数据结构拾遗(2) --红黑树的设计与实现(中)

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Insert完善

根据规则4, 新增节点必须为红; 根据规则3, 新增节点之父节点必须为黑.

示例:

(1)插入16(红色)/55(红色), 则既不用旋转, 也不用重新染色

(2)插入82(红色), 则违反了红黑规则, 需要进行动态的调整;

红黑树所需的处理

1.单旋转

新插入的X与其父P都是红色的, 而且X还是G的外部孙子;

2.双旋转

新插入的X与其父P都是红色的, 而且X还是G的内部孙子;

3.还需要处理有两个红色孩子的节点, 将右儿子变成黑色的, 我们只允许左儿子是红色的;方法: 旋转+重新染色

insert完整实现代码:

/**注意带有 ++ 注释的为新添加的代码**/template <typename Comparable>void RedBlackTree<Comparable>::insert(const Comparable &x){    current = parent = grand = great = header;    nullNode->element = x;    while (current->element != x)    {        //让祖父成为曾祖父, 父亲成为祖父, 自己成为父亲        //每个人都长了一辈        great = grand;        grand = parent;        parent = current;        current = (x < current->element) ? current->left                  : current->right;        // ++        //+ 处理1. 如果current节点有两个红孩子        if ((current->left->color == RED) && (current->right->color == RED))            handleReorient( x );    }    //如果树中包含相同的元素    if (current != nullNode)        throw DuplicateItemException();    current = new Node(x, nullNode, nullNode);    if (x < parent->element)        parent->left = current;    else        parent->right = current;    // ++    //+ 处理2. 如果新插入的节点破坏了红黑规则, 则还需做一次处理    handleReorient( x );}

/**自动平衡函数:    [1]重新染色    [2]自动旋转*/template <typename Comparable>void RedBlackTree<Comparable>::handleReorient(const Comparable & item){    // 将current节点染成红色    current->color = RED;    // 将current的left和right节点染成黑色    current->left->color = current->right->color = BLACK;    // 如果current节点的父节点也是红的 -> 单旋转 or 双旋转    if( parent->color == RED )    {        //则将其祖父(爷爷)的颜色染成红色        grand->color = RED;        //然后判断新插入的节点是否是内部孙子?        //如果是, 则增加一次旋转->构成双旋转        //if注释: 如果该节点小于爷爷, 小于爸爸, 这两种情况不同时满足        //则说明其是爷爷的内孙子        if( (item < grand->element) != (item < parent->element) )        {            // 则依grand(祖父)节点进行旋转            parent = rotate( item, grand );  // Start double rotate        }        // 则依great(曾祖父)节点进行旋转        current = rotate( item, great );        //令当前节点为黑色        current->color = BLACK;    }    //根节点必须是黑色的    header->right->color = BLACK; // Make root black}

// 自动判断并进行旋转函数template <typename Comparable>typename RedBlackTree<Comparable>::Node *RedBlackTree<Comparable>::rotate(const Comparable &item,                                 Node *theParent ){    //位于theParent的左子树    if( item < theParent->element )    {        //如果为真, 则说明theParent->left有左孩子,        //否则, 有右孩子        item < theParent->left->element ?        //如果theParent左边有一棵子树, 则以theParent->left        //为轴, 向右转        rotateWithLeftChild( theParent->left )  :  // LL        //如果theParent右边有一棵子树, 则以theParent->left        //为轴, 向左转        rotateWithRightChild( theParent->left ) ;  // LR        return theParent->left;     //返回左子树    }    else    //位于右子树    {        //如果为真, 则说明theParent->right有左孩子,往右转        //否则, 有右孩子, 往左转        item < theParent->right->element ?        rotateWithLeftChild( theParent->right ) :  // RL        rotateWithRightChild( theParent->right );  // RR        return theParent->right;    //返回右子树    }}

测试代码:

int main(){    //用NEG_INF来代表负无穷大    const int NEG_INF = -999999;    RedBlackTree<int> tree(NEG_INF);    tree.insert(50);    cout << tree.header->right->element << " " << tree.header->right->nodeColor() << endl;    cout << endl;    tree.insert(40);    cout << tree.header->right->left->element << " " << tree.header->right->left->nodeColor() << endl << endl;    //此时需要进行旋转和重新染色    tree.insert(30);    cout << tree.header->right->element << " " << tree.header->right->nodeColor() << endl ;    cout << tree.header->right->left->element << " " << tree.header->right->left->nodeColor() << endl;    cout << tree.header->right->right->element << " " << tree.header->right->right->nodeColor() << endl;    return 0;}

//由于需要用到nodeColor成员函数, 因此我们将RedBlackNode类改造如下:template <typename Comparable>class RedBlackNode{    friend class RedBlackTree<Comparable>;public:    RedBlackNode(const Comparable &theElement = Comparable(),                 RedBlackNode *theLeft = NULL,                 RedBlackNode *theRight = NULL,                 int theColor = RedBlackTree<Comparable>::BLACK)        : element(theElement), left(theLeft), right(theRight), color(theColor) {}public:    //返回当前节点的颜色, 以作测试    std::string nodeColor() const    {        return (color == RedBlackTree<Comparable>::BLACK) ? "black" : "red";    }public:    Comparable element;    RedBlackNode *left;    RedBlackNode *right;    int color;};

红黑树其他操作

template <typename Comparable>class RedBlackTree{    ...    bool isEmpty() const;    void makeEmpty();    Gref<Comparable> find(const Comparable & x) const;    Gref<Comparable> findMin() const;    Gref<Comparable> findMax() const;    //递归删除所有节点    void reclainMemory(Node *t) const;    ...};

//析构函数完善版本template <typename Comparable>RedBlackTree<Comparable>::~RedBlackTree(){    if (!isEmpty())        makeEmpty();    delete nullNode;    delete header;}

//红黑树查找:与二叉查找树类似template <typename Comparable>Gref<Comparable> RedBlackTree<Comparable>::find(const Comparable &x) const{    if (isEmpty())        return Gref<Comparable>();    nullNode->element = x;    Node *iter = header->right;    while (true)    {        if (x < iter->element)            iter = iter->left;        else if (x > iter->element)            iter = iter->right;        //如果 x == iter->element        else if (iter != nullNode)            return Gref<Comparable>(iter->element) ;        else            return Gref<Comparable>();    }}

//查找最大值: 一路向右template <typename Comparable>Gref<Comparable> RedBlackTree<Comparable>::findMax() const{    if (isEmpty())        return Gref<Comparable>();    Node *iter = header->right;    while (iter->right != nullNode)    {        // 一直向右走        iter = iter->right;    }    return Gref<Comparable>(iter->element);}

//查找最小值: 一路向左template <typename Comparable>Gref<Comparable> RedBlackTree<Comparable>::findMin() const{    if (isEmpty())        return Gref<Comparable>();    Node *iter = header->right;    while (iter->left != nullNode)    {        // 一直向左走        iter = iter->left;    }    return Gref<Comparable>(iter->element);}

template <typename Comparable>bool RedBlackTree<Comparable>::isEmpty() const{    if (header->right == nullNode)        return true;    return false;}template <typename Comparable>void RedBlackTree<Comparable>::makeEmpty(){    reclainMemory(header->right);    header->right = nullNode;}template <typename Comparable>void RedBlackTree<Comparable>::reclainMemory(Node *t) const{    //t == t->left的时候, 是当t==nullNode时    if (t != t->left)    {        reclainMemory(t->left);        reclainMemory(t->right);        delete t;    }}

Gref设计与实现

//一个包装器: 将指针封装成为引用template <typename Object>class Gref{public:    Gref(): obj(NULL) {}    explicit Gref(const Object &x)        : obj(& x) {}    const Object &get() const    {        if (isNull())            throw NullPointerException();        else            return * obj;    }    bool isNull() const    {        if (obj == NULL)            return true;        return false;    }private:    const Object * obj;};

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