之前写完了二叉搜索树,今天打算写一下AVL树,对之前的二叉搜索树进行平衡优化
AVL树(使不平衡的树,变平衡)
AVL树又称为高度平衡的二叉搜索树,它能保持二叉树的高度平衡,尽量降低二叉树的高度,减少树的平均搜索长度
AVL树的性质
- 左子树和右子树的高度之差的绝对值不超过1 (衡量是否平衡的标准)
- 树中的每个左子树和右子树都是AVL树
- 每个节点都有一个平衡因子(balance factor–bf),任一节点的平衡因子是-1,0,1。(每个节点的平衡因子等于右子树的高度减去左子树的高度 )
通过旋转,使树平衡
左单旋&右单旋
左右双旋&右左双旋
//左旋
void RotateL(Node* parent)
{
Node* subR = parent->_right;
Node* subRL = subR->_left;
parent->_right = subRL;
if (subRL)
subRL->_parent = parent;
subR->_left = parent;
Node* ppNode = parent->_parent;
parent->_parent = subR;
if (ppNode == NULL)
{
_root = subR;
subR->_parent = NULL;
}
else //(ppNode!=NULL)
{
if (parent = ppNode->_left)
{
ppNode->_left = subR;
subR->_parent = ppNode;
}
else
{
ppNode->_right = subR;
subR->_parent = ppNode;
}
}
parent->_num = subR->_num = 0;
}
//右旋
void RotateR(Node* parent)
{
Node* subL = parent->_left;
Node* subLR = subL->_right;
parent->_left= subLR;
if (subLR)
subLR->_parent = parent;
subL->_right = parent;
Node* ppNode = parent->_parent;
parent->_parent = subL;
if(ppNode == NULL)
{
_root = subL;
_root->_parent = NULL;
}
else
{
if (parent == ppNode->_left)
{
ppNode->_left = subL;
subL->_parent = ppNode;
}
else
{
ppNode->_right = subL;
subL->_parent = ppNode;
}
}
parent->_num = subL->_num = 0;
}
//左右旋
void RotateLR(Node* parent)
{
RotateL(parent->_left);
RotateR(parent);
}
//右左旋
void RotateRL(Node* parent)
{
RotateR(parent->_right);
RotateL(parent);
}
bool Insert(const K& key, V value)
{
if (_root == NULL)
{
_root = new Node(key, value);
return true;
}
//找位置
Node* cur = _root;
Node* parent = NULL;
while (cur)
{
if (key < cur->_key)
{
parent = cur;
cur = cur->_left;
}
else if (key>cur->_key)
{
parent = cur;
cur = cur->_right;
}
else
{
break;
}
}
//插入,左插/右插
cur = new Node(key, value);
if (key < parent->_key)
{
parent->_left = cur;
cur->_parent = parent;
}
else if (key>parent->_key)
{
parent->_right = cur;
cur->_parent = parent;
}
else
{
return false;
}
//更新平衡因子
while (cur != _root)
{
//如果左插--,右插++
if (cur == parent->_left)
parent->_num--;
else
parent->_num++;
//判断平衡因子
if (parent->_num == 0)
{
return true;
}
//|p|=1,子树的高度改变,继续向更新
else if (parent->_num == 1 || parent->_num == -1)
{
cur = parent;
parent = parent->_parent;
}
//|p|=2,进行平衡调整
else if (parent->_num == 2 || parent->_num == -2)
{
if (parent->_num == 2 && cur->_num == 1)
{
RotateL(parent);
}
if (parent->_num == -2 && cur->_num == -1)
{
RotateR(parent);
}
if (parent->_num == 2 && cur->_num == -1)
{
RotateRL(parent);
}
if (parent->_num == -2 && cur->_num == 1)
{
RotateLR(parent);
}
return true;
}
}
}