数据结构与算法分析 c++11 伸展树（splay tree）

一、什么是伸展树（splay tree）

伸展树（splay tree）是特殊的二叉查找树，它通过将访问过的节点旋转移到根节点处，来降低再次访问的时间。它保证从空树开始任意连续M次对树的操作最多花费O(MlogN)

时间。    

     普通AVL树的旋转和伸展树的旋转不同之处：

      AVL树的旋转操作目的是缩小左右子树的高度差，它是全局调控即目的是缩小整棵树的高度，不会针对某一个节点做优化（例如将经常访问的节点移动到根的位置或靠近根的位置）

     相反，伸展树的旋转操作目的就是把经常访问的节点移动到根的位置，它不会考虑整棵树是否平衡，旋转完成后，伸展树可能成为一个糟糕的链表，但伸展树保证在此单链表上查找经常访问的20%节点是最快的。

二、伸展树实现及测试

      SplayTree.h

#pragma once#include <iomanip>#include <iostream>template <class T>class SplayTreeNode {public:T key;                // 关键字(键值)SplayTreeNode *left;    // 左孩子SplayTreeNode *right;    // 右孩子SplayTreeNode() :left(NULL), right(NULL) {}SplayTreeNode(T value, SplayTreeNode *l, SplayTreeNode *r) :key(value), left(l), right(r) {}};template <class T>class SplayTree {private:SplayTreeNode<T> *mRoot;    // 根结点public:SplayTree();~SplayTree();// 前序遍历"伸展树"void preOrder();// 中序遍历"伸展树"void inOrder();// 后序遍历"伸展树"void postOrder();// (递归实现)查找"伸展树"中键值为key的节点SplayTreeNode<T>* search(T key);// (非递归实现)查找"伸展树"中键值为key的节点SplayTreeNode<T>* iterativeSearch(T key);// 查找最小结点：返回最小结点的键值。T minimum();// 查找最大结点：返回最大结点的键值。T maximum();// 旋转key对应的节点为根节点，并返回值为根节点。void splay(T key);// 将结点(key为节点键值)插入到伸展树中void insert(T key);// 删除结点(key为节点键值)void remove(T key);// 销毁伸展树void destroy();// 打印伸展树void print();private:// 前序遍历"伸展树"void preOrder(SplayTreeNode<T>* tree) const;// 中序遍历"伸展树"void inOrder(SplayTreeNode<T>* tree) const;// 后序遍历"伸展树"void postOrder(SplayTreeNode<T>* tree) const;// (递归实现)查找"伸展树x"中键值为key的节点SplayTreeNode<T>* search(SplayTreeNode<T>* x, T key) const;// (非递归实现)查找"伸展树x"中键值为key的节点SplayTreeNode<T>* iterativeSearch(SplayTreeNode<T>* x, T key) const;// 查找最小结点：返回tree为根结点的伸展树的最小结点。SplayTreeNode<T>* minimum(SplayTreeNode<T>* tree);// 查找最大结点：返回tree为根结点的伸展树的最大结点。SplayTreeNode<T>* maximum(SplayTreeNode<T>* tree);// 旋转key对应的节点为根节点，并返回值为根节点。SplayTreeNode<T>* splay(SplayTreeNode<T>* tree, T key);// 将结点(z)插入到伸展树(tree)中SplayTreeNode<T>* insert(SplayTreeNode<T>* &tree, SplayTreeNode<T>* z);// 删除伸展树(tree)中的结点(键值为key)，并返回被删除的结点SplayTreeNode<T>* remove(SplayTreeNode<T>* &tree, T key);// 销毁伸展树void destroy(SplayTreeNode<T>* &tree);// 打印伸展树void print(SplayTreeNode<T>* tree, T key, int direction);};/** 构造函数*/template <class T>SplayTree<T>::SplayTree() :mRoot(NULL){}/** 析构函数*/template <class T>SplayTree<T>::~SplayTree(){destroy(mRoot);}/** 前序遍历"伸展树"*/template <class T>void SplayTree<T>::preOrder(SplayTreeNode<T>* tree) const{if (tree != NULL){cout << tree->key << " ";preOrder(tree->left);preOrder(tree->right);}}template <class T>void SplayTree<T>::preOrder(){preOrder(mRoot);}/** 中序遍历"伸展树"*/template <class T>void SplayTree<T>::inOrder(SplayTreeNode<T>* tree) const{if (tree != NULL){inOrder(tree->left);cout << tree->key << " ";inOrder(tree->right);}}template <class T>void SplayTree<T>::inOrder(){inOrder(mRoot);}/** 后序遍历"伸展树"*/template <class T>void SplayTree<T>::postOrder(SplayTreeNode<T>* tree) const{if (tree != NULL){postOrder(tree->left);postOrder(tree->right);cout << tree->key << " ";}}template <class T>void SplayTree<T>::postOrder(){postOrder(mRoot);}/** (递归实现)查找"伸展树x"中键值为key的节点*/template <class T>SplayTreeNode<T>* SplayTree<T>::search(SplayTreeNode<T>* x, T key) const{if (x == NULL || x->key == key)return x;if (key < x->key)return search(x->left, key);elsereturn search(x->right, key);}template <class T>SplayTreeNode<T>* SplayTree<T>::search(T key){return search(mRoot, key);}/** (非递归实现)查找"伸展树x"中键值为key的节点*/template <class T>SplayTreeNode<T>* SplayTree<T>::iterativeSearch(SplayTreeNode<T>* x, T key) const{while ((x != NULL) && (x->key != key)){if (key < x->key)x = x->left;elsex = x->right;}return x;}template <class T>SplayTreeNode<T>* SplayTree<T>::iterativeSearch(T key){return iterativeSearch(mRoot, key);}/** 查找最小结点：返回tree为根结点的伸展树的最小结点。*/template <class T>SplayTreeNode<T>* SplayTree<T>::minimum(SplayTreeNode<T>* tree){if (tree == NULL)return NULL;while (tree->left != NULL)tree = tree->left;return tree;}template <class T>T SplayTree<T>::minimum(){SplayTreeNode<T> *p = minimum(mRoot);if (p != NULL)return p->key;return (T)NULL;}/** 查找最大结点：返回tree为根结点的伸展树的最大结点。*/template <class T>SplayTreeNode<T>* SplayTree<T>::maximum(SplayTreeNode<T>* tree){if (tree == NULL)return NULL;while (tree->right != NULL)tree = tree->right;return tree;}template <class T>T SplayTree<T>::maximum(){SplayTreeNode<T> *p = maximum(mRoot);if (p != NULL)return p->key;return (T)NULL;}/** 旋转key对应的节点为根节点，并返回值为根节点。** 注意：*   (a)：伸展树中存在"键值为key的节点"。*          将"键值为key的节点"旋转为根节点。*   (b)：伸展树中不存在"键值为key的节点"，并且key < tree->key。*      b-1 "键值为key的节点"的前驱节点存在的话，将"键值为key的节点"的前驱节点旋转为根节点。*      b-2 "键值为key的节点"的前驱节点存在的话，则意味着，key比树中任何键值都小，那么此时，将最小节点旋转为根节点。*   (c)：伸展树中不存在"键值为key的节点"，并且key > tree->key。*      c-1 "键值为key的节点"的后继节点存在的话，将"键值为key的节点"的后继节点旋转为根节点。*      c-2 "键值为key的节点"的后继节点不存在的话，则意味着，key比树中任何键值都大，那么此时，将最大节点旋转为根节点。*/template <class T>SplayTreeNode<T>* SplayTree<T>::splay(SplayTreeNode<T>* tree, T key){SplayTreeNode<T> N, *l, *r, *c;if (tree == NULL)return tree;N.left = N.right = NULL;l = r = &N;for (;;){if (key < tree->key){if (tree->left == NULL)break;if (key < tree->left->key){c = tree->left;                           /* rotate right */tree->left = c->right;c->right = tree;tree = c;if (tree->left == NULL)break;}r->left = tree;                               /* link right */r = tree;tree = tree->left;}else if (key > tree->key){if (tree->right == NULL)break;if (key > tree->right->key){c = tree->right;                          /* rotate left */tree->right = c->left;c->left = tree;tree = c;if (tree->right == NULL)break;}l->right = tree;                              /* link left */l = tree;tree = tree->right;}else{break;}}l->right = tree->left;                                /* assemble */r->left = tree->right;tree->left = N.right;tree->right = N.left;return tree;}template <class T>void SplayTree<T>::splay(T key){mRoot = splay(mRoot, key);}/** 将结点插入到伸展树中，并返回根节点** 参数说明：*     tree 伸展树的根结点*     key 插入的结点的键值* 返回值：*     根节点*/template <class T>SplayTreeNode<T>* SplayTree<T>::insert(SplayTreeNode<T>* &tree, SplayTreeNode<T>* z){SplayTreeNode<T> *y = NULL;SplayTreeNode<T> *x = tree;// 查找z的插入位置while (x != NULL){y = x;if (z->key < x->key)x = x->left;else if (z->key > x->key)x = x->right;else{cout << "不允许插入相同节点(" << z->key << ")!" << endl;delete z;return tree;}}if (y == NULL)tree = z;else if (z->key < y->key)y->left = z;elsey->right = z;return tree;}template <class T>void SplayTree<T>::insert(T key){SplayTreeNode<T> *z = NULL;// 如果新建结点失败，则返回。if ((z = new SplayTreeNode<T>(key, NULL, NULL)) == NULL)return;// 插入节点mRoot = insert(mRoot, z);// 将节点(key)旋转为根节点mRoot = splay(mRoot, key);}/** 删除结点(节点的键值为key)，返回根节点** 参数说明：*     tree 伸展树的根结点*     key 待删除结点的键值* 返回值：*     根节点*/template <class T>SplayTreeNode<T>* SplayTree<T>::remove(SplayTreeNode<T>* &tree, T key){SplayTreeNode<T> *x;if (tree == NULL)return NULL;// 查找键值为key的节点，找不到的话直接返回。if (search(tree, key) == NULL)return tree;// 将key对应的节点旋转为根节点。tree = splay(tree, key);if (tree->left != NULL){// 将"tree的前驱节点"旋转为根节点x = splay(tree->left, key);// 移除tree节点x->right = tree->right;}elsex = tree->right;delete tree;return x;}template <class T>void SplayTree<T>::remove(T key){mRoot = remove(mRoot, key);}/** 销毁伸展树*/template <class T>void SplayTree<T>::destroy(SplayTreeNode<T>* &tree){if (tree == NULL)return;if (tree->left != NULL)destroy(tree->left);if (tree->right != NULL)destroy(tree->right);delete tree;}template <class T>void SplayTree<T>::destroy(){destroy(mRoot);}/** 打印"伸展树"** key        -- 节点的键值* direction  --  0，表示该节点是根节点;*               -1，表示该节点是它的父结点的左孩子;*                1，表示该节点是它的父结点的右孩子。*/template <class T>void SplayTree<T>::print(SplayTreeNode<T>* tree, T key, int direction){if (tree != NULL){if (direction == 0)    // tree是根节点cout << setw(2) << tree->key << " is root" << endl;else                // tree是分支节点cout << setw(2) << tree->key << " is " << setw(2) << key << "'s " << setw(12) << (direction == 1 ? "right child" : "left child") << endl;print(tree->left, tree->key, -1);print(tree->right, tree->key, 1);}}template <class T>void SplayTree<T>::print(){if (mRoot != NULL)print(mRoot, mRoot->key, 0);}

main.cpp      //测试

#include <iostream>#include "SplayTree.h"using namespace std;static int arr[] = { 10,50,40,30,20,60 };#define TBL_SIZE(a) ( (sizeof(a)) / (sizeof(a[0])) )     /*伸展树测试*/int main(){int i, ilen;SplayTree<int>* tree = new SplayTree<int>();cout << "== 依次添加: ";ilen = TBL_SIZE(arr);for (i = 0; i<ilen; i++){cout << arr[i] << " ";tree->insert(arr[i]);}cout << "\n== 前序遍历: ";tree->preOrder();cout << "\n== 中序遍历: ";tree->inOrder();cout << "\n== 后序遍历: ";tree->postOrder();cout << endl;cout << "== 最小值: " << tree->minimum() << endl;cout << "== 最大值: " << tree->maximum() << endl;cout << "== 树的详细信息: " << endl;tree->print();i = 10;cout << "\n== 旋转节点(" << i << ")为根节点";tree->splay(i);cout << "\n== 树的详细信息: " << endl;tree->print();// 销毁二叉树tree->destroy();return 0;}