算法学习

平衡二叉查找树

平衡二叉查找树是非常早出现的平衡树，由于全部子树的高度差不超过1，所以操作平均为O(logN)。

平衡二叉查找树和BS树非常像，插入和删除操作也基本一样。可是每一个节点多了一个高度的信息。在每次插入之后都要更新树的每一个节点的高度。发现不平衡之后就要进行旋转。

单旋转

单旋转是碰到左左或者右右的情况下所使用的方法。
比如：

---

  3
    
     2
       
         1

这样的情况就须要旋转，由于3是根节点，它的左子树高度为0，右子树高度为2。相差超过1了。所以要进行旋转。而这是右右的情况，所以是单旋转。

---

       2
      / 
     1   3

这样子旋转过后就能够了~

双旋转

双旋转也非常easy，但代码操作会略微麻烦一点：

---

  2
    
     4
    /
   3

遇到这样的情况就是双旋转，由于3是在2 4之间的。
旋转过后：

---

  3
 / 
2   4

这样子就能够了。。

事实上非常多时候情况比这个复杂，可是本质都是这样子操作的。

实现代码：

//
//  AVL.h
//  AVL
//
//  Created by Alps on 14-8-7.
//  Copyright (c) 2014年 chen. All rights reserved.
//

#ifndef AVL_AVL_h
#define AVL_AVL_h

#define ElementType int

struct TreeNode;
typedef TreeNode* AVL;
typedef TreeNode* Position;

Position Find(ElementType key, AVL A);
Position FindMax(AVL A);
Position FindMin(AVL A);

AVL Insert(ElementType key, AVL A);
AVL Delete(ElementType key, AVL A);

struct TreeNode{
    ElementType element;
    AVL left;
    AVL right;
    int height;
};


#endif

上面的代码是AVL.h文件。

//
//  main.cpp
//  AVL
//
//  Created by Alps on 14-8-7.
//  Copyright (c) 2014年 chen. All rights reserved.
//

#include <iostream>
#include "AVL.h"

int Height(AVL A){//求节点高度
    if (A == NULL) {
        return -1;
    }else{
        return A->height;
    }
}
int MAX(int a, int b){//返回两数中的大数
    return a>b?
a:b;
}

AVL SingleRotateWithRight(AVL A){//右单旋转
    AVL tmp = A->right;
    A->right = tmp->left;
    tmp->left = A;
    A->height = MAX(Height(A->left), Height(A->right))+1;
    tmp->height = MAX(Height(tmp->left), Height(tmp->right))+1;
    return tmp;
}

AVL DoubleRotateWithRight(AVL A){//右双旋转
    AVL tmp = A->right;
    AVL tmp1 = tmp->left;
    tmp->left = tmp1->right;
    A->right = tmp1->left;
    tmp1->right = tmp;
    tmp1->left = A;
    tmp->height = MAX(Height(tmp->left), Height(tmp->right))+1;
    A->height = MAX(Height(A->left), Height(A->right))+1;
    tmp1->height = MAX(Height(tmp1->left), Height(tmp1->right))+1;
    return tmp1;
}

AVL SingleRotateWithLeft(AVL A){//左单旋转
    AVL tmp = A->left;
    A->left = tmp->right;
    tmp->right = A;
    A->height = MAX(Height(A->left), Height(A->right))+1;
    tmp->height = MAX(Height(tmp->left), Height(tmp->right))+1;
    return tmp;
}

AVL DoubleRotateWithLeft(AVL A){//左双旋转
    AVL tmp = A->left;
    AVL tmp1 = tmp->right;
    tmp->right = tmp1->left;
    A->left = tmp1->right;
    tmp1->left = tmp;
    tmp1->right = A;
    tmp->height = MAX(Height(tmp->left), Height(tmp->right))+1;
    A->height = MAX(Height(A->left), Height(A->right))+1;
    tmp1->height = MAX(Height(tmp1->left), Height(tmp1->right))+1;
    return tmp1;
}



AVL Insert(ElementType key, AVL A){//插入元素
    if (A == NULL) {
        A = (AVL)malloc(sizeof(struct TreeNode));
        A->element = key;
        A->height = 0;
        A->right = NULL;
        A->left = NULL;
//        return A;
    }else{
        if (key > A->element) {//假设大于当前节点，向右子树插入
            A->right = Insert(key, A->right);
            if (Height(A->right) - Height(A->left) == 2) {
                if (key > A->right->element) {//假设插入到节点的右子树的右方，右单旋转
                    A = SingleRotateWithRight(A);
                }else{
                    A = DoubleRotateWithRight(A);//插入到当前节点右子树的左方，右双旋转
                }
            }
            
        }else
            if (key < A->element) {
               A->left = Insert(key, A->left);
                if (Height(A->left) - Height(A->right) == 2) {
                    if (key < A->left->element) {//左单旋转
                        A = SingleRotateWithLeft(A);
                    }else{
                        A = DoubleRotateWithLeft(A);
                    }
                }
            }
    }
    
    A->height = MAX(Height(A->left), Height(A->right))+1;
    return A;
}

Position FindMax(AVL A){//找到当前树的最大值
    AVL tmp = A;
    if (A == NULL) {
        return NULL;
    }else{
        while (tmp->right != NULL) {
            tmp = tmp->right;
        }
    }
    return tmp;
}

Position FindMin(AVL A){//找到当前树的最小值
    AVL tmp = A;
    if (A == NULL) {
        return NULL;
    }else{
        while (tmp->left != NULL) {
            tmp = tmp->left;
        }
    }
    return tmp;
}

Position Find(ElementType key,AVL A){//查找节点，返回节点指针
    AVL tmp = A;
    if (A == NULL) {
        return NULL;
    }else{
        while (tmp != NULL && tmp->element != key) {
            if (key > tmp->element) {
                tmp = tmp->right;
            }else{
                tmp = tmp->left;
            }
        }
    }
    return tmp;
}

AVL Delete(ElementType key, AVL A){//删除节点
    
    if (A == NULL || Find(key, A) == NULL) {
        return NULL;
    }else{
        
        if (key == A->element) {//假设找到了要删除的节点
            AVL tmp;
            if (A->left && A->right) {//假设要删除的节点有左右子树
                tmp = FindMin(A->left);//用当前节点左子树的最小值替换
                A->element = tmp->element;
                A->left = Delete(A->element, A->left);//删掉左子树最小值节点
            }else{
                tmp = A;
                if (A->left) {
                    A = A->left;//<span style="font-family: Arial, Helvetica, sans-serif;">假设仅仅存在左子树，直接返回它的左子树节点</span>

                }else{
                    if (A->right) {
                        A = A->right; //<span style="font-family: Arial, Helvetica, sans-serif;">假设仅仅存在右子树。直接返回它的右子树节点</span>

                    }else{
                        A = NULL;//删除的是叶子节点，直接赋值为NULL
                    }
                }
                free(tmp);
                tmp = NULL;
                return A;//返回删除后的节点
            }
        }else{
            if (key > A->element) {//假设大于，去右子树
                A->right = Delete(key, A->right);
                if (Height(A->left) - Height(A->right) == 2) {
                    if (A->left->right != NULL && (Height(A->left->right) > Height(A->left->left))) {//假设当前节点不平衡。且节点左孩子存在右孩子，双旋转
                        A = DoubleRotateWithLeft(A);
                    }else{
                        A = SingleRotateWithLeft(A);//否则单旋转
                    }
                }
//                A->height = MAX(Height(A->left), Height(A->right));
            }else{
                if (key < A->element) {
                    A->left = Delete(key, A->left);
                    if (Height(A->right) - Height(A->left) == 2) {
                        if (A->right->left != NULL && (Height(A->right->left) > Height(A->right->right))) {//
                            A = DoubleRotateWithRight(A);
                        }else{
                            A = SingleRotateWithRight(A);
                        }
                    }
//                    A->height = MAX(Height(A->left), Height(A->right));
                }
            }
        }
    }
    A->height = MAX(Height(A->left), Height(A->right))+1;
    return A;
}


int main(int argc, const char * argv[])
{
    AVL A = NULL;
    A = Insert(3, A);
    printf("%d %d
",A->element,A->height);
    A = Insert(2, A);
    printf("%d %d
",A->left->element,A->height);
    A = Insert(1, A);
    printf("%d %d
",A->left->element,A->left->height);
    A = Insert(4, A);
    A = Insert(5, A);
    printf("%d %d
",A->right->element,A->right->height);
    A = Insert(6, A);
    printf("%d %d
",A->element,A->height);
    A = Insert(7, A);
    A = Insert(16, A);
    A = Insert(15, A);
    printf("%d %d
",A->right->element,A->right->height);
    A = Insert(14, A);
    printf("%d %d
",A->right->element,A->right->height);
    A = Delete(16, A);
    printf("%d %d
",A->right->element,A->right->height);
    A = Delete(6, A);
    A = Delete(5, A);
    printf("%d %d
",A->right->element,A->right->height);
    return 0;
}