目录
一、概念
二、红黑树的性质
三、红黑树的定义
四、红黑树的插入操作
情况一(叔叔节点存在且为红色)——变色+向上调整:
情况二(叔叔节点不存在或为黑色)——旋转+变色:
2.1叔叔节点不存在
2.2叔叔节点为黑色
插入的代码实现:
五、红黑树的验证
六、红黑树完整代码
一、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。
二、红黑树的性质
- 每个结点不是红色就是黑色
- 根节点是黑色的
- 如果一个节点是红色的,则它的两个孩子结点必须是黑色的
- 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
- 每个叶子结点都是黑色的(此处的叶子结点指的是空结点)
三、红黑树的定义
enum Colour
{RED,BLACK
};template<class K, class V>
struct RBTreeNode
{RBTreeNode(const pair<K, V>& kv):_kv(kv),_right(nullptr),_left(nullptr),_parent(nullptr),_col(RED)//默认插入节点为红色,如果为黑色,就会对其他路径也造成影响{}pair<K, V> _kv;RBTreeNode<K, V>* _right;RBTreeNode<K, V>* _left;RBTreeNode<K, V>* _parent;Colour _col;
};
C++STL中的set和map底层就是使用红黑树实现的,而map是存放键值对的,所以我们给红黑树的节点中的值存放一个键值对,以及左右孩子的指针和指向父节点的指针,还有一个存放颜色的标记。
四、红黑树的插入操作
红黑树的插入首先和普通二叉搜索树的插入操作一样,新建一个节点,左节点的值小于根,右节点的值大于根,找到位置进行插入。插入后应如果破坏了红黑树的性质,就需要进行调整。
因为新节点的默认颜色是红色,因此:如果其双亲节点的颜色是黑色,没有违反红黑树任何
性质,则不需要调整;但当新插入节点的双亲节点颜色为红色时,就违反了性质三不能有连
在一起的红色节点,此时需要对红黑树分情况来讨论:
我们给出一个约定:cur为当前节点,p为父亲节点,g为祖父节点,u为叔叔节点
情况一(叔叔节点存在且为红色)——变色+向上调整:
将p和u改成黑色,将g改为红色
此时有三种情况:
1、g没有父亲节点,直接变成黑色就可以,插入结束;
2、g有父亲节点,且父亲为黑色,插入结束;
3、g有父亲节点,且父亲为红色(违反了红色节点不能连续的性质),需要向上调整。
情况二(叔叔节点不存在或为黑色)——旋转+变色:
2.1叔叔节点不存在
如果cur在parent的左边——右旋:
cur在parent的右边——先左旋再右旋:
2.2叔叔节点为黑色
如果cur在parent的左边——右旋:
cur在parent的右边——先左旋再右旋:
以上插入操作是p在g节点左边的情况,p在g节点右边的情况与以上插入过程类似,仅仅是镜像翻转一下。
插入的代码实现:
左旋代码:
void RotateL(Node* parent){Node* cur = parent->_right;Node* curleft = cur->_left;parent->_right = curleft;cur->_left = parent;if (curleft)curleft->_parent = parent;Node* ppnode = parent->_parent;parent->_parent = cur;if (parent == _root){cur->_parent = nullptr;_root = cur;}else{if (ppnode->_left == parent){ppnode->_left = cur;}else{ppnode->_right = cur;}cur->_parent = ppnode;}}
右旋代码:
void RotateR(Node* parent){Node* cur = parent->_left;Node* curright = cur->_right;parent->_left = curright;cur->_right = parent;if (curright)curright->_parent = parent;Node* ppnode = parent->_parent;parent->_parent = cur;if (parent == _root){cur->_parent = nullptr;_root = cur;}else{if (ppnode->_left == parent){ppnode->_left = cur;}else{ppnode->_right = cur;}cur->_parent = ppnode;}}
插入代码:
bool insert(const pair<K, V>& kv){//如果root为空if (_root == nullptr){_root = new Node(kv);_root->_col = BLACK;return true;}//插入Node* cur = _root;Node* parent = cur;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);//插入节点if (parent->_kv.first < kv.first){parent->_right = cur;}else{parent->_left = cur;}cur->_parent = parent;//插入完毕,开始调整颜色while (parent && parent->_col == RED){Node* grandfather = parent->_parent;//叔叔在右if (grandfather->_left == parent){Node* uncle = grandfather->_right;//叔叔存在且为红色——变色if (uncle && uncle->_col == RED){parent->_col = BLACK;uncle->_col = BLACK;grandfather->_col = RED;//向上更新cur = grandfather;parent = cur->_parent;}//叔叔不存在或者为黑色——旋转+变色else{//右单旋即可if (parent->_left == cur){RotateR(grandfather);//变色parent->_col = BLACK;grandfather->_col = RED;}//先左单旋,后右单旋else{RotateL(parent);RotateR(grandfather);//变色cur->_col = BLACK;grandfather->_col = RED;}break;}}//叔叔在左else{Node* uncle = grandfather->_left;//uncle存在且为红色——变色if (uncle && uncle->_col == RED){parent->_col = BLACK;uncle->_col = BLACK;grandfather->_col = RED;//向上更新cur = grandfather;parent = cur->_parent;}//uncle不存在或为黑色——旋转+变色else{//左单旋即可if (parent->_right == cur){RotateL(grandfather);//变色grandfather->_col = RED;parent->_col = BLACK;}//先右单旋,再左单旋else{RotateR(parent);RotateL(grandfather);//变色cur->_col = BLACK;grandfather->_col = RED;}break;}}}_root->_col = BLACK;return true;}
五、红黑树的验证
bool isBalance(){return _isBalance(_root);}bool checkcolour(Node* root, int benckmark, int blackcount){if (root == nullptr){if (blackcount != benckmark)return false;return true;}if (root->_col == RED && root->_parent && root->_parent->_col == RED)return false;if (root->_col == BLACK)++benckmark;return checkcolour(root->_left, benckmark, blackcount)&& checkcolour(root->_right, benckmark, blackcount);}bool _isBalance(Node* root){if (root == nullptr)return true;if (root->_col != BLACK)return false;Node* cur = root;//求树中最左路径黑色节点的个数while (cur){if (cur->_col == BLACK)++blackcount;cur = cur->_left;}return checkcolour(_root, 0, blackcount);}
六、红黑树完整代码
#pragma once#include <iostream>
#include <vector>
using namespace std;enum Colour
{RED,BLACK
};template<class K, class V>
struct RBTreeNode
{RBTreeNode(const pair<K, V>& kv):_kv(kv),_right(nullptr),_left(nullptr),_parent(nullptr),_col(RED)//默认插入节点为红色,如果为黑色,就会对其他路径也造成影响{}pair<K, V> _kv;RBTreeNode<K, V>* _right;RBTreeNode<K, V>* _left;RBTreeNode<K, V>* _parent;Colour _col;
};
/*
* 红黑树插入思路——关键在于uncle节点:
* 分为两大类:
* 一、如果uncle存在且为红色——仅仅变色即可
*
* g(黑) g(红)
* p(红) u(红) -------> p(黑) u(黑) ------->继续向上更新
* c(红) c(红)
*
*
* 二、如果uncle不存在或为黑色——旋转加变色
*
* 情况一: g(黑) p(红)
* p(红) NULL/黑 -------> c(红) g(黑)
* c(红)
*
* 仅仅右旋即可,g变成红色; p变成黑色; break;
*
* 情况二: g(黑) g(黑) c(红)
* p(红) NULL/黑 -------> 先左旋 c(红) -------> p(红) g(黑)
* c(红) p(红)
*
* c变成黑色,g变成红色,break;
*
* 情况三:情况一的对称图形
* 情况四:情况二的对称图形
*
*/
template<class K, class V>
class RBTree
{typedef RBTreeNode<K, V> Node;
public:RBTree():_root(nullptr){}void InOrder(){cout << "InOrder: ";_InOrder(_root);cout << endl;}bool insert(const pair<K, V>& kv){//如果root为空if (_root == nullptr){_root = new Node(kv);_root->_col = BLACK;return true;}//插入Node* cur = _root;Node* parent = cur;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);//插入节点if (parent->_kv.first < kv.first){parent->_right = cur;}else{parent->_left = cur;}cur->_parent = parent;//插入完毕,开始调整颜色while (parent && parent->_col == RED){Node* grandfather = parent->_parent;//叔叔在右if (grandfather->_left == parent){Node* uncle = grandfather->_right;//叔叔存在且为红色——变色if (uncle && uncle->_col == RED){parent->_col = BLACK;uncle->_col = BLACK;grandfather->_col = RED;//向上更新cur = grandfather;parent = cur->_parent;}//叔叔不存在或者为黑色——旋转+变色else{//右单旋即可if (parent->_left == cur){RotateR(grandfather);//变色parent->_col = BLACK;grandfather->_col = RED;}//先左单旋,后右单旋else{RotateL(parent);RotateR(grandfather);//变色cur->_col = BLACK;grandfather->_col = RED;}break;}}//叔叔在左else{Node* uncle = grandfather->_left;//uncle存在且为红色——变色if (uncle && uncle->_col == RED){parent->_col = BLACK;uncle->_col = BLACK;grandfather->_col = RED;//向上更新cur = grandfather;parent = cur->_parent;}//uncle不存在或为黑色——旋转+变色else{//左单旋即可if (parent->_right == cur){RotateL(grandfather);//变色grandfather->_col = RED;parent->_col = BLACK;}//先右单旋,再左单旋else{RotateR(parent);RotateL(grandfather);//变色cur->_col = BLACK;grandfather->_col = RED;}break;}}}_root->_col = BLACK;return true;}bool isBalance(){return _isBalance(_root);}private:bool checkcolour(Node* root, int benckmark, int blackcount){if (root == nullptr){if (blackcount != benckmark)return false;return true;}if (root->_col == RED && root->_parent && root->_parent->_col == RED)return false;if (root->_col == BLACK)++benckmark;return checkcolour(root->_left, benckmark, blackcount)&& checkcolour(root->_right, benckmark, blackcount);}bool _isBalance(Node* root){if (root == nullptr)return true;if (root->_col != BLACK)return false;Node* cur = root;while (cur){if (cur->_col == BLACK)++blackcount;cur = cur->_left;}return checkcolour(_root, 0, blackcount);}void RotateL(Node* parent){Node* cur = parent->_right;Node* curleft = cur->_left;parent->_right = curleft;cur->_left = parent;if (curleft)curleft->_parent = parent;Node* ppnode = parent->_parent;parent->_parent = cur;if (parent == _root){cur->_parent = nullptr;_root = cur;}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;if (curright)curright->_parent = parent;Node* ppnode = parent->_parent;parent->_parent = cur;if (parent == _root){cur->_parent = nullptr;_root = cur;}else{if (ppnode->_left == parent){ppnode->_left = cur;}else{ppnode->_right = cur;}cur->_parent = ppnode;}}void _InOrder(Node* root){if (root == nullptr)return;_InOrder(root->_left);cout << root->_kv.first << " ";_InOrder(root->_right);}
private:Node* _root;int blackcount = 0;
};
测试:
运行结果:
之后更新红黑树的应用,用红黑树封装map和set。