【数据结构】红黑树的插入(Insert)

前言：

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。

红黑树的基本概念：

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(不能有连续的两个红节点)

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NIL节点（空节点））

红黑树的插入写法要根据红黑树插入的各种情况来分析：

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。





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





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




上面已经把没种情况列出来了，其他相反的情况类似，反过来写一下就行了，具体细节过程见代码，

#ifndef __RBTREE_H__
#define __RBTREE_H__

enum colour
{
RED,
BLACK,
};

template<class K,class V>
struct RBTreeNode
{
int _col;
K _key;
V _value;
RBTreeNode<K, V>* _left;
RBTreeNode<K, V>* _right;
RBTreeNode<K, V>* _parent;

RBTreeNode(const K& key, const V& value)
:_key(key)
, _value(value)
, _col(RED)
, _left(NULL)
, _right(NULL)
, _parent(NULL)
{}

};

template<class K,class V>
class RBTree
{
typedef RBTreeNode<K, V> Node;
public:
RBTree()
:_root(NULL)
{}

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;
}

//插入位置以后，如何调整
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)
{
//情况1-> 不需要旋转
if (cur == parent->_left)
{
grandfather->_col = RED;
parent->_col = BLACK;
uncle->_col = BLACK;

cur = grandfather;
parent = cur->_parent;
}

//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2,情况3
else if (uncle == NULL || (uncle && uncle->_col == BLACK))
{
//情况3
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1
uncle = grandfather->_left;
if (uncle && uncle->_col == RED)
{
//不需要旋转
if (cur == parent->_right)
{
uncle->_col = BLACK;
grandfather->_col = RED;
parent->_col = BLACK;

cur = grandfather;
parent = cur->_parent;
}

//旋转
else if (cur == parent->_left)
{
uncle->_col = BLACK;
grandfather->_col = RED;
parent->_col = BLACK;
RotateR(parent);

cur = grandfather;
parent = cur->_parent;
}
}

else if (uncle == NULL || (uncle && uncle->_col == BLACK))
{
//情况2，3
if (cur == parent->_left)
{
RotateR(parent);
}
parent->_col = BLACK;
grandfather->_col = RED;
RotateL(grandfather);
break;
}
}
}
_root->_col = BLACK;
return true;
}

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);
}

void InOrder()
{
_InOrder(_root);
}

protected:
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);
}

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

_InOrder(root->_left);
cout << root->_key << " ";
_InOrder(root->_right);
}
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->_key > parent->_parent->_key)
{
parent->_parent->_right = parent;
}
}

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;

}

protected:
Node* _root;
};

void testRBtree()
{
RBTree<int, int> rbt;
int arr[8] = { 2, 5, 12, 16, 18, 26, 3, 1 };
for (int i = 0; i < 8; i++)
{
rbt.Insert(arr[i], i);
}

rbt.InOrder();
cout << endl;

cout << "isRBTree? ->:" << rbt.isRBTree() << endl;

}

#endif //__RBTREE_H__