d25747719da38c01b3fa9b4f.gif​ 

目录

前言

红黑树的概念

红黑树的性质

节点的定义

红黑树的插入操作 

 检测操作:

情况一: cur为红,p为红,g为黑,u存在且为红

情况二: cur为红,p为红,g为黑,u不存在/u存在且为黑 

Insert 代码

 红黑树的验证

红黑树与AVL树的比较

 完整代码


前言

红黑树的概念

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。

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。(不存在连续的红节点)
  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(每条路径都存在相同数量的黑色节点)  
  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)

节点的定义

 

红黑树的插入操作 

红黑树是在二叉搜索树的基础上加上其平衡限制条件,因此红黑树的插入可分为两步:

  1. 按照二叉搜索的树规则插入新节点
  2. 检测新节点插入后,红黑树的性质是否造到破坏 

 检测操作:

约定:cur为当前节点,p为父节点,g为祖父节点,u为叔叔节点

情况一: cur为红,p为红,g为黑,u存在且为红

 

 

情况二: cur为红,p为红,g为黑,u不存在/u存在且为黑 

情况二的双旋情况:

Insert 代码

bool Insert(const pair<K, V>& kv)
{
	if (_root == nullptr)
	{
		_root = new Node(kv);
		_root->_col = BLACK;  //根节点默认黑色
		return true;
	}

	Node* cur = _root;
	Node* parent = nullptr;
	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;  //新增节点给红色
	if (parent->_kv.first > kv.first)
	{
		parent->_left = cur;
	}
	else
	{
		parent->_right = cur;
	}
	cur->_parent = parent;

	// 检测新节点插入后,红黑树的性质是否造到破坏
	//父亲的颜色是黑色也结束
	while (parent && parent->_col == RED)
	{
		//关键看叔叔
		Node* grandfather = parent->_parent;
		if (parent == grandfather->_left)
		{
			Node* uncle = grandfather->_right;
			//如果叔叔存在也为红->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
			{
				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;
			}
		}
		else
		{
			Node* uncle = grandfather->_left; 
			//如果叔叔存在也为红->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
			{
				//    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;
			}
		}
	}
	//始终保持根为黑
	_root->_col = BLACK;

	return true;
}

 红黑树的验证

 红黑树的检测分为两步:

  1. 检测其是否满足二叉搜索树(中序遍历是否为有序序列)
  2. 检测其是否满足红黑树的性质

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

		int refNum = 0;    //取其中一条路径作为参考值
		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)
		{
			//cout << blackNum << endl;
			if (refNum != blackNum)
			{
				cout << "存在黑色节点数量不相等的路径" << 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);
	}

红黑树与AVL树的比较

红黑树和AVL树都是高效的平衡二叉树,增删改查的时间复杂度都是O(logN),红黑树不追求绝对平衡,其只需保证最长路径不超过最短路径的2倍,相对而言,降低了插入和旋转的次数, 所以在经常进行增删的结构中性能比AVL树更优,而且红黑树实现比较简单,所以实际运用中红 黑树更多。 

 完整代码

#pragma once 

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)
		, _col(RED)
	{}
};

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;  //根节点默认黑色
			return true;
		}

		Node* cur = _root;
		Node* parent = nullptr;
		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;  //新增节点给红色
		if (parent->_kv.first > kv.first)
		{
			parent->_left = cur;
		}
		else
		{
			parent->_right = cur;
		}
		cur->_parent = parent;

		// 检测新节点插入后,红黑树的性质是否造到破坏
		//父亲的颜色是黑色也结束
		while (parent && parent->_col == RED)
		{
			//关键看叔叔
			Node* grandfather = parent->_parent;
			if (parent == grandfather->_left)
			{
				Node* uncle = grandfather->_right;
				//如果叔叔存在也为红->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
				{
					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;
				}
			}
			else
			{
				Node* uncle = grandfather->_left; 
				//如果叔叔存在也为红->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
				{
					//    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;
				}
			}
		}
		//始终保持根为黑
		_root->_col = BLACK;

		return true;
	}

	void RotateR(Node* parent)
	{
		Node* subL = parent->_left;
		Node* subLR = subL->_right;

		parent->_left = subLR;
		if (subLR) //节点可能为空
			subLR->_parent = parent;

		subL->_right = parent; //旧父节点变成subL的右节点

		Node* ppNode = parent->_parent;  //该不平衡节点可能不是根节点,所以要找到它的父节点
		parent->_parent = subL;

		if (parent == _root)   //如果该节点是根节点
		{
			_root = subL;
			_root->_parent = nullptr;
		}
		else  //不平衡节点只是一棵子树
		{
			if (ppNode->_left == parent)  //如果旧父节点等于爷爷节点的左节点,新父节点为爷爷节点的左节点
			{
				ppNode->_left = subL;
			}
			else
			{
				ppNode->_right = subL;
			}
			subL->_parent = ppNode;	//新父节点指向爷爷节点。
		}
	}

	void RotateL(Node* parent)
	{
		Node* subR = parent->_right;
		Node* subRL = subR->_left;

		parent->_right = subRL;
		if (subRL)
			subRL->_parent = parent;

		subR->_left = parent;
		Node* ppNode = parent->_parent;

		parent->_parent = subR;

		if (parent == _root)
		{
			_root = subR;
			_root->_parent = nullptr;
		}
		else
		{
			if (ppNode->_right == parent)
			{
				ppNode->_right = subR;
			}
			else
			{
				ppNode->_left = subR;
			}
			subR->_parent = ppNode;
		}
	}

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

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

		int refNum = 0;    //取其中一条路径作为参考值
		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)
		{
			//cout << blackNum << endl;
			if (refNum != blackNum)
			{
				cout << "存在黑色节点数量不相等的路径" << 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);
	}


	void _InOrder(Node* root)
	{
		if (root == nullptr)
			return;

		_InOrder(root->_left);
		cout << root->_kv.first << ":" << root->_kv.second << endl;
		_InOrder(root->_right);
	}

private:
	Node* _root = nullptr;
	size_t _size = 0;
};


void RBTreeTest1()
{
	//int a[] = { 8, 3, 1, 10, 6, 4, 7, 14, 13 };
	int a[] = { 4, 2, 6, 1, 3, 5, 15, 7, 16, 14,8, 3, 1, 10, 6, 4, 7, 14, 13 };
	RBTree<int, int> t1;
	for (auto e : a)
	{
		t1.Insert({ e,e });

		//cout << "Insert:" << e << "->" << t1.IsBalance() << endl;
	}

	t1.InOrder();

	cout << t1.IsBalance() << endl; 
}
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