《数据结构与算法分析：C语言描述》复习——第四章“树”——伸展树

2014.06.15 20:42

简介：

　　伸展树是一种介于普通二叉搜索树和AVL树之间的，比较平衡的一种二叉搜索树。它不像AVL树那样总是高度平衡，虽然单次操作的就可能耗费O(n)时间，但连续M次基本操作的时间复杂度能做到O(M * log(N))，M当然不能和1太接近。这种复杂度叫做均摊复杂度，英文叫amortiized complexity。当我学这门课的时候，这个概念我根本不理解，伸展树我也没有动手写过。后来在多个编程题目中的算法分析中，逐渐明白了均摊复杂度的意义。伸展树的一大特点，就是每次要访问一个节点，一定要想办法把它弄到根节点。这个办法，就是“旋转”。如果你明白AVL树的旋转，这个自然也不必多说了。不过，伸展树除了左单旋转、右单旋转、左右双旋转、右左双旋转之外，还用到了左左双旋转，右右双旋转。AVL旋转是为了恢复树的平衡，伸展树的旋转则是为了把一个节点一直旋转到根节点。在我费劲力气实现了AVL树之后，我以为能通过删掉很多代码把它变成伸展树，结果除了不再记录树的高度之外，我没删掉多少代码，还额外增加了两个旋转。事实证明伸展树也是个好想但不好写的结构。

图示：

　　照理说增删改查都应该给出图示，但我只想用图示说明一个节点是如何被旋转到根节点的。旋转不是为了恢复平衡，但旋转的确可以让一棵树变得更平衡，这也是伸展树高效率的关键——越旋转越平衡。

　　

实现：

// My implementation for splay tree, modified from my avl tree.
#include <iostream>
#include <string>
#include <vector>
using namespace std;

struct TreeNode {
    int val;
    TreeNode *left;
    TreeNode *right;
    TreeNode *parent;
    TreeNode(int _val): val(_val), left(nullptr), right(nullptr), parent(nullptr) {};
};

class SplayTree {
public:
    SplayTree() {
        m_root = nullptr;
    }
    
    bool empty() {
        return m_root == nullptr;
    }
    
    void clear() {
        _deleteTree(m_root);
    }
    
    void insertNode(const int &val) {
        if (m_root == nullptr) {
            m_root = new TreeNode(val);
            return;
        }

        TreeNode *ptr = _findNode(val);
        
        if (val == ptr->val) {
            return;
        }
        
        if (val < ptr->val) {
            ptr->left = new TreeNode(val);
            ptr->left->parent = ptr;
            ptr = ptr->left;
        } else {
            ptr->right = new TreeNode(val);
            ptr->right->parent = ptr;
            ptr = ptr->right;
        }
        _splayNode(ptr);
    }
    
    void deleteNode(const int &val) {
        if (m_root == nullptr) {
            return;
        }
        
        TreeNode *par, *cur;
        
        cur = _findNode(val);
        if (cur == nullptr || cur->val != val) {
            return;
        }
        // the node is splayed to the root, cur must be m_root.
        par = cur->parent;
        
        TreeNode *ptr;
        
        if (cur->left != nullptr) {
            ptr = _shiftLeft(cur);
            return;
        }
    
        if (cur->right != nullptr) {
            ptr = _shiftRight(cur);
            return;
        }
    
        if (par == nullptr) {
            delete cur;
            m_root = nullptr;
        } else if (val < par->val) {
            delete cur;
            par->left = nullptr;
        } else {
            delete cur;
            par->right = nullptr;
        }
    }
    
    void updateNode(const int &old_val, const int &new_val) {
        deleteNode(old_val);
        insertNode(new_val);
    }
    
    bool contains(const int &val) {
        TreeNode *ptr = _findNode(val);
        return ptr == nullptr ? false : ptr->val == val ? true : false;
    }
    
    string preorderTraversal() {
        string result;
        _preorderTraversalRecursive(m_root, result);
        return result;
    }
    
    string inorderTraversal() {
        string result;
        _inorderTraversalRecursive(m_root, result);
        return result;
    }
    
    string postorderTraversal() {
        string result;
        _postorderTraversalRecursive(m_root, result);
        return result;
    }
    
    ~SplayTree() {
        clear();
    }
private:
    TreeNode *m_root;
    
    void _deleteTree(TreeNode *&root) {
        if (root == nullptr) {
            return;
        }
        _deleteTree(root->left);
        _deleteTree(root->right);
        delete root;
        root = nullptr;
    }
    
    TreeNode* _findNode(const int &val) {
        TreeNode *ptr;
        
        ptr = m_root;
        while (ptr != nullptr) {
            if (val == ptr->val) {
                return ptr;
            }
            if (val < ptr->val) {
                if (ptr->left != nullptr) {
                    ptr = ptr->left;
                } else {
                    return ptr;
                }
            } else {
                if (ptr->right != nullptr) {
                    ptr = ptr->right;
                } else {
                    return ptr;
                }
            }
        }
        if (ptr->val == val) {
            _splayNode(ptr);
            return m_root;
        }

        return ptr;
    }
    
    void _preorderTraversalRecursive(const TreeNode  *root, string &result) {
        result.push_back('{');
        if (root == nullptr) {
            // '#' represents NULL.
            result.push_back('#');
        } else {
            result.append(to_string(root->val));
            _preorderTraversalRecursive(root->left, result);
            _preorderTraversalRecursive(root->right, result);
        }
        result.push_back('}');
    }
    
    void _inorderTraversalRecursive(const TreeNode  *root, string &result) {
        result.push_back('{');
        if (root == nullptr) {
            // '#' represents NULL.
            result.push_back('#');
        } else {
            _inorderTraversalRecursive(root->left, result);
            result.append(to_string(root->val));
            _inorderTraversalRecursive(root->right, result);
        }
        result.push_back('}');
    }
    
    void _postorderTraversalRecursive(const TreeNode  *root, string &result) {
        result.push_back('{');
        if (root == nullptr) {
            // '#' represents NULL.
            result.push_back('#');
        } else {
            _postorderTraversalRecursive(root->left, result);
            _postorderTraversalRecursive(root->right, result);
            result.append(to_string(root->val));
        }
        result.push_back('}');
    }
    
    TreeNode *_shiftLeft(TreeNode *root) {
        TreeNode *cur, *par;
        
        // root and root->left is guaranteed to be non-empty.
        par = root;
        cur = par->left;
        
        while (cur->right != nullptr) {
            par = cur;
            cur = cur->right;
        }
        root->val = cur->val;
        
        if (cur->left != nullptr) {
            return _shiftLeft(cur);
        }
        
        if (cur->right != nullptr) {
            return _shiftRight(cur);
        }
        
        if (cur == par->left) {
            delete par->left;
            par->left = nullptr;
        } else {
            delete par->right;
            par->right = nullptr;
        }

        return par;
    }

    TreeNode *_shiftRight(TreeNode *root) {
        TreeNode *cur, *par;
        
        // root and root->right is guaranteed to be non-empty.
        par = root;
        cur = par->right;
        
        while (cur->left != nullptr) {
            par = cur;
            cur = cur->left;
        }
        root->val = cur->val;
        
        if (cur->left != nullptr) {
            return _shiftLeft(cur);
        }
        
        if (cur->right != nullptr) {
            return _shiftRight(cur);
        }
        
        if (cur == par->left) {
            delete par->left;
            par->left = nullptr;
        } else {
            delete par->right;
            par->right = nullptr;
        }
        
        return par;
    }
    
    void _singleRotationLeft(TreeNode *cur) {
        // Subtree A is deeper than subtree B.
        // Before rotation:
        //     X
        //    / \
        //   Y   C
        //  / \
        // A   B
        // ----------
        // After rotation:
        //   Y
        //  / \
        // A   X
        //    / \
        //   B   C
        TreeNode *par = cur->parent;
        TreeNode *B;
        TreeNode *X, *Y;
        
        X = cur;
        Y = cur->left;
        B = Y->right;
        
        Y->right = X;
        X->parent = Y;
        X->left = B;
        if (B != nullptr) {
            B->parent = Y;
        }
        
        if (par == nullptr) {
            m_root = Y;
        } else if (par->left == cur) {
            par->left = Y;
        } else {
            par->right = Y;
        }
        Y->parent = par;
    }
    
    void _singleRotationRight(TreeNode *cur) {
        // Subtree C is deeper than subtree B.
        // Before rotation:
        //   X
        //  / \
        // A   Y
        //    / \
        //   B   C
        // ----------
        // After rotation:
        //     Y
        //    / \
        //   X   C
        //  / \
        // A   B
        TreeNode *par = cur->parent;
        TreeNode *B;
        TreeNode *X, *Y;
        
        X = cur;
        Y = cur->right;
        B = Y->left;
        
        Y->left = X;
        X->parent = Y;
        X->right = B;
        if (B != nullptr) {
            B->parent = X;
        }
        
        if (par == nullptr) {
            m_root = Y;
        } else if (par->left == cur) {
            par->left = Y;
        } else {
            par->right = Y;
        }
        Y->parent = par;
    }
    
    void _doubleRotationLeftLeft(TreeNode *cur) {
        // This is only for splay tree, not AVL.
        // Before rotation:
        //       X
        //      / \
        //     Y   D
        //    / \
        //   Z   C
        //  / \
        // A   B
        // ----------
        // After rotation:
        //   Z
        //  / \
        // A   Y
        //    / \
        //   B   X
        //      / \
        //     C   D
        TreeNode *par = cur->parent;
        TreeNode *B, *C;
        TreeNode *X, *Y, *Z;
        
        X = cur;
        Y = X->left;
        Z = Y->left;
        B = Z->right;
        C = Y->right;
        
        Z->right = Y;
        Y->parent = Z;
        Y->right = X;
        X->parent = Y;
        Y->left = B;
        if (B != nullptr) {
            B->parent = Y;
        }
        X->left = C;
        if (C != nullptr) {
            C->parent = X;
        }
        
        if (par == nullptr) {
            m_root = Z;
        } else if (par->left == cur) {
            par->left = Z;
        } else {
            par->right = Z;
        }
        Z->parent = par;
    }
    
    void _doubleRotationLeftRight(TreeNode *cur) {
        // Subtree Z is deeper than subtree A. Single rotation won't work, so let's use this one instead.
        // Before rotation:
        //     X
        //    / \
        //   Y   D
        //  / \
        // A   Z
        //    / \
        //   B   C
        // ----------
        // After rotation:
        //      Z
        //    /   \
        //   Y     X
        //  / \   / \
        // A   B C   D
        TreeNode *par = cur->parent;
        TreeNode *B, *C;
        TreeNode *X, *Y, *Z;
        
        X = cur;
        Y = X->left;
        Z = Y->right;
        B = Z->left;
        C = Z->right;
        
        Z->left = Y;
        Y->parent = Z;
        Z->right = X;
        X->parent = Z;
        Y->right = B;
        if (B != nullptr) {
            B->parent = X;
        }
        X->left = C;
        if (C != nullptr) {
            C->parent = X;
        }
        
        if (par == nullptr) {
            m_root = Z;
        } else if (par->left == cur) {
            par->left = Z;
        } else {
            par->right = Z;
        }
        Z->parent = par;
    }
    
    void _doubleRotationRightLeft(TreeNode *cur) {
        // Subtree Z is deeper than subtree D. Single rotation won't work, so let's use this one instead.
        // Before rotation:
        //   X
        //  / \
        // A   Y
        //    / \
        //   Z   D
        //  / \
        // B   C
        // ----------
        // After rotation:
        //      Z
        //    /   \
        //   X     Y
        //  / \   / \
        // A   B C   D
        TreeNode *par = cur->parent;
        TreeNode *B, *C;
        TreeNode *X, *Y, *Z;
        
        X = cur;
        Y = X->right;
        Z = Y->left;
        B = Z->left;
        C = Z->right;
        
        Z->left = X;
        X->parent = Z;
        Z->right = Y;
        Y->parent = Z;
        X->right = B;
        if (B != nullptr) {
            B->parent = X;
        }
        Y->left = C;
        if (C != nullptr) {
            C->parent = Y;
        }
        
        if (par == nullptr) {
            m_root = Z;
        } else if (par->left == cur) {
            par->left = Z;
        } else {
            par->right = Z;
        }
        Z->parent = par;
    }
    
    void _doubleRotationRightRight(TreeNode *cur) {
        // This is only for splay tree, not AVL.
        // Before rotation:
        //   X
        //  / \
        // A   Y
        //    / \
        //   B   Z
        //      / \
        //     C   D
        // ----------
        // After rotation:
        //       Z
        //      / \
        //     Y   D
        //    / \
        //   X   C
        //  / \
        // A   B
        TreeNode *par = cur->parent;
        TreeNode *B, *C;
        TreeNode *X, *Y, *Z;
        
        X = cur;
        Y = X->right;
        Z = Y->right;
        B = Y->left;
        C = Z->left;
        
        Z->left = Y;
        Y->parent = Z;
        Y->left = X;
        X->parent = Y;
        X->right = B;
        if (B != nullptr) {
            B->parent = X;
        }
        Y->right = C;
        if (C != nullptr) {
            C->parent = Y;
        }
        
        if (par == nullptr) {
            m_root = Z;
        } else if (par->left == cur) {
            par->left = Z;
        } else {
            par->right = Z;
        }
        Z->parent = par;
    }
    
    void _splayNode(TreeNode *cur) {
        if (m_root == nullptr || cur == nullptr) {
            return;
        }
        
        TreeNode *par, *grand;
        
        while (cur != nullptr && cur->parent != nullptr) {
            par = cur->parent;
            grand = par->parent;
            if (grand == nullptr) {
                if (par->left == cur) {
                    _singleRotationLeft(par);
                } else {
                    _singleRotationRight(par);
                }
                return;
            }
            if (grand->left == par) {
                if (par->left == cur) {
                    _doubleRotationLeftLeft(grand);
                } else {
                    _doubleRotationLeftRight(grand);
                }
            } else {
                if (par->left == cur) {
                    _doubleRotationRightLeft(grand);
                } else {
                    _doubleRotationRightRight(grand);
                }
            }
        }
    }
};

int main()
{
    SplayTree tree;
    
    tree.clear();
    tree.insertNode(1);
    cout << tree.preorderTraversal() << endl;
    tree.insertNode(2);
    cout << tree.preorderTraversal() << endl;
    tree.insertNode(3);
    cout << tree.preorderTraversal() << endl;
    tree.insertNode(4);
    cout << tree.preorderTraversal() << endl;
    tree.insertNode(5);
    cout << tree.preorderTraversal() << endl;
    tree.insertNode(6);
    cout << tree.preorderTraversal() << endl;
    // until now the tree is skewed.
    // look at this step.
    tree.insertNode(-1);
    cout << tree.preorderTraversal() << endl;
    tree.deleteNode(6);
    cout << tree.preorderTraversal() << endl;
    tree.deleteNode(5);
    cout << tree.preorderTraversal() << endl;
    tree.deleteNode(4);
    cout << tree.preorderTraversal() << endl;
    tree.deleteNode(3);
    cout << tree.preorderTraversal() << endl;
    tree.deleteNode(2);
    cout << tree.preorderTraversal() << endl;
    tree.deleteNode(1);
    cout << tree.preorderTraversal() << endl;

    return 0;
}