RBTree(红黑树)

http://www.linuxidc.com/Linux/2017-01/139950.htm

http://blog.csdn.net/spch2008/article/details/9338923

红黑树

       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；(从每个叶子到根的所有路径上不能有两个连续的红色节点)
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。

这些约束强制了红黑树的关键性质: 从根到叶子的最长的可能路径不多于最短的可能路径的两倍长。结果是这个树大致上是平衡的。因为操作比如插入、删除和查找某个值的最坏情况时间都要求与树的高度成比例，这个在高度上的理论上限允许红黑树在最坏情况下都是高效的，而不同于普通的二叉查找树。

要知道为什么这些特性确保了这个结果，注意到性质4导致了路径不能有两个毗连的红色节点就足够了。最短的可能路径都是黑色节点，最长的可能路径有交替的红色和黑色节点。因为根据性质5所有最长的路径都有相同数目的黑色节点，这就表明了没有路径能多于任何其他路径的两倍长。

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);
    }

三、左单旋  右边过多

    //左单旋 右边过多  红黑树放弃了追求完全平衡，追求大致平衡
    void RotateL(Node* &root) 
    {
        Node* subR = root->_right;
        Node* subRL = subR->_left;

        root->_right = subRL;
        if (subRL)
        {
            subRL->_parent = root;
        }

        subR->_left = root;
        subR->_parent = root->_parent;
        root->_parent = subR;
        root = subR;

        if (root->_parent == NULL)
        {
            _root = root;
        }
        else if (root->_parent->_key  >  root->_key)
        {
            root->_parent->_left = root;
        }
        else if ( root->_parent->_key < root->_key )
        {
            root->_parent->_right = root;
        }
    }

四、右单旋 左边过多

 //右单旋 左边过多  红黑树放弃了追求完全平衡，追求大致平衡
    void RotateR(Node*& root)
    {
        Node* subL = root->_left;
        Node* subLR = subL->_right;

        root->_left = subLR;
        if (subLR)
        {
            subLR->_parent = root;
        }

        subL->_right = root;
        subL->_parent = root->_parent;
        root->_parent = subL;

        root = subL;

        if (root->_parent == NULL)
        {
            _root = parent;
        }
        else if (root->_parent->_key  >  root->_key)
        {
            root->_parent->_left = root;
        }
        else if (root->_parent->_key  <  root->_key)
        {
            root->_parent->_right = root;
        }
    }

五、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。



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)
                    {
						RotateR(parent);
                        uncle->_col = BLACK;
                        grandfather->_col = RED;
                        parent->_col = BLACK;

                        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。

#include<iostream>
using namespace std;

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)
    {

    }
    //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)
                    {
                        RotateR(parent);
                        uncle->_col = BLACK;
                        grandfather->_col = RED;
                        parent->_col = BLACK;

                        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;
    }

    //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);
    }

    //中序遍历
    void InOrderTraverse()
    {
        _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)
        {
            _InOrder(root->_left);
            cout << root->_key << " ";
            _InOrder(root->_right);
        }
    }

    //左单旋 右边过多  红黑树放弃了追求完全平衡，追求大致平衡
    void RotateL(Node* &root)
    {
        Node* subR = root->_right;
        Node* subRL = subR->_left;

        root->_right = subRL;
        if (subRL)
        {
            subRL->_parent = root;
        }

        subR->_left = root;
        subR->_parent = root->_parent;
        root->_parent = subR;
        root = subR;

        if (root->_parent == NULL)
        {
            _root = root;
        }
        else if (root->_parent->_key  >  root->_key)
        {
            root->_parent->_left = root;
        }
        else if ( root->_parent->_key < root->_key )
        {
            root->_parent->_right = root;
        }
    }

    //右单旋 左边过多  红黑树放弃了追求完全平衡，追求大致平衡
    void RotateR(Node*& root)
    {
        Node* subL = root->_left;
        Node* subLR = subL->_right;

        root->_left = subLR;
        if (subLR)
        {
            subLR->_parent = root;
        }

        subL->_right = root;
        subL->_parent = root->_parent;
        root->_parent = subL;

        root = subL;

        if (root->_parent == NULL)
        {
            _root = parent;
        }
        else if (root->_parent->_key  >  root->_key)
        {
            root->_parent->_left = root;
        }
        else if (root->_parent->_key  <  root->_key)
        {
            root->_parent->_right = root;
        }
    }
protected:
    Node* _root;
};

#include "Queue.h"
void TestRBtree()
{
    RBTree<int, int>RBT;
    int Arr[10]= { 1, 2, 5, 12, 16, 18, 26, 3, 99, 100 };

    for (int i = 0; i < 10; i++)
    {
        RBT.Insert(Arr[i], i);
    }
    RBT.InOrderTraverse();
    cout << endl;
    cout << "is RBTree ?: " << RBT.isRBTree() << endl;

}

int main()
{
    TestRBtree();
    return 0;
}