红黑树C++实现

看算法导论看的有点晕，代码的缩进让我直接抓狂了，网上找了一个比较靠谱的解法，链接

 

现在具体分析一下插入操作：
1. 首先父亲节点如果是黑色节点的话,不需要处理了,多一个红色节点不会有任何影响.
2. 如果父亲节点是红色的,而叔父节点也是红色,那就是把父亲和叔父节点同时改成黑色,祖父改为红色，然后递归的向上传递.
3. 如果父亲节点是红色的,而叔父节点是黑色的. 后面的话比较拗口.父亲是祖父的左孩子.
   (1)如果自己是父亲的左孩子,那么修改父亲为黑色，祖父为红色，然后右转祖父
   (2)如果自己是父亲的右孩子,那么先把自己指向父亲,再做左转自己，执行(1).

4. 如果父亲是祖父的右孩子

    (1)如果自己是父亲的右孩子,那么修改父亲为黑色，祖父为红色，然后左转祖父
    (2)如果自己是父亲的左孩子,那么先把自己指向父亲,再做右转自己，执行(1).

（以上转自上述网页，并根据网页中的代码对描述作了适当的修改）。

对应的代码如下：

template <class T>void RBTree<T>::RBInsert(RBTreeNode<T>*node){RBTreeNode<T> *y = RBTreeNode<T>::NIL,*x = root;while (x!=RBTreeNode<T>::NIL){y = x;if(y->value == node->value)return;if(y->value >node->value)x = x->left;else x = x->right;}node->father = y;if(y==RBTreeNode<T>::NIL)//node 为rootroot = node;else if(node->value < y->value)//left childy->left = node;else y->right = node;node->left = node->right = RBTreeNode<T>::NIL;node->color = RED;RBInsertFixUp(node);}template<class T>void RBTree<T>::RBInsertFixUp(RBTreeNode<T>*node){while (node!= root && node->father->color == RED){RBTreeNode<T>* pUncle = GetUncle(node),*pGrandFather = GetGrandFather(node);if (pUncle!= RBTreeNode<T>::NIL && pUncle->color == RED){node->father->color = BLACK;pUncle->color = BLACK;pGrandFather->color = RED;node = pGrandFather;}else{//if uncle = blackif (node->father == pGrandFather->left){//father is left childif(node == node->father->right){//right childnode = node->father;LeftRotate(node);}node->father->color = BLACK;pGrandFather->color = RED;RightRotate(pGrandFather);}else{//father is right childif(node == node->father->left){//left childnode = node->father;RightRotate(node);}node->father->color = BLACK;pGrandFather->color = RED;LeftRotate(pGrandFather);}}}root->color = BLACK;}

 

 

现在来看看红黑树的删除操作：

在这里先暂且复习一下树中节点x的后继操作：

1. 如果x含有右子，则返回右子中的最小值（不停地遍历右子的左枝）

2. 没有右子，令x的父节点为y，如果x是y的左子，则直接返回y

3. x是y的右子，则迭代{x = y；y=y->father;}直到y为空或者x是y的右子为止。

代码如下：

template<class T>const TreeNode<T>* Tree<T>::TreeSuccessor(const TreeNode<T> * const node){if(!node)return NULL;if (node->right){ TreeNode<T>* minNode = node; while(minNode ->left) minNode = minNode->left; return minNode; }TreeNode<T>* temp = node->father;const TreeNode<T>*tempf = node;while(temp && tempf == temp->right){tempf = temp;temp = temp->father;}return temp;}

 

另外简单证明一下二叉树节点node的后继无左子的定理：

反证法：

假设node的后继successor有左子left：

1. 如果node.value<left.value

    则node的后继为left而非successor

2. 如果node.value>left.value

    则left不可能为successor的左子树（因为如果left的value比node要小，则left必在node的左子树中，而node的后继一定大于node，一定不在node的左子树中）

矛盾。因此定理成立。

 

 

现在分析一下红黑树的删除操作，以下为了表示简便，F代表父节点，S代表兄弟节点，x表示当前节点：

1. 如果x不为根并且x为黑色，执行2，否则执行6。

2. 如果x为右子，执行5，否则：

     如果S为黑，执行3，否则：

        置S为黑，F为红，左旋F，并更新S为F的右子

3. 如果S的两子均为黑，置S为红，x更新为F，执行1；
    否则执行4

4. 这里分两种情况：

    4.1 如果S的右子为黑，S的左子也置黑，S置红，S更新为F的右子，否则直接执行4.2

    4.2 S的颜色置为F的颜色，F置黑，S右子置黑，左旋F，x更新为root，执行6

5. 和2-4对称，只需要把left换为right即可

6. 置x为黑色（x为根或者x为红色）

 

这个说起来还是很绕口，其实就是删除一个红节点不影响树的红黑性质，但是删除树的黑节点则会影响以下三点：

1 具有等同黑高的性质

2 如果删除了x，而x的孩子和父亲都为红色，则违反了红节点只能有黑孩子的性质

3 如果x只有一个红孩子，则x为根，删除了x会破坏根为黑色的性质。

根据以上的讨论，需要检查被删除节点的孩子的情况。

 

所有的代码如下：

#ifndef HIT_RB_TREE#define HIT_RB_TREEusing std::cout;using std::endl;namespace HITRB_TREE{enum Color{BLACK = 0,RED};//enum void{SUCCEED = 0,NOTFOUND,NULLNODE,DUPLICATENODE};template<class T>struct RBTreeNode{T value;RBTreeNode<T> *father;RBTreeNode<T> *left;RBTreeNode<T> *right;Color color;RBTreeNode():value(static_cast<T>(0)),father(NULL),left(NULL),right(NULL),color(BLACK){}RBTreeNode(T va,RBTreeNode<T> *f = NULL,RBTreeNode<T> *l = NULL,RBTreeNode<T> *r=NULL,Color col=BLACK):value(va),father(f),left(l),right(r),color(col){}};template<class T>class RBTree{public:RBTree(){pGuard = new RBTreeNode<T>;pGuard->father = pGuard->left = pGuard->right = pGuard;root = pGuard;}~RBTree(){DistroyTree(root);delete pGuard;pGuard = NULL;}void RBInsert(RBTreeNode<T>*node);void RBInsert(const T &va);void Remove(const T& va);void Remove(RBTreeNode<T>*pNode);RBTreeNode<T>* RBSearch(const T&);void DistroyTree(RBTreeNode<T> *pNode);const RBTreeNode<T>* Successor(const RBTreeNode<T>*node);const RBTreeNode<T>* MaxNode(const RBTreeNode<T>*node);const RBTreeNode<T>* const GetGuardNode()const{return pGuard;}protected:private:void LeftRotate(RBTreeNode<T>*node);void RightRotate(RBTreeNode<T>*node);void RBInsertFixUp(RBTreeNode<T>*node);void RBRemoveFixUp(RBTreeNode<T>*pNode);RBTreeNode<T>* NewNode(const T &value){RBTreeNode<T> *pNode = new RBTreeNode<T>(value,pGuard,pGuard,pGuard);return pNode;}RBTreeNode<T>* GetGrandFather(const RBTreeNode<T>*node)const{if(node!=pGuard && node->father!=pGuard)return node->father->father;else return pGuard;}RBTreeNode<T>* GetUncle(const RBTreeNode<T>*node)const{RBTreeNode<T>* pTemp = GetGrandFather(node);if(pTemp == pGuard)return pGuard;if(node->father == pTemp->left)return pTemp->right;else return pTemp->left;}RBTreeNode<T>* GetSibling(const RBTreeNode<T>*node)const{if(node == node->father->left)return node->father->right;else return node->father->left;}RBTreeNode<T> *root;RBTreeNode<T> *pGuard;};//function definitiontemplate<class T>void RBTree<T>::DistroyTree(RBTreeNode<T> *pNode){if(pNode != pGuard){DistroyTree(pNode->left);DistroyTree(pNode->right);delete pNode;pNode = NULL;}}template<class T>void RBTree<T>::LeftRotate(RBTreeNode<T>*node){if(node->right == pGuard){cout<<"[WARNING +1000] node["<<node->value<<"] does not have any right node, cant be"<<" left rotate!"<<endl;return;}//y为node的右子RBTreeNode<T> *y = node->right;//把node的右子更新为其右子的左子node->right = y->left;//对更新的数据做处理，如果右子的左子不为空，则把其父节点修正为nodeif(y->left != pGuard)y->left->father = node;//把y的父节点更新为node的父节点y->father = node->father;//node为root的情况if(node->father == pGuard)root = y;//把y链接到node的父节点的正确位置上（左子还是右子）else if(node == node->father->left)node->father->left = y;else node->father->right = y;//更新y和node的链接关系y->left = node;node->father = y;}template<class T>void RBTree<T>::RightRotate(RBTreeNode<T>*node){if(node->left == pGuard){cout<<"[WARNING +1000] node["<<node->value<<"] does not have any left node, cant be"<<" right rotate!"<<endl;return;}//y为node的左子RBTreeNode<T> *y = node->left;//把node的左子更新为其左子的右子node->left = y->right;//对更新的数据做处理，如果左子的右子不为空，则把其父节点修正为nodeif(y->right != pGuard)y->right->father = node;//把y的父节点更新为node的父节点y->father = node->father;//node为root的情况if(node->father == pGuard)root = y;//把y链接到node的父节点的正确位置上（左子还是右子）else if(node == node->father->left)node->father->left = y;else node->father->right = y;//更新y和node的链接关系y->right = node;node->father = y;}template <class T>void RBTree<T>::RBInsert(RBTreeNode<T>*node){RBTreeNode<T> *y = pGuard,*x = root;while (x!=pGuard){y = x;if(y->value == node->value)return ;//DUPLICATENODE;if(y->value >node->value)x = x->left;else x = x->right;}node->father = y;if(y==pGuard)//node 为rootroot = node;else if(node->value < y->value)//left childy->left = node;else y->right = node;node->left = node->right = pGuard;node->color = RED;RBInsertFixUp(node);return;// SUCCEED;}template<class T>void RBTree<T>::RBInsertFixUp(RBTreeNode<T>*node){while (node->father->color == RED){RBTreeNode<T>* pUncle = GetUncle(node),*pGrandFather = GetGrandFather(node);if (pUncle->color == RED){node->father->color = BLACK;pUncle->color = BLACK;pGrandFather->color = RED;node = pGrandFather;}else{//if uncle = blackif (node->father == pGrandFather->left){//father is left childif(node == node->father->right){//right childnode = node->father;LeftRotate(node);}node->father->color = BLACK;pGrandFather->color = RED;RightRotate(pGrandFather);}else{//father is right childif(node == node->father->left){//left childnode = node->father;RightRotate(node);}node->father->color = BLACK;pGrandFather->color = RED;LeftRotate(pGrandFather);}}}root->color = BLACK;}template <class T>void RBTree<T>::RBInsert(const T&va){RBInsert(NewNode(va));return;// SUCCEED;}template<class T>const RBTreeNode<T>* RBTree<T>::Successor(const RBTreeNode<T>*node){//调用之间应该检查node是否为MaxNodeif(node == pGuard)return pGuard;const RBTreeNode<T>*pTemp = node;if (pTemp->right!=pGuard){//find the minim element in right sub-treewhile (pTemp->left!=pGuard){pTemp = pTemp->left;}return pTemp;}RBTreeNode<T>*pTempFather = node->father;while (pTempFather != pGuard && node == pTempFather->right){pTemp = pTempFather;pTempFather = pTempFather->father;}return pTempFather;}template<class T>const RBTreeNode<T>* RBTree<T>::MaxNode(const RBTreeNode<T>*node){if(node == NULL)node = root;const RBTreeNode<T>*pTemp = node;while (pTemp->right!=pGuard)pTemp = pTemp->right;return pTemp;}template<class T>void RBTree<T>::Remove(RBTreeNode<T>*pNode){//null node if(pNode == NULL || pNode == pGuard)return;// NULLNODE;//temp point definitionRBTreeNode<T> *pTemp=pGuard,*pTempNode = pGuard;if (pNode->left == pGuard || pNode->right == pGuard)pTempNode = pNode;else pTempNode = const_cast<RBTreeNode<T>*>( Successor(pNode) );if(pTempNode->left != pGuard)pTemp = pTempNode->left;//maybe NILelse pTemp = pTempNode->right;//maybe successor's right node!pTemp->father = pTempNode->father;//different form BST, need not checkif(pTempNode->father == pGuard)root = pTemp;//set the root to NILelse if(pTempNode == pTempNode->father->left)pTempNode->father->left = pTemp;else pTempNode->father->right = pTemp;if(pNode != pTempNode){//has two children, and the pTempNode is pointing to successor.//now need to copy the successor's data to nodepNode->color = pTempNode->color;pNode->father = pTempNode->father;pNode->left = pTempNode->left;pNode->right = pTempNode->right;pNode->value = pTempNode->value;}if(pTempNode->color == BLACK)//change the black height, so need to fix up!RBRemoveFixUp(pTemp);delete pTempNode;pTempNode = NULL;return;// SUCCEED;}template<class T>void RBTree<T>::RBRemoveFixUp(RBTreeNode<T>*pNode){while (pNode != root && pNode->color == BLACK){RBTreeNode<T> *pFather = pNode->father;RBTreeNode<T> *pSibling = pFather->right;if (pNode == pNode->father->left){if (pSibling->color == RED){pSibling->color = BLACK;pFather->color = RED;LeftRotate(pFather);pSibling = pFather->right;}if (pSibling->left->color == BLACK &&pSibling->right->color == BLACK){pSibling->color = RED;pNode = pFather;}else{if (pSibling->right->color == BLACK)//sibling's left is red{pSibling->left->color = BLACK;pSibling->color = RED;RightRotate(pSibling);pSibling = pFather->right;}//right is red, left is unsurepSibling->color = pFather->color;pFather->color = BLACK;pSibling->right->color = BLACK;LeftRotate(pFather);pNode = root;}}else{if (pSibling->color == RED){pSibling->color = BLACK;pFather->color = RED;RightRotate(pFather);pSibling = pFather->left;}if (pSibling->left->color == BLACK &&pSibling->right->color == BLACK){pSibling->color = RED;pNode = pFather;}else{if (pSibling->left->color == BLACK)//sibling's left is red{pSibling->right->color = BLACK;pSibling->color = RED;LeftRotate(pSibling);pSibling = pFather->left;}//two children of sibling are both redpSibling->color = pFather->color;pFather->color = BLACK;pSibling->left->color = BLACK;RightRotate(pFather);pNode = root;}}}pNode->color = BLACK;}template<class T>void RBTree<T>::Remove(const T& value){//temp point definitionRBTreeNode<T> *pTemp=pGuard,*pTempNode = pGuard,*pNode = root;//find the right positionRBSearch(value);if(pNode == pGuard)return;// NOTFOUND;if (pNode->left == pGuard || pNode->right == pGuard)pTempNode = pNode;else pTempNode = const_cast<RBTreeNode<T>*>( Successor(pNode) );if(pTempNode->left != pGuard)pTemp = pTempNode->left;//maybe NILelse pTemp = pTempNode->right;//maybe successor's right node!pTemp->father = pTempNode->father;//different form BST, need not checkif(pTempNode->father == pGuard)root = pTemp;//set the root to NILelse if(pTempNode == pTempNode->father->left)pTempNode->father->left = pTemp;else pTempNode->father->right = pTemp;if(pNode != pTempNode){//has two children, and the pTempNode is pointing to successor.//now need to copy the successor's data to nodepNode->color = pTempNode->color;pNode->father = pTempNode->father;pNode->left = pTempNode->left;pNode->right = pTempNode->right;pNode->value = pTempNode->value;}if(pTempNode->color == BLACK)//change the black height, so need to fix up!RBRemoveFixUp(pTemp);delete pTempNode;pTempNode = NULL;return;// SUCCEED;}template<class T>RBTreeNode<T>* RBTree<T>::RBSearch(const T&value){RBTreeNode<T> *pNode = root;while (pNode!=pGuard){if(pNode->value >value)pNode = pNode->left;else if (pNode->value < value)pNode = pNode->right;else if(pNode->value == value)break;}return pNode;//maybe NIL}}//end of namespace#endif