文章目录
- 红黑树
- 概念
- 性质(条件限制)
- 节点的定义
- 红黑树的结构
- 红黑树的插入
- cur为红,p为红,g为黑,u存在且为红
- cur为红,p为红,g为黑,u不存在或u为黑,插入到p对应的一边
- cur为红,p为红,g为黑,u不存在或u存在且为黑,插入到与p相反的一边
- 示例代码
- 红黑树的验证
- 红黑树与AVL树的比较
- 完整代码
红黑树
概念
和AVL树一样,红黑树也是一种二叉搜索树,是解决二叉搜索树不平衡的另一种方案,他在每个节点上增加一个存储位,用于表示节点的颜色,是Red或者Black
红黑树的核心思想是通过一些着色的条件限制,达到一种最长路径不超过最短路径的两倍的状态
所以说红黑树并不是严格平衡的树,而是一种近似平衡
例如
性质(条件限制)
红黑树一共有五条性质,由此来保证最长路径不超过最短路径的两倍
- 每个节点都有颜色,不是黑色就是红色
- 根节点是黑色的
- 如果一共节点是红色,那么他的子节点一定是黑色(不会出现两个红色节点连接的情况)
- 对于每个节点,以这个节点到所有后代的任意路径上,均包含相同数目的黑色节点
- 每个叶子节点(空节点)是黑色的(为了满足第四条性质,某些情况下如果没有第五条第四条会失效)
节点的定义
// 颜色
enum Color {RED,BLACK
};template<class T>
struct RBTreeNode {RBTreeNode<T>* _left;RBTreeNode<T>* _right;RBTreeNode<T>* _parent;T _data;Color _col;RBTreeNode(const T& data) :_left(nullptr),_right(nullptr),_parent(nullptr),_data(data),_col(RED){}
};
我们定义颜色时,使用枚举类型,可以方便且明了的看到颜色
除此之外我们默认插入节点是红色的,因为一旦插入节点是黑色,就会违反第四条规则,如果要满足的话,就要走到每一条路径上插入对应的黑色节点,代价巨大
当插入节点是红色时,有可能会违反第三条规则,但是我们可以通过变色,旋转等操作在局部进行改变,这样就能使之仍然满足条件
红黑树的结构
为了后续利用红黑树封装map和set,我们对红黑树增加一个头节点,为了和根节点进行区分,我们将头节点赋为黑色,并且让头节点的parent指向根节点,left指向红黑树的最小节点,right指向最大节点,如图
红黑树的插入
红黑树插入时也是按照二叉搜索树的规则进行插入,并在此基础上加上平衡条件,因此插入也就分为两步
- 按照二叉搜索树的规则插入新节点
- 插入节点后检测规则是否被破坏
因为插入红节点时只有可能破坏第三条规则,因此我们只需要判断父节点是否为红色即可
然后我们分情况讨论
为了方便叙述,我们约定cur为插入节点,p为父节点,g为祖父节点,u为叔叔节点
cur为红,p为红,g为黑,u存在且为红
画出来是这样的
这时我们需要将g改为红色,p和u改为黑色即可,这样既能保证红色不连续,黑色数量一致,如图
但是如果g是是子树,那么g一定有父节点,当g的父节点也是红色时,也就同样需要向上调整了
cur为红,p为红,g为黑,u不存在或u为黑,插入到p对应的一边
画出来是这样的
u的情况有两种
- u节点不存在,说明cur一定是新插入的节点,因为要保证左右两个路径的黑色节点的数量相同
- u节点存在,说明cur节点是由下至上调整的红色,原因也是左右路径的黑色节点要相同
对于这两种情况的调整方法是相同的,如果p是g的左节点,cur为p的左节点,则右单旋,如果p是g的右节点,cur为p的右节点,则左单旋
同时p要变成黑色,g要变成红色
变成如下状态
那么因为最上面的根节点颜色没有变化,也就不需要继续向上调整了
cur为红,p为红,g为黑,u不存在或u存在且为黑,插入到与p相反的一边
如图
这种情况需要针对p进行单旋,如果p为g的左节点,cur为p的右节点,则对p左单旋,反之则为右单旋,此时就会变成第二种情况,再继续处理即可
第一次处理的结果如下
示例代码
template<class K, class T, class KeyOfT>
class RBTree {typedef RBTreeNode<T> Node;
public:pair<Node*, bool> Insert(const T& data) {// 插入根节点直接返回if (_root == nullptr) {_root = new Node(data);_root->_col = BLACK;return make_pair(_root, true);}Node* parent = nullptr;Node* cur = _root;KeyOfT kot;// 平衡二叉树找到插入位置while (cur) {if (kot(cur->_data) < kot(data)) {parent = cur;cur = cur->_right;} else if (kot(cur->_data) > kot(data)) {parent = cur;cur = cur->_left;} else {return make_pair(cur, false);}}// 新建节点cur = new Node(data);Node* newnode = cur;cur->_col = RED;// 连接父节点if (kot(parent->_data) < kot(data)) {parent->_right = cur;cur->_parent = parent;} else {parent->_left = cur;cur->_parent = parent;}// 如果父节点存在且父节点为红色则需要调整while (parent && parent->_col == RED) {Node* grandfather = parent->_parent;if (parent == grandfather->_left) {// g// p u// c // 判断u是否存在和他的颜色Node* uncle = grandfather->_right;// 如果存在且为红色if (uncle && uncle->_col == RED) {// 变色parent->_col = BLACK;uncle->_col = BLACK;grandfather->_col = RED;// 向上调整cur = grandfather;parent = cur->_parent;} else {// 如果不存在或u为黑色,需要判断同侧还是异侧// 如果是同侧if (cur == parent->_left) {// g// p// cRotateR(grandfather); // 右旋// 调整颜色parent->_col = BLACK;grandfather->_col = RED;} else {// g// p// cRotateL(parent);RotateR(grandfather);cur->_col = BLACK;grandfather->_col = RED;}break;}} else { // p = g->rNode* uncle = grandfather->_left;// g// u p// c// 判断u是否存在和他的颜色// 如果存在且为红色if (uncle && uncle->_col == RED) {// 变色parent->_col = BLACK;uncle->_col = BLACK;grandfather->_col = RED;// 向上调整cur = grandfather;parent = cur->_parent;} else {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;}}}_root->_col = BLACK;return make_pair(newnode, true);}void RotateL(Node* parent) {Node* subR = parent->_right;Node* subRL = subR->_left;parent->_right = subRL;subR->_left = parent;Node* parentParent = parent->_parent;parent->_parent = subR;if (subRL)subRL->_parent = parent;else {if (parentParent->_left == parent) {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 (_root == parent) {_root = subL;subL->_parent = nullptr;} else {if (parentParent->_left == parent) {parentParent->_left = subL;} else {parentParent->_right = subL;}subL->_parent = parentParent;}}
private:Node* _root = nullptr;
};
红黑树的验证
红黑树要验证需要验证两个部分
- 检测是否中序遍历是有序序列
- 检测是否满足红黑树的性质
这里我们就不讲红黑树的删除了,完成红黑树的验证之后就算作已经完成了任务,接下来会使用红黑树模拟实现map和set
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,但是红黑树不追求绝对的平衡,降低了插入和旋转的次数,因此性能比AVL更优,而且红黑树比AVL树的实现更加简单,所以实际中运用红黑树更多
完整代码
#pragma once
#include<utility>
#include<iostream>
using namespace std;
// 颜色
enum Color {RED,BLACK
};template<class T>
struct RBTreeNode {RBTreeNode<T>* _left;RBTreeNode<T>* _right;RBTreeNode<T>* _parent;T _data;Color _col;RBTreeNode(const T& data):_left(nullptr), _right(nullptr), _parent(nullptr), _data(data), _col(RED) {}
};template<class K, class T, class KeyOfT>
class RBTree {typedef RBTreeNode<T> Node;
public:pair<Node*, bool> Insert(const T& data) {if (_root == nullptr) {_root = new Node(data);_root->_col = BLACK;return make_pair(_root, true);}Node* parent = nullptr;Node* cur = _root;KeyOfT kot;// 平衡二叉树找到插入位置while (cur) {if (kot(cur->_data) < kot(data)) {parent = cur;cur = cur->_right;} else if (kot(cur->_data) > kot(data)) {parent = cur;cur = cur->_left;} else {return make_pair(cur, false);}}// 新建节点cur = new Node(data);Node* newnode = cur;cur->_col = RED;// 连接父节点if (kot(parent->_data) < kot(data)) {parent->_right = cur;cur->_parent = parent;} else {parent->_left = cur;cur->_parent = parent;}// 如果父节点存在且父节点为红色则需要调整while (parent && parent->_col == RED) {Node* grandfather = parent->_parent;if (parent == grandfather->_left) {// g// p u// c // 判断u是否存在和他的颜色Node* uncle = grandfather->_right;// 如果存在且为红色if (uncle && uncle->_col == RED) {// 变色parent->_col = BLACK;uncle->_col = BLACK;grandfather->_col = RED;// 向上调整cur = grandfather;parent = cur->_parent;} else {// 如果不存在或u为黑色,需要判断同侧还是异侧// 如果是同侧if (cur == parent->_left) {// g// p// cRotateR(grandfather); // 右旋// 调整颜色parent->_col = BLACK;grandfather->_col = RED;} else {// g// p// cRotateL(parent);RotateR(grandfather);cur->_col = BLACK;grandfather->_col = RED;}break;}} else { // p = g->rNode* uncle = grandfather->_left;// g// u p// c// 判断u是否存在和他的颜色// 如果存在且为红色if (uncle && uncle->_col == RED) {// 变色parent->_col = BLACK;uncle->_col = BLACK;grandfather->_col = RED;// 向上调整cur = grandfather;parent = cur->_parent;} else {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;}}}_root->_col = BLACK;return make_pair(newnode, true);}void RotateL(Node* parent) {Node* subR = parent->_right;Node* subRL = subR->_left;parent->_right = subRL;subR->_left = parent;Node* parentParent = parent->_parent;parent->_parent = subR;if (subRL)subRL->_parent = parent;else {if (parentParent->_left == parent) {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 (_root == parent) {_root = subL;subL->_parent = nullptr;} else {if (parentParent->_left == parent) {parentParent->_left = subL;} else {parentParent->_right = subL;}subL->_parent = parentParent;}}void InOrder() {_InOrder(_root);cout << endl;}void _InOrder(Node* root) {if (root == nullptr)return;_InOrder(root->_left);cout << root->_data << ' ';_InOrder(root->_right);}bool Check(Node* root, int blacknum, const int refVal) {if (root == nullptr) {if (blacknum != refVal) {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 << "存在连续的红节点" << endl;return false;}if (root->_col == BLACK) {++blacknum;}return Check(root->_left, blacknum, refVal) && Check(root->_right, blacknum, refVal);}bool IsBalance() {if (_root == nullptr)return true;if (_root->_col == RED)return false;int refVal = 0; // 参考值Node* cur = _root;while (cur) {if (cur->_col == BLACK) {++refVal;}cur = cur->_left;}int blacknum = 0;return Check(_root, blacknum, refVal);}int Height() {return _Height(_root);}int _Height(Node* root) {if (root == nullptr)return 0;int leftH = _Height(root->_left);int rightH = _Height(root->_right);return leftH + rightH;}size_t Size() {return _Size(_root);}size_t _Size(Node* root) {if (root == nullptr)return 0;return _Size(root->_left) + _Size(root->_right) + 1;}Node* Find(const K& key) {Node* cur = _root;while (cur) {if (cur->_data < key) {cur = cur->_right;}else if (cur->_data > key) {cur = cur->_left;}else {return cur;}}return nullptr;}private:Node* _root = nullptr;
};