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AVL树

AVL树的概念

二叉搜索树在顺序有序或接近有序的情况下，而插入搜索树将退化为单叉树，此时查找的时间复杂度为O(n)，效率低下。
两位俄罗斯的数学家G.M.Adelson-Velskii和E.M.Landis在1962年发明了一种解决上述问题的方法：当向二叉搜索树中插入新节点后，保证每个节点的左右子树高度差的绝对值不超过1，即可降低树的高度，减少平均搜索长度。因此，AVL树也被叫做高度平衡二叉搜索树，插入，查找，删除在平均和最坏情况下的时间复杂度都是O($lo{g}_{2}n$)。

AVL树节点的定义

	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; //balance factor 平衡因子AVLTreeNode(const pair<K, V>& kv):_left(nullptr),_right(nullptr),_parent(nullptr),_kv(kv),_bf(0){}};

注意：实现AVL树平衡因子不是必须的，只不过有了平衡因子帮助我们更便捷地控制整棵树。

AVL树的插入

1. 根据二叉搜索树的规则插入新节点

bool Insert(const pair<K, V> & kv){root为空，特殊处理if (_root == nullptr){_root = new Node(kv);return true;}Node* curr = _root;Node* parent = nullptr;while (curr){if (curr->_kv.first < kv.first){parent = curr;curr = curr->_right;}else if (curr->_kv.first > kv.first){parent = curr;curr = curr->_left;}else{return false;}}将新节点和其父亲节点链接起来Node* newnode = new Node(kv);if (parent->_kv.first < kv.first)parent->_right = newnode;elseparent->_left = newnode;newnode->_parent = parent;......}

2. 不断向上更新平衡因子
   1. 当前平衡因子为0，说明插入之前平衡因子为1 / -1，插入之后不改变树的高度，不会影响其他祖先节点，此时更新结束。
   2. 当前平衡因子为1 / -1，说明插入之前平衡因子为0，插入之后当前节点地高度发生变化，会影响其他祖先节点，但是不违反规则，需要向上对祖先节点进行更新，直至当前节点为root。
   3. 当前平衡因子为 2 / -2，此时当前节点所在地子树违反了平衡规则，需要进行处理–>旋转。

while (parent)
{if (parent->_left == newnode){parent->_bf--;}else{parent->_bf++;}if (parent->_bf == 0)break;else if (parent->_bf == -1 || parent->_bf == 1){newnode = parent;parent = parent->_parent;}else if (parent->_bf == -2 || parent->_bf == 2){旋转处理}else{assert(false);}
}

AVL树的旋转

右单旋

void RotatoR(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 (parent == _root){_root = subL;subL->_parent = nullptr;}else{if (ppnode->_left == parent)ppnode->_left = subL;elseppnode->_right = subL;subL->_parent = ppnode;}parent->_bf = 0;subL->_bf = 0;
}

左单旋

void RotatoL(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 (parent == _root){_root = subR;subR->_parent = nullptr;}else{if (ppnode->_left == parent)ppnode->_left = subR;elseppnode->_right = subR;subR->_parent = ppnode;}parent->_bf = 0;subR->_bf = 0;
}

左右双旋

旋转之前，45的平衡因子可能是-1/0/1，旋转完成之后，根据情况对其他节点的平衡因子进行调整

void RotatoLR(Node* parent)
{Node* subL = parent->_left;Node* subLR = subL->_right;int bf = subLR->_bf;RotatoL(subL);RotatoR(parent);subLR->_bf = 0;if (bf == 0){subL->_bf = 0;parent->_bf = 0;}else if (bf == 1){subL->_bf = -1;parent->_bf = 0;}else if (bf == -1){subL->_bf = 0;parent->_bf = 1;}
}

右左双旋

void RotatoRL(Node* parent)
{Node* subR = parent->_right;Node* subRL = subR->_left;int bf = subRL->_bf;subRL->_bf = 0;RotatoR(subR);RotatoL(parent);if (bf == 0){subR->_bf = 0;parent->_bf = 0;}else if (bf == 1){subR->_bf = 0;parent->_bf = -1;}else if (bf == -1){subR->_bf = 1;parent->_bf = 0;}
}

代码实现

namespace xxx
{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; //balance factor 平衡因子AVLTreeNode(const pair<K, V>& kv):_left(nullptr),_right(nullptr),_parent(nullptr),_kv(kv),_bf(0){}};template<class K, class V>class AVLTree{typedef AVLTreeNode<K, V> Node;public:bool Insert(const pair<K, V> & kv){if (_root == nullptr){_root = new Node(kv);return true;}Node* curr = _root;Node* parent = nullptr;while (curr){if (curr->_kv.first < kv.first){parent = curr;curr = curr->_right;}else if (curr->_kv.first > kv.first){parent = curr;curr = curr->_left;}else{return false;}}Node* newnode = new Node(kv);if (parent->_kv.first < kv.first)parent->_right = newnode;elseparent->_left = newnode;newnode->_parent = parent;while (parent){if (parent->_left == newnode){parent->_bf--;}else{parent->_bf++;}if (parent->_bf == 0)break;else if (parent->_bf == -1 || parent->_bf == 1){newnode = parent;parent = parent->_parent;}else if (parent->_bf == -2 || parent->_bf == 2){if (parent->_bf == 2 && newnode->_bf == 1){RotatoL(parent);}else if (parent->_bf == -2 && newnode->_bf == -1){RotatoR(parent);}else if (parent->_bf == -2 && newnode->_bf == 1){RotatoLR(parent);}else if (parent->_bf == 2 && newnode->_bf == -1){RotatoRL(parent);}else{assert(false);}break;}else{assert(false);}}return true;}void RotatoL(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 (parent == _root){_root = subR;subR->_parent = nullptr;}else{if (ppnode->_left == parent)ppnode->_left = subR;elseppnode->_right = subR;subR->_parent = ppnode;}parent->_bf = 0;subR->_bf = 0;}void RotatoR(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 (parent == _root){_root = subL;subL->_parent = nullptr;}else{if (ppnode->_left == parent)ppnode->_left = subL;elseppnode->_right = subL;subL->_parent = ppnode;}parent->_bf = 0;subL->_bf = 0;}void RotatoLR(Node* parent){Node* subL = parent->_left;Node* subLR = subL->_right;int bf = subLR->_bf;RotatoL(subL);RotatoR(parent);subLR->_bf = 0;if (bf == 0){subL->_bf = 0;parent->_bf = 0;}else if (bf == 1){subL->_bf = -1;parent->_bf = 0;}else if (bf == -1){subL->_bf = 0;parent->_bf = 1;}}void RotatoRL(Node* parent){Node* subR = parent->_right;Node* subRL = subR->_left;int bf = subRL->_bf;subRL->_bf = 0;RotatoR(subR);RotatoL(parent);if (bf == 0){subR->_bf = 0;parent->_bf = 0;}else if (bf == 1){subR->_bf = 0;parent->_bf = -1;}else if (bf == -1){subR->_bf = 1;parent->_bf = 0;}}void InOrder(){_InOrder(_root);cout << endl;}bool IsAVLTree(){return _IsAVLTree(_root);}private:bool _IsAVLTree(Node* root){if (root == nullptr)return true;int leftH = Height(root->_left);int rightH = Height(root->_right);return abs(leftH - rightH) <= 1&& _IsAVLTree(root->_left)&& _IsAVLTree(root->_right);}int Height(Node* node){if (node == nullptr)return 0;int leftH = Height(node->_left);int rightH = Height(node->_right);return 1 + max(leftH, rightH);}void _InOrder(Node* root){if (root == nullptr)return;_InOrder(root->_left);cout << root->_kv.second << ' ';_InOrder(root->_right);}Node* _root = nullptr;};
}

总结

AVL树是一棵绝对平衡的二叉搜索树，其要求每个节点的左右子树高度差的绝对值都不超过1，这样可以保证查询时高效的时间复杂度，即$lo{g}_2(N)$。但是如果要对AVL树做一些结构修改的操作，性能非常低下，比如：插入时要维护其绝对平衡，旋转的次数比较多。因此：如果需要一种查询高效且有序的数据结构，而且数据的个数为静态的(即不会改变)，可以考虑AVL树，但一个结构经常修改，就不太适合。