AVL树,即平衡二叉树,是一种在搜索二叉树上进行改进的数据结构,搜索二叉树能够控制节点在树中位置的数据结构,能够做到建立数据的关联性。
对于单个元素搜索的一般场景下时间复杂度为$log_2N$,但是极端场景下:
搜索树的时间复杂度会退化到$O(log_2N)$
此时平衡二叉树被提出,能够在插入元素时动态地调整元素位置,使得二叉树的形状尽量“丰满”,达到左右子树较为平衡的状态,即左右子树高度差不超过1。
我们设置平衡因子bf对每个节点是否平衡进行判断,需要平衡的节点进行相应的操作,总结下来有四种情况需要进行调整。
左单旋指的是在插入节点后parent的bf = 2,Cur的bf = 1的情况,具体来说,我们需要做的调整如下:
具体操作结合代码进行讲解:
void RotateL(Node* parent)
{
Node* SubR = parent->_right;//记录主要调节节点
Node* SubRL = SubR->_left;//记录主要调节节点
parent->_right = SubRL;//连接操作
SubR->_left = parent;//进行连接
Node* pparent = parent->_parent;//改变子树根节点需要对该子树的父亲进行记录
parent->_parent = SubR;//三叉树要记得对父亲节点进行记录
if (SubRL)
{
SubRL->_parent = parent;//非空树进行连接
}
if (_root == parent)
{
_root = SubR;//若是根节点直接进行更改
SubR->_parent = nullptr;
}
else//判断子树原根节点是其父亲节点的左孩子还是右孩子,对其进行连接
{
if(pparent->_right == parent)
{
pparent->_right = SubR;
}
else
{
pparent->_left = SubR;
}
SubR->_parent = pparent;
}
parent->_bf = SubR->_bf = 0;//修改平衡因子
}
右单旋指的是在插入节点后parent的bf = -2,Cur的bf = -1的情况,具体来说,我们需要做的调整如下:
void RotateR(Node* parent)
{
Node* SubL = parent->_left;
Node* SubLR = SubL->_right;//记录主要调节节点
parent->_left = SubLR;
Node* pparent = parent->_parent;
parent->_parent = SubL;//改变子树根节点需要对该子树的父亲进行记录
SubL->_right = parent;
if (SubLR)
SubLR->_parent = parent;//非空树进行连接
}
if (_root == parent)
{
_root = SubL;//根节点直接进行修改
SubL->_parent = nullptr;
}
else//非根节点需要将子树根节点与其父亲进行连接
{
if (pparent->_right == parent)
{
pparent->_right = SubL;
}
else
{
pparent->_left = SubL;
}
SubL->_parent = pparent;
}
parent->_bf = SubL->_bf = 0;//修改平衡因子
}
双旋,也就是需要两次旋转才可以对树进行平衡,大体思路即对树进行两次旋转,先完成一个局部旋转,使整棵树的情况简单下来,再对整个数进行更改,先左后右实际上是新插入节点再较高左子树的右侧进行了插入,具体调整如下:
接下来结合代码进行具体讲解:
void RotateLR(Node* parent)
{
Node* SubL = parent->_left;
Node* SubLR = SubL->_right;
int bf = SubLR->_bf;//记录
RotateL(parent->_left);//复用代码
RotateR(parent);
if (bf == 0)//新增节点本身导致的不平衡,也就是原来为空的情况
{
SubL->_bf = SubLR->_bf = parent->_bf = 0;
}
else if (bf == 1)//插入节点右边高
{
parent->_bf = 0;
SubLR->_bf = 0;
SubL->_bf = -1;
}
else if (bf == -1)//插入节点左边高
{
parent->_bf = 0;
SubLR->_bf = 1;
SubL->_bf = 0;
}
else
{
assert(false);
}
}
双旋的第二种情况就是再右边高的子树左边进行插入,具体操作如下:
接下来结合具体代码进行分析
void RotateRL(Node* parent)
{
Node* SubR = parent->_right;
Node* SubRL = SubR->_left;
int bf = SubRL->_bf;
RotateR(parent->_right);
RotateL(parent);
if (bf == 0)
{
SubR->_bf = SubRL->_bf = parent->_bf = 0;
}
else if(bf == 1)
{
parent->_bf = -1;
SubRL->_bf = 0;
SubR->_bf = 0;
}
else if (bf == -1)
{
parent->_bf = 0;
SubRL->_bf = 0;
SubR->_bf = 1;
}
else
{
assert(false);
}
}
注:以上所示图的仅为编号,相对位置关系仅在初始状况中成立
#pragma once
template<class K ,class V>
struct AVLTreeNode
{
AVLTreeNode<K, V>* _left;
AVLTreeNode<K, V>* _right;
AVLTreeNode<K, V>* _parent;
pair<K, V> _kv;
int _bf;
AVLTreeNode(const pair<K,V> kv)
:_kv(kv)
,_left(nullptr)
,_right(nullptr)
,_parent(nullptr)
,_bf(0)
{}
};
template<class K, class V>
class AVLtree
{
public:
typedef AVLTreeNode<K, V> Node;
void RotateL(Node* parent)
{
Node* SubR = parent->_right;
Node* SubRL = SubR->_left;
parent->_right = SubRL;
SubR->_left = parent;
Node* pparent = parent->_parent;
parent->_parent = SubR;
if (SubRL)
{
SubRL->_parent = parent;
}
if (_root == parent)
{
_root = SubR;
SubR->_parent = nullptr;
}
else
{
if(pparent->_right == parent)
{
pparent->_right = SubR;
}
else
{
pparent->_left = SubR;
}
SubR->_parent = pparent;
}
parent->_bf = SubR->_bf = 0;
}
void RotateR(Node* parent)
{
Node* SubL = parent->_left;
Node* SubLR = SubL->_right;
parent->_left = SubLR;
Node* pparent = parent->_parent;
parent->_parent = SubL;
SubL->_right = parent;
if (SubLR)
SubLR->_parent = parent;
if (_root == parent)
{
_root = SubL;
SubL->_parent = nullptr;
}
else
{
if (pparent->_right == parent)
{
pparent->_right = SubL;
}
else
{
pparent->_left = SubL;
}
SubL->_parent = pparent;
}
parent->_bf = SubL->_bf = 0;
}
void RotateRL(Node* parent)
{
Node* SubR = parent->_right;
Node* SubRL = SubR->_left;
int bf = SubRL->_bf;
RotateR(parent->_right);
RotateL(parent);
if (bf == 0)
{
SubR->_bf = SubRL->_bf = parent->_bf = 0;
}
else if(bf == 1)
{
parent->_bf = -1;
SubRL->_bf = 0;
SubR->_bf = 0;
}
else if (bf == -1)
{
parent->_bf = 0;
SubRL->_bf = 0;
SubR->_bf = 1;
}
else
{
assert(false);
}
}
void RotateLR(Node* parent)
{
Node* SubL = parent->_left;
Node* SubLR = SubL->_right;
int bf = SubLR->_bf;
RotateL(parent->_left);
RotateR(parent);
if (bf == 0)
{
SubL->_bf = SubLR->_bf = parent->_bf = 0;
}
else if (bf == 1)
{
parent->_bf = 0;
SubLR->_bf = 0;
SubL->_bf = -1;
}
else if (bf == -1)
{
parent->_bf = 0;
SubLR->_bf = 1;
SubL->_bf = 0;
}
else
{
assert(false);
}
}
bool insert(const pair<K, V> kv)
{
if (_root == nullptr)
{
_root = new Node(kv);
return true;
}
Node* parent = nullptr;
Node* cur = _root;
while (cur)
{
if (cur->_kv.first < kv.first)
{
parent = cur;
cur = cur->_right;
}
else if (cur->_kv.first > kv.first)
{
parent = cur;
cur = cur->_left;
}
else
{
return false;
}
}
cur = new Node(kv);
if (parent->_kv.first > cur->_kv.first)
{
parent->_left = cur;
cur->_parent = parent;
}
else
{
parent->_right = cur;
cur->_parent = parent;
}
while (parent)
{
if (cur == parent->_left)
{
parent->_bf--;
}
else
{
parent->_bf++;
}
if (parent->_bf == 0)
{
break;
}
else if (parent->_bf == 1 || parent->_bf == -1)
{
cur = parent;
parent = parent->_parent;
}
else if(parent->_bf == 2 || parent->_bf == -2)
{
if (parent->_bf == 2 && cur->_bf == 1)
{
RotateL(parent);
}
else if (parent->_bf == -2 && cur->_bf == -1)
{
RotateR(parent);
}
else if (parent->_bf == 2 && cur->_bf == -1)
{
RotateRL(parent);
}
else if (parent->_bf == -2 && cur->_bf == 1)
{
RotateLR(parent);
}
break;
}
}
return true;
}
void Inorder()
{
_Inorder(_root);
}
bool balance_test()
{
return _balance(_root);
}
private:
void _Inorder(Node*root)
{
if (root == nullptr)
{
return;
}
_Inorder(root->_left);
cout<<root->_kv.first<<endl;
_Inorder(root->_right);
}
int _Height(Node* root)
{
if (root == nullptr)
return 0;
int height_left = _Height(root->_left);
int height_right = _Height(root->_right);
return height_left > height_right ? height_left + 1 : height_right + 1;
}
bool _balance(Node* root)
{
if (root == nullptr)
{
return true;
}
int height_left = _Height(root->_left);
int height_right = _Height(root->_right);
if(height_right- height_left!=root->_bf)
{
cout<<root->_kv.first<<"平衡因子异常" << endl;
return false;
}
return abs(height_right - height_left) < 2 && _Height(root->_left) && _Height(root->_right);
}
Node* _root = nullptr;
};
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原创声明:本文系作者授权腾讯云开发者社区发表,未经许可,不得转载。
如有侵权,请联系 cloudcommunity@tencent.com 删除。