高級數據結構實現——自頂向下伸展樹

【0】README

1） 本文部分內容轉自 數據結構與算法分析，旨在理解 高級數據結構實現——自頂向下伸展樹 的基礎知識； 
2） 源代碼部分思想借鑒了數據結構與算法分析，有一點干貨原創代碼，for original source code, please visithttps://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】自頂向下伸展樹相關

1）problem+solution

• 1.1）problem： 普通伸展樹的展開操作需要從根沿樹往下的一次遍歷， 以及而后的從底向上的一次遍歷。（詳情，參見：http://blog.csdn.net/pacosonswjtu/article/details/50525435） 這可以通過保存一些父指針來完成， 或者通過將訪問路徑存儲到一個棧中來完成。 但遺憾的 是， 這兩種方法均需要大量的開銷 ；
• 1.2）solution： 本節中， 我們指出如何在初始訪問路徑上施行一些旋轉。結果得到在時間中更快的過程，只用到 O（1）的額外空間， 但卻保持了 O(logN) 的攤還時間界；（干貨——伸展樹是基於AVL樹的， 在AVL的基礎上引入伸展樹的目的是保持他的攤還時間界為 O（logN））

2）對伸展樹的自頂向下的旋轉操作：（單旋轉+一字型旋轉+之字型旋轉）

• 2.1）這種伸展方式會把樹切成三份，L樹，M樹，R樹，考慮的情況有：單旋轉，“一字型”旋轉，“之字形”旋轉。起初左樹（L） 和 右樹（R）均為空（NULL）； 
   
   
  

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3）看個荔枝：

• 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 rotate
TopDownSplayTree 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_rotate
TopDownSplayTree 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  TopDownSplayTree
TopDownSplayTree 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 TopDownSplayTree
TopDownSplayTree 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");
    }
}