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用c++实现红黑树的插入、删除、遍历操作

由 自闭症网瘾萝莉.ら 提交于 2019-12-10 10:00:44

红黑树是一棵二叉搜索树，它在每个节点上增加了一个存储位来表示节点的颜色，可以是Red或Black。通过对任何一条从根到叶子简单路径上的颜色来约束，红黑树保证最长路径不超过最短路径的两倍，因而近似于平衡。

红黑树是满足下面红黑性质的二叉搜索树：

  1. 每个节点，不是红色就是黑色的；
  2. 根节点是黑色的；
  3. 如果一个节点是红色的，则它的两个子节点是黑色的；
  4. 对每个节点，从该节点到其所有后代叶节点的简单路径上，均包含相同数目的黑色节点；
  5. 每个叶子节点都是黑色的（这里的叶子节点是指的空节点）

思考：为什么满足上面的颜色约束性质，红黑树能保证最长路径不超过最短路径的两倍？

如图：所能增加的红节点数最多和黑节点数目一样多，故红黑树能保证最长路径不超过最短路径的两倍。
在这里插入图片描述
一、判断是否是红黑树：


//判断是否是红黑树
	bool isRBTree()
	{
		int BlackNodeNum = 0;
		int curBlackNodeNum = 0;
		Node* cur = _root;
 
		while (cur)
		{
			if (cur->_col == BLACK)
			{
				BlackNodeNum++;
			}
 
			cur = cur->_left;
		}
		return _isRBTree(_root, BlackNodeNum, curBlackNodeNum);

}

bool _isRBTree(Node* root, int BlackNodeNum, int curBlackNodeNum)
	{
		if (root == NULL)
		{
			return true;
		}
 
		if (root->_col == BLACK)
		{
			curBlackNodeNum++;
		}
 
		if (BlackNodeNum == curBlackNodeNum)
		{
			if (root->_parent == NULL)
			{
				return true;
			}
			else if (root->_col == RED && root->_col == root->_parent->_col)
			{
				return false;
			}
			else
			{
				return true;
			}
		}
 
		return _isRBTree(root->_left, BlackNodeNum, curBlackNodeNum) && _isRBTree(root->_right, BlackNodeNum, curBlackNodeNum);
	}

二、红黑树的中序遍历：

//中序遍历
	void InOrder()
	{
		_InOrder(_root);
	}

void _InOrder(Node* root)
	{
		if (root == NULL)
		{
			return;
		}
 
		_InOrder(root->_left);
		cout << root->_key << " ";
		_InOrder(root->_right);
	}




void _InOrder(Node* root)
	{
		if (root == NULL)
		{
			return;
		}
 
		_InOrder(root->_left);
		cout << root->_key << " ";
		_InOrder(root->_right);
	}

三、红黑树的删除：

//删除 
	bool remove(const K& key){
		Node *cur = NULL;
		Node *parent = NULL;
		cur = _root;
		while(cur != NULL){
			if(key < cur->_key) cur = cur->_left;
			if(key > cur->_key) cur = cur->_right;
			if(key == cur->_key){
				break;
			} 
		}
		
		if(cur == NULL) return false;
		
		Node *instead = NULL;
		instead = cur;
		
		if(instead->_left != NULL){
		    instead = instead->_left;
			while(instead != NULL){  
			   parent = instead;
			   instead = instead->_right;   
		    }
		    cur->_key = parent->_key;
		    cur->_value = parent->_value;
		    parent->_parent->_right = NULL;
		    free(parent);
				
		}
		else if(instead->_right != NULL){
			instead = instead->_right;
			while(instead != NULL){
				parent = instead;
				instead = instead->_left;
			}
			cur->_key = parent->_key;
			cur->_value = parent->_value;
			parent->_parent->_left = NULL;
			free(parent);
	 
		}
		else if(instead->_left == NULL && instead->_right == NULL){
			Node *gradfather = NULL;
			gradfather = instead->_parent;
			if(gradfather->_key > instead->_key) gradfather->_left = NULL;
			else if(gradfather->_key < instead->_key) gradfather->_right = NULL;
			free(instead);
			
		}
		 
	}

四、左单旋
在这里插入图片描述

//左单旋
	void RotateL(Node*& parent)
	{
		Node* subR = parent->_right;
		Node* subRL = subR->_left;
 
		parent->_right = subRL;
		if (subRL)
		{
			subRL->_parent = parent;
		}
 
		subR->_left = parent;
		subR->_parent = parent->_parent;
		parent->_parent = subR;
		parent = subR;
 
		if (parent->_parent == NULL)
		{
			_root = parent;
		}
		else if (parent->_parent->_key > parent->_key)
		{
			parent->_parent->_left = parent;
		}
		else if ( parent->_parent->_key<parent->_key )
		{
			parent->_parent->_right = parent;
		}
	}

五、右单旋
在这里插入图片描述

//右单旋
	void RotateR(Node*& parent)
	{
		Node* subL = parent->_left;
		Node* subLR = subL->_right;
 
		parent->_left = subLR;
		if (subLR)
		{
			subLR->_parent = parent;
		}
 
		subL->_right = parent;
		subL->_parent = parent->_parent;
		parent->_parent = subL;
 
		parent = subL;
 
		if (parent->_parent == NULL)
		{
			_root = parent;
		}
		else if (parent->_parent->_key > parent->_key)
		{
			parent->_parent->_left = parent;
		}
		else if (parent->_parent->_key < parent->_key)
		{
			parent->_parent->_right = parent;
		}
 
	}

六、插入的三种情况
ps:cur为当前节点，p为父节点，g为祖父节点，u为叔叔节点
1.第一种情况
cur为红，p为红，g为黑，u存在且为红，则将p,u改为黑，g改为红，然后把g当成cur，继续向上调整。
在这里插入图片描述
2.第二种情况
cur为红，p为红，g为黑，u不存在/u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
在这里插入图片描述
3.第三种情况
cur为红，p为红，g为黑，u不存在/u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
在这里插入图片描述
上面已经把每种情况基本列出来了，其他相反的情况类似，反过来写一下就行了，具体详细过程参考代码。


//红黑树的插入操作
	bool Insert(const K& key, const V& value)
	{
		if (_root == NULL)
		{
			_root = new Node(key, value);
			_root->_col = BLACK;
			return true;
		}
 
		Node* parent = NULL;
		Node* cur = _root;
		while (cur)
		{
			if (cur->_key > key)
			{
				parent = cur;
				cur = cur->_left;
			}
			else if (cur->_key < key)
			{
				parent = cur;
				cur = cur->_right;
			}
			else
			{
				return false;
			}
		}
 
		//插入位置  
		if (parent->_key >key)
		{
			cur = new Node(key, value);
			parent->_left = cur;
			cur->_parent = parent;
		}
		else if (parent->_key < key)
		{
			cur = new Node(key, value);
			parent->_right = cur;
			cur->_parent = parent;
		}
 
		//插入以后，进行调整  
		while (cur != _root && parent->_col == RED)
		{
			Node* grandfather = parent->_parent;
			Node* uncle = NULL;
			//左边的情况  
			if (parent == grandfather->_left)
			{
				//情况一  
				uncle = grandfather->_right;
				if (uncle && uncle->_col == RED)
				{
					//1. 不需要旋转  
					if (cur == parent->_left)
					{
						grandfather->_col = RED;
						parent->_col = BLACK;
						uncle->_col = BLACK;
 
						cur = grandfather;
						parent = cur->_parent;
					}
 
					//2.需要旋转  
					else if (cur == parent->_right)
					{
						RotateL(parent);
						grandfather->_col = RED;
						parent->_col = BLACK;
						uncle->_col = BLACK;
 
						cur = grandfather;
						parent = cur->_parent;
					}
 
				}
 
				//情况二，三 
				else if (uncle == NULL || (uncle && uncle->_col == BLACK))
				{
					if (cur == parent->_right)
					{
						RotateL(parent);
					}
					parent->_col = BLACK;
					grandfather->_col = RED;
					RotateR(grandfather);
					break;
				}
			}
			//右边的情况  
			else if (parent == grandfather->_right)
			{
				//情况一
				uncle = grandfather->_left;
				if (uncle && uncle->_col == RED)
				{
					//1.不需要旋转  
					if (cur == parent->_right)
					{
						uncle->_col = BLACK;
						grandfather->_col = RED;
						parent->_col = BLACK;
 
						cur = grandfather;
						parent = cur->_parent;
					}
 
					//2.需要旋转  
					else if (cur == parent->_left)
					{
						uncle->_col = BLACK;
						grandfather->_col = RED;
						parent->_col = BLACK;
						RotateR(parent);
 
						cur = grandfather;
						parent = cur->_parent;
					}
				}
				//情况二，三
				else if (uncle == NULL || (uncle && uncle->_col == BLACK))
				{
					if (cur == parent->_left)
					{
						RotateR(parent);
					}
					parent->_col = BLACK;
					grandfather->_col = RED;
					RotateL(grandfather);
					break;
				}
			}
		}
		_root->_col = BLACK;
		return true;
	}

红黑树和AVL树的比较：

红黑树和AVL树都是高效的平衡二叉树，增删查改的时间复杂度都是O(lg(N))，红黑树的不追求完全平衡，保证最长路径不超过最短路径的2倍，相对而言，降低了旋转的要求，所以性能会优于AVL树，所以实际运用中红黑树更多。

完整代码及测试用例：

#ifndef __RBTree_h__
#define __RBTree_h__

#include<iostream>
using namespace std;
 
enum colour
{
	RED,
	BLACK,
};
 
template<class K, class V>
struct RBTreeNode
{
	int _col;
	K _key;
	V _value;
	long long _value_cot;
	long long _sum_cot;
	RBTreeNode<K, V>* _left;
	RBTreeNode<K, V>* _right;
	RBTreeNode<K, V>* _parent;
 
	RBTreeNode(const K& key, const V& value):
		_col(RED),
		_key(key),
	    _value(value),
		_value_cot(1),
		_sum_cot(0),
		_left(NULL),
		_right(NULL),
		_parent(NULL)
	{}
 
};
 
 
template<class K, class V>
class RBTree
{
	typedef RBTreeNode<K, V> Node;
public:
	RBTree()
		:_root(NULL)
	{}
 
	//红黑树的插入操作
	bool Insert(const K& key, const V& value)
	{
		if (_root == NULL)
		{
			_root = new Node(key, value);
			_root->_col = BLACK;
			return true;
		}
 
		Node* parent = NULL;
		Node* cur = _root;
		while (cur)
		{
			if (cur->_key > key)
			{
				parent = cur;
				cur = cur->_left;
			}
			else if (cur->_key < key)
			{
				parent = cur;
				cur = cur->_right;
			}
			else
			{
				cur->_value_cot++;
				return true;
			}
		}
 
		//插入位置  
		if (parent->_key >key)
		{
			cur = new Node(key, value);
			parent->_left = cur;
			cur->_parent = parent;
		}
		else if (parent->_key < key)
		{
			cur = new Node(key, value);
			parent->_right = cur;
			cur->_parent = parent;
		}
 
		//插入以后，进行调整  
		while (cur != _root && parent->_col == RED)
		{
			Node* grandfather = parent->_parent;
			Node* uncle = NULL;
			//左边的情况  
			if (parent == grandfather->_left)
			{
				//情况一  
				uncle = grandfather->_right;
				if (uncle && uncle->_col == RED)
				{
					//1. 不需要旋转  
					if (cur == parent->_left)
					{
						grandfather->_col = RED;
						parent->_col = BLACK;
						uncle->_col = BLACK;
 
						cur = grandfather;
						parent = cur->_parent;
					}
 
					//2.需要旋转  
					else if (cur == parent->_right)
					{
						RotateL(parent);
						grandfather->_col = RED;
						parent->_col = BLACK;
						uncle->_col = BLACK;
 
						cur = grandfather;
						parent = cur->_parent;
					}
 
				}
 
				//情况二，三 
				else if (uncle == NULL || (uncle && uncle->_col == BLACK))
				{
					if (cur == parent->_right)
					{
						RotateL(parent);
					}
					parent->_col = BLACK;
					grandfather->_col = RED;
					RotateR(grandfather);
					break;
				}
			}
			//右边的情况  
			else if (parent == grandfather->_right)
			{
				//情况一
				uncle = grandfather->_left;
				if (uncle && uncle->_col == RED)
				{
					//1.不需要旋转  
					if (cur == parent->_right)
					{
						uncle->_col = BLACK;
						grandfather->_col = RED;
						parent->_col = BLACK;
 
						cur = grandfather;
						parent = cur->_parent;
					}
 
					//2.需要旋转  
					else if (cur == parent->_left)
					{
						uncle->_col = BLACK;
						grandfather->_col = RED;
						parent->_col = BLACK;
						RotateR(parent);
 
						cur = grandfather;
						parent = cur->_parent;
					}
				}
				//情况二，三
				else if (uncle == NULL || (uncle && uncle->_col == BLACK))
				{
					if (cur == parent->_left)
					{
						RotateR(parent);
					}
					parent->_col = BLACK;
					grandfather->_col = RED;
					RotateL(grandfather);
					break;
				}
			}
		}
		_root->_col = BLACK;
		return true;
	}
	
	//删除 
	bool remove(const K& key){
		Node *cur = NULL;
		Node *parent = NULL;
		cur = _root;
		while(cur != NULL){
			if(key < cur->_key) cur = cur->_left;
			if(key > cur->_key) cur = cur->_right;
			if(key == cur->_key){
				break;
			} 
		}
		
		if(cur == NULL) return false;
		
		Node *instead = NULL;
		instead = cur;
		
		if(instead->_left != NULL){
		    instead = instead->_left;
			while(instead != NULL){  
			   parent = instead;
			   instead = instead->_right;   
		    }
		    cur->_key = parent->_key;
		    cur->_value = parent->_value;
		    parent->_parent->_right = NULL;
		    free(parent);
				
		}
		else if(instead->_right != NULL){
			instead = instead->_right;
			while(instead != NULL){
				parent = instead;
				instead = instead->_left;
			}
			cur->_key = parent->_key;
			cur->_value = parent->_value;
			parent->_parent->_left = NULL;
			free(parent);
	 
		}
		else if(instead->_left == NULL && instead->_right == NULL){
			Node *gradfather = NULL;
			gradfather = instead->_parent;
			if(gradfather->_key > instead->_key) gradfather->_left = NULL;
			else if(gradfather->_key < instead->_key) gradfather->_right = NULL;
			free(instead);
			
		}
		 
	}
    
    
    
    
    //查询第 k 大 
    K query(long long k){
    	_updata_value_cot(_root);
    	
    	if(k < _root->_sum_cot) return false;
    	Node *cur = NULL;
    	cur = _root;
    	long long cur_left_sum = 0;
    	while(cur != NULL){
    		if(cur->_left == NULL)  cur_left_sum = 0;
    		else cur_left_sum = cur->_left->_sum_cot;
			
			if(k == cur_left_sum + 1) return cur->_key;
    		else if(k < cur_left_sum + 1){
    			cur = cur->left;
			}
			else if(k > cur_left_sum + 1){
				cur = cur->_right;
				k -= cur_left_sum + 1;
			}
		}
	}
	
	
	
	//中序遍历
	void InOrder()
	{
		_InOrder(_root);
	}
	
	
	
	
	
	
 
protected:
	
	void _InOrder(Node* root)
	{
		if (root == NULL)
		{
			return;
		}
 
		_InOrder(root->_left);
		for(int i = 0; i < root->_value_cot; i++)
		cout << root->_key << " ";
		_InOrder(root->_right);
	}
 
	//左单旋
	void RotateL(Node*& parent)
	{
		Node* subR = parent->_right;
		Node* subRL = subR->_left;
 
		parent->_right = subRL;
		if (subRL)
		{
			subRL->_parent = parent;
		}
 
		subR->_left = parent;
		subR->_parent = parent->_parent;
		parent->_parent = subR;
		parent = subR;
 
		if (parent->_parent == NULL)
		{
			_root = parent;
		}
		else if (parent->_parent->_key > parent->_key)
		{
			parent->_parent->_left = parent;
		}
		else if ( parent->_parent->_key<parent->_key )
		{
			parent->_parent->_right = parent;
		}
	}
 
	//右单旋
	void RotateR(Node*& parent)
	{
		Node* subL = parent->_left;
		Node* subLR = subL->_right;
 
		parent->_left = subLR;
		if (subLR)
		{
			subLR->_parent = parent;
		}
 
		subL->_right = parent;
		subL->_parent = parent->_parent;
		parent->_parent = subL;
 
		parent = subL;
 
		if (parent->_parent == NULL)
		{
			_root = parent;
		}
		else if (parent->_parent->_key > parent->_key)
		{
			parent->_parent->_left = parent;
		}
		else if (parent->_parent->_key < parent->_key)
		{
			parent->_parent->_right = parent;
		}
 
	}
	
	
    long long updata_value_cot(Node *root){
    	
    	if(root == NULL) return 0;
    	root->_sum_cot = 1;
    	root->_sum_cot += updata_value_cot(root->_left) + updata_value_cot(root->_right);
    	  	
	}
	
	
    
    	
	
protected:
	Node* _root;       
};

#endif   /*__RBTree_h__*/ 

#include<bits/stdc++.h>

#include "RBTree.h"


void TestRBtree()
{
	RBTree<int, int>RBT;
	int n;
	
    cin>>n;
	for (int i = 0; i < n; i++)
	{
		RBT.Insert(rand(), i);
	}
	
	RBT.InOrder();
	cout << endl;
	
	cin>>n;
	RBT.remove(n);
	
	
	RBT.InOrder();
	cout << endl;
 
	
}
 
int main()
{
	TestRBtree();
	system("pause");
	return 0;
}
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