算法导论 第13章 13-2 红黑树上的连接操作

题目

解答：

         a) 由题意可知：现在红黑树节点的域中增加了黑高度域bh。回顾习题13.3-3和13.4-5，题目要求我们标出图中每个节点的黑高度，具体见下面：

        习题13.3-3 假设开始时所有子树黑高度为k

         题目习题13.4-5 删除过程并不能假设所有子树黑高度为k，但是可以设A节点，即x所指向的节点（注意A现在是双黑色）的黑高度为k来算出其他节点黑高度，具体如下：

根据图示我们可以很清楚的发现：在插入过程中，只有在进行调整时的情况1下会使新插入节点B的祖父C的黑高度增加1，其他几种情况均不会出现黑高度改变的情况，那么我们只用将情况1的代码稍作修改即可，如下：

if (uncle != nil && uncle->color == red){//情况1，叔叔节点存在且为红色curr->parent->color = black;uncle->color = black;curr->parent->parent->color = red;curr = curr->parent->parent;++curr->bh;}

而在删除过程中，在进行调整时的情况2中节点A原本为两重黑，后来只剩下一重，那么A的父节点B黑高度减1；在情况4中，节点A的父节点B在旋转后黑高度减1，而它的兄弟节点D的黑高度增1，其他情况均不变，那么我们只需将两种情况下的代码改为如下样子即可：

情况2：

if (brother->left->color == black && brother->right->color == black){//情况2，兄弟是黑色，且两孩子也是黑色，将当前节点和兄弟去一重黑色brother->color = red;curr = curr->parent;--curr->bh;}

情况4：

else{//情况4brother->color = curr->parent->color;curr->parent->color = black;brother->left->color = black;rightRotate(curr->parent);++brother->bh;--brother->left->bh;curr = root;}

        b) 只需沿着T1一直向右遍历，遇到黑高度为bh[T2]的黑节点即可，该过程记为locateMaxNodeOfBh，很明显，查找节点时间复杂度为O(lgn)，稍后直接给出代码。

        c) 题目说是在不破坏二叉查找树的前提下，而不是红黑树。根据题目已知，我们可以运用locateMaxNodeOfBh过程在Ty中找到黑高度和T2相等的最大节点，设为w，然后以x为衔接点进行连接，此时T2作为x右子树，w成为x左子树，x成为w父节点的右孩子，完毕。稍后给出在不破坏红黑树前提下的链接，该过程记为joinRight，时间为O(lgn)，因为要调整。

       d) 红色，直接调用insertFixup过程，从x开始自底向上调整，显然时间为O(lgn)。

     

       e) 同b)理，只需沿着T2一直向左遍历即可，遇到黑高度为bn[T1]的黑节点即可，该过程记为locateMinNodeOfBh。

 

       f) 在执行连接之前组要找到连接点，采用locateMinNodeOfBh过程或者locateMaxNodeOfBh过程，时间为O(lgn)，连接完毕后，从连接点开始自底向上插入调整，时间亦为O(lgn)，故整个连接过程的时间为O(lgn)。

下面是该红黑树加强版的C++代码：

/*******红黑树加强版********#节点成员添加了黑高度bh数据成员#相应的insertFixup和eraseFixup函数添加了对黑高度的调整#添加了红黑树连接操作jion(Left/Right)成员函数以及配套的功能函数*/#include<iostream>#include<iomanip>using namespace std;enum COLOR { red, black };//枚举，定义颜色template <typename T> class RBTree;template <typename T>class node{private:friend class RBTree<T>;node *parent;node *left;node *right;T key;int bh;//黑高度COLOR color;node(){}//默认构造函数，只供创建nil时调用public:node(const T &k, COLOR c = red) :key(k), color(c), bh(1),parent(NULL), left(NULL), right(NULL){}T& getKey(){ return key; }const T& getKey()const { return key; }//省略指针域的getter和setter};template <typename T>class RBTree{private:static node<T> *nil;//哨兵，静态成员，被整个RBTree类所共有node<T> *root;RBTree(const RBTree&);//禁止复制构造RBTree operator=(const RBTree&);//禁止赋值void leftRotate(node<T>*);//左旋void rightRotate(node<T>*);//右旋void insertFixup(node<T>*);//插入节点后红黑性质调整void eraseFixup(node<T>*);//删除节点后红黑性质调整public:RBTree() :root(nil){root->parent = nil;root->left = nil;root->right = nil;root->color = black;root->bh = -1;//哨兵的黑高度设为-1.}RBTree(node<T> *rbt) :root(rbt){}//复制构造函数，用于创建子红黑树对象void insert(const T&);//插入void create();//创建红黑树void erase(const T&);//删除//将两棵树(T & rbt)和一个值k链接，链接在右边，其中key[T]<=k<=key[rbt]void joinRight(const T &k, RBTree *rbt);//将两棵树(T & rbt)和一个值k链接，链接在左边，其中key[T]>=k>=key[rbt]void joinLeft(const T &k, RBTree *rbt);void preTraversal()const;//先根遍历void inTraversal()const;//中根遍历void destroy();//销毁红黑树node<T>* locateMaxNodeOfBh(int bh)const;//在红黑树中查找某一黑高度的最大值节点node<T>* locateMinNodeOfBh(int bh)const;//在红黑树中查找某一黑高度的最小值节点node<T>* locate(const T&)const;//查找node<T>* minMum()const;//最小值node<T>* maxMum()const;//最大值node<T>* successor(const T&)const;//找后继node<T>* predecessor(const T&)const;//前驱bool empty()const{ return root == nil; }//判空};template <typename T> node<T> *RBTree<T>::nil = new node<T>;//定义静态成员niltemplate <typename T>void RBTree<T>::leftRotate(node<T> *curr){if (curr->right != nil){//存在右孩子时才能左旋node<T> *rchild = curr->right;curr->right = rchild->left;if (rchild->left != nil)rchild->left->parent = curr;rchild->parent = curr->parent;if (curr->parent == nil)root = rchild;else if (curr == curr->parent->left)curr->parent->left = rchild;else curr->parent->right = rchild;curr->parent = rchild;rchild->left = curr;}}template <typename T>void RBTree<T>::rightRotate(node<T> *curr){if (curr->left != nil){//存在左孩子时才能右旋node<T> *lchild = curr->left;curr->left = lchild->right;if (lchild->right != nil)lchild->right->parent = curr;lchild->parent = curr->parent;if (curr->parent == nil)root = lchild;else if (curr == curr->parent->left)curr->parent->left = lchild;else curr->parent->right = lchild;lchild->right = curr;curr->parent = lchild;}}template <typename T>void RBTree<T>::insert(const T &k){node<T> *pkey = new node<T>(k),*p = nil, *curr = root;while (curr != nil){//找插入位置p = curr;//记住当前节点父亲if (k < curr->key)//往左找curr = curr->left;else curr = curr->right;//向右找}pkey->parent = p;if (p == nil)//插入的是第一个节点root = pkey;else if (k < p->key)p->left = pkey;else p->right = pkey;pkey->left = pkey->right = nil;insertFixup(pkey);//调整}template <typename T>void RBTree<T>::insertFixup(node<T> *curr){while (curr->parent->color == red){//父亲为红节点时才需要进入循环调整if (curr->parent == curr->parent->parent->left){//父亲是祖父左孩子node<T> *uncle = curr->parent->parent->right;if (uncle != nil && uncle->color == red){//情况1，叔叔节点存在且为红色curr->parent->color = black;uncle->color = black;curr->parent->parent->color = red;curr = curr->parent->parent;++curr->bh;}else if (curr == curr->parent->right){//情况2，叔叔节点为黑色，且当前节点是父亲右孩子curr = curr->parent;leftRotate(curr);//将父节点左旋，以转变为情况3}else{//情况3，叔叔节点为黑色，且当前节点是父亲左孩子curr->parent->color = black;curr->parent->parent->color = red;rightRotate(curr->parent->parent);}}else{//父亲是祖父右孩子，与上面三种情况对称node<T> *uncle = curr->parent->parent->left;if (uncle != nil && uncle->color == red){//情况1curr->parent->color = black;uncle->color = black;curr->parent->parent->color = red;curr = curr->parent->parent;++curr->bh;}else if (curr == curr->parent->left){//情况2curr = curr->parent;rightRotate(curr);}else{//情况3curr->parent->color = black;curr->parent->parent->color = red;leftRotate(curr->parent->parent);}}}root->color = black;//跳出循环时将根节点置为黑色}template <typename T>void RBTree<T>::create(){T k;cout << "Enter element(s),CTRL+Z to end" << endl;//换行后CTRL+Z结束输入while (cin >> k)insert(k);cin.clear();}template <typename T>void RBTree<T>::preTraversal()const{node<T> *curr = root;if (curr != nil){cout << curr->key << " : ";if (curr->color == red) cout << setw(12) << "red";else cout << setw(12) << "black";cout << "black height: " << curr->bh << endl;RBTree LEFT(curr->left);//继续左子树先根遍历LEFT.preTraversal();RBTree RIGHT(curr->right);RIGHT.preTraversal();}}template <typename T>void RBTree<T>::inTraversal()const{node<T> *curr = root;if (curr != nil){RBTree LEFT(curr->left);LEFT.inTraversal();cout << left << curr->key << " : ";if (curr->color == red) cout << left << setw(12) << "red";else cout << left << setw(12) << "black";cout << "black height: " << curr->bh << endl;RBTree RIGHT(curr->right);//继续右子树中根遍历RIGHT.inTraversal();}}template <typename T>node<T>* RBTree<T>::successor(const T &k)const{node<T> *curr = locate(k);if (curr->right != nil){//若右子树不为空，则后继为右子树最小值RBTree RIGHT(curr->right);return RIGHT.minMum();}node<T> *p = curr->parent;while (p != nil && curr == p->right){//否则为沿右指针一直向上直到第一个拐弯处节点curr = p;p = p->parent;}return p;}template <typename T>node<T>* RBTree<T>::minMum()const{node<T> *curr = root;while (curr->left != nil)curr = curr->left;return curr;}template <typename T>node<T>* RBTree<T>::maxMum()const{node<T> *curr = root;while (curr->right != nil)curr = curr->right;return curr;}template <typename T>node<T>* RBTree<T>::predecessor(const T &k)const{node<T> *curr = locate(k);if (curr->left != nil){//若左子树不为空，则前驱为左子树最大值RBTree LEFT(curr->left);return LEFT.maxMum();}node<T> *p = curr->parent;while (p != nil && curr == p->left){//否则为沿左指针一直往上的第一个拐弯处节点curr = p;p = p->parent;}return p;}template <typename T>void RBTree<T>::erase(const T &k){node<T> *curr = locate(k), *pdel, *child;if (curr->left == nil || curr->right == nil)//决定删除节点pdel = curr;//若当前节点至多有一个孩子，则删除它else pdel = successor(k);//否则若有两孩子，则删除其后继if (pdel->left != nil)//记下不为空的孩子child = pdel->left;else child = pdel->right;child->parent = pdel->parent;if (pdel->parent == nil)//若删除的是根节点root = child;else if (pdel == pdel->parent->left)//否则若被删节点是其父亲左孩子pdel->parent->left = child;else pdel->parent->right = child;if (curr != pdel)curr->key = pdel->key;//若被删的是后继，则将后继值赋给当前节点if (pdel->color == black)//被删节点为黑色时才调整eraseFixup(child);delete pdel;//释放所占内存}template <typename T>void RBTree<T>::eraseFixup(node<T> *curr){while (curr != root && curr->color == black){//当前不为根，且为黑色if (curr == curr->parent->left){//若其是父亲左孩子node<T> *brother = curr->parent->right;//兄弟节点肯定存在if (brother->color == red){//情况1，兄弟是红色，转变为情况2,3,4brother->color = black;curr->parent->color = red;leftRotate(curr->parent);brother = curr->parent->right;}if (brother->left->color == black && brother->right->color == black){//情况2，兄弟是黑色，且两孩子也是黑色，将当前节点和兄弟去一重黑色brother->color = red;curr = curr->parent;--curr->bh;}else if (brother->right->color == black){//情况3，兄弟左孩子为红，右孩子为黑，转变为情况4brother->color = red;brother->left->color = black;rightRotate(brother);brother = curr->parent->right;}else{//情况4，右孩子为黑色，左孩子随意brother->color = curr->parent->color;curr->parent->color = black;brother->right->color = black;leftRotate(curr->parent);++brother->bh;--brother->left->bh;curr = root;}}else{//若其是父亲右孩子，与上面四中情况对称node<T> *brother = curr->parent->left;if (brother->color == red){//情况1brother->color = black;curr->parent->color = red;rightRotate(curr->parent);brother = curr->parent->left;}if (brother->right->color == black && brother->left->color == black){//情况2brother->color = red;curr = curr->parent;--curr->bh;}else if (brother->left->color == black){//情况3brother->color = red;brother->right->color = black;leftRotate(brother);brother = curr->parent->left;}else{//情况4brother->color = curr->parent->color;curr->parent->color = black;brother->left->color = black;rightRotate(curr->parent);++brother->bh;--brother->left->bh;curr = root;}}}curr->color = black;//结束循环时将当前节点置为黑色}template <typename T>node<T>* RBTree<T>::locate(const T &k)const{node<T> *curr = root;while (curr != nil && curr->key != k){if (k < curr->key)curr = curr->left;else curr = curr->right;}return curr;}template <typename T>node<T>* RBTree<T>::locateMaxNodeOfBh(int bh)const{//查找红黑树上黑高度为bh的最大值节点node<T> *curr = root;if (curr == nil || curr->bh < bh) return nil;//树为空或者此树黑高度太低while (curr->bh != bh)curr = curr->right;return curr;}template <typename T>node<T>* RBTree<T>::locateMinNodeOfBh(int bh)const{//查找红黑树上黑高度为bh的最小值节点，与locateMaxNodeOfBh对称node<T> *curr = root;if (curr == nil || curr->bh < bh) return nil;while (curr->bh != bh)curr = curr->left;return curr;}template <typename T>void RBTree<T>::joinRight(const T &k, RBTree *rbt){//将较矮的红黑树rbt以及节点k合并到较高红黑树T上,其中key[T]<=k<=key[rbt]node<T> *r = locateMaxNodeOfBh(rbt->root->bh);//找到链接点node<T> *curr = new node<T>(k);//默认赋予红色node<T> *pr = r->parent;curr->left = r;//树T的连接点r成为当前新建节点curr的左孩子r->parent = curr;curr->right = rbt->root;//curr的右孩子是矮树根rbt->root->parent = curr;curr->parent = pr;//curr的父节点设为r的父节点curr->bh = rbt->root->bh + 1;//设置节点k的黑高度if (pr != nil)//如果两棵树黑高度不相同pr->right = curr;else//否则curr为新根root = curr;insertFixup(curr);//调整}template <typename T>void RBTree<T>::joinLeft(const T &k, RBTree *rbt){//将较矮的红黑树rbt以及节点k合并到较高红黑树T上,其中key[T]>=k>=key[rbt],与joinRight对称node<T> *r = locateMinNodeOfBh(rbt->root->bh);node<T> *curr = new node<T>(k);//默认赋予红色node<T> *pr = r->parent;curr->right = r;r->parent = curr;curr->left = rbt->root;rbt->root->parent = curr;curr->parent = pr;curr->bh = rbt->root->bh + 1;if (pr != nil)//如果两棵树黑高度不相同pr->left = curr;elseroot = curr;insertFixup(curr);}template <typename T>void RBTree<T>::destroy(){while (root != nil){cout << "erase: " << root->key << endl;erase(root->key);}delete nil;}int main(){//26 17 41 14 21 30 47 10 16 19 23 28 38 7 12 15 20 35 39 3例1//11 2 14 1 7 15 5 8 4例2//58 89 767 67 79 779 3787 88 89例3RBTree<int> rbt1;cout << "-------------create rbt1------------" << endl;rbt1.create();cout << "------------inTraversal rbt1--------" << endl;rbt1.inTraversal();cout << "------------preTraversal rbt1-------" << endl;rbt1.preTraversal();RBTree<int> rbt2;cout << "-------------create rbt2------------" << endl;rbt2.create();cout << "------------inTraversal rbt2--------" << endl;rbt2.inTraversal();cout << "------------preTraversal rbt2-------" << endl;rbt2.preTraversal();rbt1.joinRight(50, &rbt2);cout << "------------inTraversal joinRight--------" << endl;rbt1.inTraversal();cout << "------------preTraversal joinRight-------" << endl;rbt1.preTraversal();rbt1.destroy();getchar();return 0;}