高级数据结构实现——自顶向下伸展树

【0】README

1） 本文部分内容转自 数据结构与算法分析，旨在理解 高级数据结构实现——自顶向下伸展树 的基础知识；
2） 源代码部分思想借鉴了数据结构与算法分析，有一点干货原创代码，for original source code, please visit https://github.com/pacosonTang/dataStructure-algorithmAnalysis/tree/master/chapter12/p345_topdown_splay_tree
3） you can also refer to the link http://www.cnblogs.com/huangxincheng/archive/2012/08/04/2623455.html
4） for basic splay tree , please visit http://blog.csdn.net/PacosonSWJTU/article/details/50525435

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【1】自顶向下伸展树相关

1）problem+solution

• 1.1）problem： 普通伸展树的展开操作需要从根沿树往下的一次遍历， 以及而后的从底向上的一次遍历。（详情，参见： http://blog.csdn.net/pacosonswjtu/article/details/50525435） 这可以通过保存一些父指针来完成， 或者通过将访问路径存储到一个栈中来完成。 但遗憾的 是， 这两种方法均需要大量的开销 ；
• 1.2）solution： 本节中， 我们指出如何在初始访问路径上施行一些旋转。结果得到在时间中更快的过程，只用到 O（1）的额外空间， 但却保持了 O(logN) 的摊还时间界；（干货——伸展树是基于AVL树的， 在AVL的基础上引入伸展树的目的是保持他的摊还时间界为 O（logN））

2）对伸展树的自顶向下的旋转操作：（单旋转+一字型旋转+之字型旋转）

• 2.1）这种伸展方式会把树切成三份，L树，M树，R树，考虑的情况有：单旋转，“一字型”旋转，“之字形”旋转。起初左树（L） 和 右树（R）均为空（NULL）；
  
  
  

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3）看个荔枝：

• 3.1）splay + deleting opeartions：
  

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• 3.2）inserting opeartions：
  

4）source code at a glance

#include "topdown_splay_tree.h" // allocate memory for new node.Node makeNode(int value){    Node node;    node = (Node)malloc(sizeof(struct Node));    if(!node)    {        Error("failed makeNode, for out of space !");        return NULL;    }    node->left = NULL;    node->right = NULL;     node->value = value;    return node;}// left left single rotateTopDownSplayTree left_left_single_rotate(TopDownSplayTree root){           TopDownSplayTree temp;    temp = root; // 1st step    root = root->left; // 2nd step    temp->left = root->right; // 3rd step    root->right = temp; // 4th step    return root;}// right_right_single_rotateTopDownSplayTree right_right_single_rotate(TopDownSplayTree root){           TopDownSplayTree temp;    temp = root; // 1st step    root = root->right; // 2nd step    temp->right = root->left; // 3rd step    root->left = temp; // 4th step      return root;}// performing splay operations TopDownSplayTree topdown_splay(int value, TopDownSplayTree middle){    struct Node plusTree;       Node leftTreeMax;       Node rightTreeMin;    leftTreeMax = &plusTree;    rightTreeMin = &plusTree;    while(value != middle->value)    {           if(middle->value < value)  // the new node is greater.        {                if(middle->right == NULL)            {                break;            }            else if(middle->right->value < value && middle->right->right)            {                middle = right_right_single_rotate(middle);            }                       leftTreeMax->right = middle;            leftTreeMax = middle;            middle = middle->right;                 leftTreeMax->right = NULL;        }        if(middle->value > value) // the new node is less.        {                   if(middle->left == NULL)            {                break;            }            else if(middle->left->value > value && middle->left->left)            {                middle = left_left_single_rotate(middle);            }            rightTreeMin->left = middle;            rightTreeMin = middle;            middle = middle->left;            rightTreeMin->left = NULL;        }       }    leftTreeMax->right = middle->left;    rightTreeMin->left = middle->right;    middle->left = plusTree.right;    middle->right = plusTree.left;    return middle;}// delete the root of the  TopDownSplayTreeTopDownSplayTree deleteNode(int value, TopDownSplayTree root){    TopDownSplayTree newroot;    if(root == NULL)     {        return root;    }    else // the splay tree is not null    {        root = topdown_splay(value, root);        if(root->value == value)  // find the node with given value.        {             if(root->left == NULL)            {                newroot = root->right;            }            else            {                newroot = root->left;                // perform splay again with value towards the left subtree which is not null.                newroot = topdown_splay(value, newroot);                newroot->right = root->right;             }            free(root);            root = newroot;        }           }       return root;}// insert the node with value into the TopDownSplayTreeTopDownSplayTree insert(int value, TopDownSplayTree root){    TopDownSplayTree node;    node = makeNode(value);          if(root == NULL) // the splay tree is null    {        return node;    }    else // the splay tree is not null    {        root = topdown_splay(value, root);        if(root->value > value)          {            node->left = root->left;            node->right = root;            root->left = NULL;            root = node;                    }        else if(root->value < value)          {             node->right = root->right;             root->right = NULL;             node->left = root;                       root = node;        }        else                    {            return root;        }    }       return root;}// test for insert operation.int main1(){    TopDownSplayTree root;      int data[] = {5, 11, 23, 10, 17};    int size = 5;    int i;    printf("\n === executing insert with {5, 11, 23, 10, 17} in turn.=== \n");    root = NULL;    for(i=0; i<size; i++)    {        root = insert(data[i], root);        printPreorder(1, root);     }           printf("\n === executing insert with 8 in turn.=== \n");    root = insert(8, root);    printPreorder(1, root);         printf("\n === executing insert with 18 in turn.=== \n");    root = insert(18, root);    printPreorder(1, root);    return 0;}// test for splay operation and deleting operation.int main(){       TopDownSplayTree root;    TopDownSplayTree temp;    printf("\n ====== test for splaying operation====== \n");    printf("\n === original tree is as follows.=== \n");    root = makeNode(12); // root = 12    temp = root;    temp->left = makeNode(5);    temp->right = makeNode(25);    temp = temp->right;  // root = 25    temp->left = makeNode(20);    temp->right = makeNode(30);    temp = temp->left;  // root = 20    temp->left = makeNode(15);    temp->right = makeNode(24);    temp = temp->left;  // root = 15    temp->left = makeNode(13);    temp->right = makeNode(18);    temp = temp->right;  // root = 18    temp->left = makeNode(16);      printPreorder(1, root);    printf("\n === executing splay operation with finding value=19.=== \n");    root = topdown_splay(19, root);    printPreorder(1, root);      printf("\n === executing deleting operation with value=15.=== \n");    root = deleteNode(15, root);    printPreorder(1, root);      return 0;}// analog print node values in the binominal tree, which involves preorder traversal. void printPreorder(int depth, TopDownSplayTree root){               int i;    if(root)     {               for(i = 0; i < depth; i++)            printf("    ");        printf("%d\n", root->value);                    printPreorder(depth + 1, root->left);                                           printPreorder(depth + 1, root->right);    }     else    {        for(i = 0; i < depth; i++)            printf("    ");        printf("NULL\n");    }}