DS-红黑树(RBTree)

一.红黑树

1.1 红黑树的起源

当对对AVL树做一些结构修改的操作时候，性能较为低下，比如：插入时要维护其绝对平衡，旋转的次数比较多，更差的是在删除时，有可能一直要让旋转持续到根的位置。
因此1972年Rudolf Bayer提出的对称二叉B树（Symmetric Binary B-Trees），随后在1978年Leo J. Guibas和Robert Sedgewick的工作中进一步发展和完善，最终形成了现代意义上的红黑树，它通过简单的规则和较少的旋转操作实现了有效的自平衡，广泛应用于各类需要高效查找、插入和删除操作的场合，例如在Java集合框架中的TreeMap和TreeSet类，以及哈希表中解决冲突时采用的链表+红黑树混合结构。

1.2 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两倍，因而是接近平衡的。

红黑树具有以下性质：

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)

1.3 红黑树节点的定义

enum Color
{RED,BLACK
};template<typename T>
struct RBTreeNode
{RBTreeNode(const T& data): _left(nullptr), _right(nullptr), _parent(nullptr), _data(data), _col(RED){}RBTreeNode<T>* _left;RBTreeNode<T>* _right;RBTreeNode<T>* _parent; //父节点T _data;Color _col; //颜色
};

1.4 红黑树的插入

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:

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

bool insert(const T& data)
{if (_root == nullptr){_root = new Node(data);return true;}Node* grandparent = nullptr;Node* uncle = nullptr;Node* parent = nullptr;Node* cur = _root;while (cur){parent = cur;if (data < cur->_data){cur = cur->_left;}else if (data > cur->_data){cur = cur->_right;}else{return false;}}cur = new Node(data);if (data < parent->_data){parent->_left = cur;}else if (data > parent->_data){parent->_right = cur;}else{assert(false);}cur->_parent = parent;while (parent && parent->_col == RED){grandparent = parent->_parent;if (grandparent){if (parent == grandparent->_left){uncle = grandparent->_right;}else{uncle = grandparent->_left;}}else{break;}if (uncle && uncle->_col == RED){parent->_col = BLACK;uncle->_col = BLACK;grandparent->_col = RED;cur = grandparent;}else{cur = Rotate(grandparent, parent, cur);}parent = cur->_parent;_root->_col = BLACK;}return true;
}

大致思路如下:
以下简称插入节点为cur，cur的父节点为parent，parent的父节点为grandparent，parent的兄弟节点为uncle。

先先按照二叉搜索树的规则将节点插入到红黑树中，节点颜色为红色。
插入后，红黑树的性质可能遭到破坏，此时就要根据红黑树的性质进行检测。
cur插入后，如果parent颜色为黑色，则没有破坏红黑树的性质，插入结束。
若parent为红色，则此时有两种情况(uncle为红，uncle为黑/uncle不存在)
uncle为红时，将parent和uncle变为黑色，grandparent变为红色，然后把grandparent视为cur，继续向上调整。
uncle为黑/uncle不存在时，进行旋转。 (旋转在1.5处详细解释)

1.5 红黑树的旋转

此处的旋转与AVL树的旋转思路较为相似。

当uncle为黑/uncle不存在时，parent为cur的（左/右）孩子且parent为grandparent的（左/右）孩子，进行（右单旋）/（左单旋）。
右单旋

void RotateR(Node* parent)
{Node* ppnode = parent->_parent;Node* cur = parent->_left;Node* cur_right = cur->_right;if (ppnode){if (ppnode->_left == parent){ppnode->_left = cur;}else if (ppnode->_right == parent){ppnode->_right = cur;}else{assert(false);}}else{_root = cur;}cur->_parent = ppnode;parent->_left = cur_right;if (cur_right){cur_right->_parent = parent;}cur->_right = parent;parent->_parent = cur;
}

左单旋

void RotateL(Node* parent)
{Node* ppnode = parent->_parent;Node* cur = parent->_right;Node* cur_left = cur->_left;if (ppnode){if (ppnode->_right == parent){ppnode->_right = cur;}else if (ppnode->_left == parent){ppnode->_left = cur;}else{assert(false);}}else{_root = cur;}cur->_parent = ppnode;parent->_right = cur_left;if (cur_left){cur_left->_parent = parent;}cur->_left = parent;parent->_parent = cur;
}

当uncle为黑/uncle不存在时，parent为cur的（右/左）孩子且parent为grandparent的（左/右）孩子，进行（右单旋）/（左单旋）。

Node* Rotate(Node* grandparent, Node* parent, Node* cur)
{if (parent == grandparent->_left){if (cur == parent->_left){RotateR(grandparent);parent->_col = RED;grandparent->_col = BLACK;cur->_col = BLACK;return parent;}else{RotateL(parent);RotateR(grandparent);parent->_col = BLACK;grandparent->_col = BLACK;cur->_col = RED;return cur;}}else{if (cur == parent->_left){RotateR(parent);RotateL(grandparent);parent->_col = BLACK;grandparent->_col = BLACK;cur->_col = RED;return cur;}else{RotateL(grandparent);parent->_col = RED;grandparent->_col = BLACK;cur->_col = BLACK;return parent;}}

1.6 红黑树的特点与应用

红黑树是一棵不追求绝对平衡的二叉搜索树，其只需保证最长路径不超过最短路径的2倍，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。
红黑树在C++ STL库中 map/set 等结构中充当底层结构，据说在java中哈希表中的哈希桶下的链表长度超过一定的阈值时，也会转换为红黑树提高效率。使得在处理大量冲突键时，极大的缓解了链表过长导致的哈希表查找效率退化。

1.7 完整代码

#pragma once
#include <iostream>
#include <assert.h>
using namespace std;enum Color
{RED,BLACK
};template<typename T>
struct RBTreeNode
{RBTreeNode(const T& data): _left(nullptr), _right(nullptr), _parent(nullptr), _data(data), _col(RED){}RBTreeNode<T>* _left;RBTreeNode<T>* _right;RBTreeNode<T>* _parent;T _data;Color _col;
};template<typename T>
class RBTree
{
public:typedef RBTreeNode<T> Node;RBTree():_root(nullptr){}bool insert(const T& data){if (_root == nullptr){_root = new Node(data);return true;}Node* grandparent = nullptr;Node* uncle = nullptr;Node* parent = nullptr;Node* cur = _root;while (cur){parent = cur;if (data < cur->_data){cur = cur->_left;}else if (data > cur->_data){cur = cur->_right;}else{return false;}}cur = new Node(data);if (data < parent->_data){parent->_left = cur;}else if (data > parent->_data){parent->_right = cur;}else{assert(false);}cur->_parent = parent;while (parent && parent->_col == RED){grandparent = parent->_parent;if (grandparent){if (parent == grandparent->_left){uncle = grandparent->_right;}else{uncle = grandparent->_left;}}else{break;}if (uncle && uncle->_col == RED){parent->_col = BLACK;uncle->_col = BLACK;grandparent->_col = RED;cur = grandparent;}else{cur = Rotate(grandparent, parent, cur);}parent = cur->_parent;_root->_col = BLACK;}return true;}Node* get_Root(){return _root;}bool checkColour(Node* root, int blacknum, int benchmark){if (root == nullptr){if (blacknum != benchmark)return false;return true;}if (root->_col == BLACK){++blacknum;}if (root->_col == RED && root->_parent && root->_parent->_col == RED){cout << root->_data << "出现连续红色节点" << endl;return false;}return checkColour(root->_left, blacknum, benchmark)&& checkColour(root->_right, blacknum, benchmark);}bool isBalance(){return isBalance(_root);}int height(){return height(_root);}
private:Node* _root;void RotateR(Node* parent){Node* ppnode = parent->_parent;Node* cur = parent->_left;Node* cur_right = cur->_right;if (ppnode){if (ppnode->_left == parent){ppnode->_left = cur;}else if (ppnode->_right == parent){ppnode->_right = cur;}else{assert(false);}}else{_root = cur;}cur->_parent = ppnode;parent->_left = cur_right;if (cur_right){cur_right->_parent = parent;}cur->_right = parent;parent->_parent = cur;}void RotateL(Node* parent){Node* ppnode = parent->_parent;Node* cur = parent->_right;Node* cur_left = cur->_left;if (ppnode){if (ppnode->_right == parent){ppnode->_right = cur;}else if (ppnode->_left == parent){ppnode->_left = cur;}else{assert(false);}}else{_root = cur;}cur->_parent = ppnode;parent->_right = cur_left;if (cur_left){cur_left->_parent = parent;}cur->_left = parent;parent->_parent = cur;}Node* Rotate(Node* grandparent, Node* parent, Node* cur){if (parent == grandparent->_left){if (cur == parent->_left){RotateR(grandparent);parent->_col = RED;grandparent->_col = BLACK;cur->_col = BLACK;return parent;}else{RotateL(parent);RotateR(grandparent);parent->_col = BLACK;grandparent->_col = BLACK;cur->_col = RED;return cur;}}else{if (cur == parent->_left){RotateR(parent);RotateL(grandparent);parent->_col = BLACK;grandparent->_col = BLACK;cur->_col = RED;return cur;}else{RotateL(grandparent);parent->_col = RED;grandparent->_col = BLACK;cur->_col = BLACK;return parent;}}}bool isBalance(Node* root){if (root == nullptr)return true;if (root->_col != BLACK){return false;}// 基准值int benchmark = 0;Node* cur = _root;while (cur){if (cur->_col == BLACK)++benchmark;cur = cur->_left;}return checkColour(root, 0, benchmark);}int height(Node* root){if (root == nullptr)return 0;int leftHeight = height(root->_left);int rightHeight = height(root->_right);return leftHeight > rightHeight ? leftHeight + 1 : rightHeight + 1;}
};