算法学习 - 平衡二叉查找树实现(AVL树)
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\n",A->element,A->height);
A = Insert(2, A);
printf("%d %d\n",A->left->element,A->height);
A = Insert(1, A);
printf("%d %d\n",A->left->element,A->left->height);
A = Insert(4, A);
A = Insert(5, A);
printf("%d %d\n",A->right->element,A->right->height);
A = Insert(6, A);
printf("%d %d\n",A->element,A->height);
A = Insert(7, A);
A = Insert(16, A);
A = Insert(15, A);
printf("%d %d\n",A->right->element,A->right->height);
A = Insert(14, A);
printf("%d %d\n",A->right->element,A->right->height);
A = Delete(16, A);
printf("%d %d\n",A->right->element,A->right->height);
A = Delete(6, A);
A = Delete(5, A);
printf("%d %d\n",A->right->element,A->right->height);
return 0;
}
