AVL樹

    AVL樹是一種二叉平衡查找樹，被廣泛使用。

    AVL樹的性質是：任意節點的左右子樹高度差不超過1。

    在進行節點插入與刪除時，AVL樹首先進行與二叉平衡查找樹相同的插入、刪除過程，然後自底向上進行回溯，回溯的過程中重新計算節點的左右子樹高度差，高度差超過2的節點進行平衡。

    AVL樹的平衡操作通過旋轉來完成。平衡時只需要考慮4種情況，既LL、RR、LR、RL，具體情形參見《數據結構與算法 - 面向對象的C++設計模式》。另外TAOCP一書對AVL樹有更詳盡的討論。

    下面給出AVL樹的一個示例代碼，演示AVL樹如何插入、刪除。代碼僅供參考，細節考慮不周處請忽略。

 

    avltree.cpp

#include <stdio.h> #include <stdlib.h> #include <memory.h> #include <Windows.h> /* 樹節點 */ typedef struct tagNode { tagNode * pParent; tagNode * pLeft; tagNode * pRight; int nKey; int nValue; int nLHeight; int nRHeight; } Node; int hmax( Node * pNode ) { if ( pNode == NULL ) { return 0; } return pNode->nLHeight > pNode->nRHeight ? pNode->nLHeight : pNode->nRHeight; } int hdiff( Node * pNode ) { if ( pNode == NULL ) { return 0; } return pNode->nRHeight - pNode->nLHeight; } /* 左旋轉 */ void rotate_left( Node ** ppTree, Node * pNode ) { Node * pRight; if ( ppTree == NULL || *ppTree == NULL || pNode == NULL || pNode->pRight == NULL ) { return; } pRight = pNode->pRight; /* 修改right的parent指針 */ pRight->pParent = pNode->pParent; /* 修改父節點的child指針 */ if ( pNode == *ppTree ) { *ppTree = pRight; } else { if ( pNode == pNode->pParent->pLeft ) { pNode->pParent->pLeft = pRight; } else { pNode->pParent->pRight = pRight; } } /* 修改節點的parent指針 */ pNode->pParent = pRight; /* 修改節點右child指針 */ pNode->pRight = pRight->pLeft; /* 修改節點右child的parent指針 */ if ( pNode->pRight != NULL ) { pNode->pRight->pParent = pNode; } /* 修改right的left指針 */ pRight->pLeft = pNode; pNode->nRHeight = pRight->nLHeight; pRight->nLHeight = hmax( pNode ) +1; if ( pRight->pParent != NULL ) { if ( pRight == pRight->pParent->pLeft ) { pRight->pParent->nLHeight = hmax( pRight ) +1; } else { pRight->pParent->nRHeight = hmax( pRight ) +1; } } return; } /* 右旋轉 */ void rotate_right( Node ** ppTree, Node * pNode ) { Node * pLeft; if ( ppTree == NULL || *ppTree == NULL || pNode == NULL || pNode->pLeft == NULL ) { return; } pLeft = pNode->pLeft; /* 修改left的parent指針 */ pLeft->pParent = pNode->pParent; /* 修改父節點的child指針 */ if ( pNode == *ppTree ) { *ppTree = pLeft; } else { if ( pNode == pNode->pParent->pLeft ) { pNode->pParent->pLeft = pLeft; } else { pNode->pParent->pRight = pLeft; } } /* 修改節點的parent指針 */ pNode->pParent = pLeft; /* 修改節點右child指針 */ pNode->pLeft = pLeft->pRight; /* 修改節點右child的parent指針 */ if ( pNode->pLeft != NULL ) { pNode->pLeft->pParent = pNode; } /* 修改right的left指針 */ pLeft->pRight = pNode; pNode->nLHeight = pLeft->nRHeight; pLeft->nRHeight = hmax( pNode ) +1; if ( pLeft->pParent != NULL ) { if ( pLeft == pLeft->pParent->pLeft ) { pLeft->pParent->nLHeight = hmax( pLeft ) +1; } else { pLeft->pParent->nRHeight = hmax( pLeft ) +1; } } return; } void balance( Node ** ppTree, Node * pNode ) { Node * pChild = NULL; /* 平衡 */ while ( pNode != NULL ) { if ( pNode->pLeft != NULL ) { pNode->nLHeight = hmax( pNode->pLeft ) +1; } if ( pNode->pRight != NULL ) { pNode->nRHeight = hmax( pNode->pRight ) +1; } if ( hdiff(pNode) > 1 ) { pChild = pNode->pRight; if ( hdiff(pChild) < 0 ) { rotate_right( ppTree, pChild ); rotate_left( ppTree, pNode ); } else { rotate_left( ppTree, pNode ); } } else if ( hdiff(pNode) < -1 ) { pChild = pNode->pLeft; if ( hdiff(pChild) > 0 ) { rotate_left( ppTree, pChild ); rotate_right( ppTree, pNode ); } else { rotate_right( ppTree, pNode ); } } pNode = pNode->pParent; } return; } /* 插入 */ void avl_insert( Node ** ppTree, Node * pNode ) { Node * pParent = NULL; Node * pChild = NULL; if ( ppTree == NULL || pNode == NULL ) { return; } if ( *ppTree == NULL ) { *ppTree = pNode; return; } pChild = *ppTree; while ( pChild != NULL ) { pParent = pChild; if ( pNode->nKey < pChild->nKey ) { pChild = pChild->pLeft; } else { pChild = pChild->pRight; } } if ( pNode->nKey < pParent->nKey ) { pParent->pLeft = pNode; } else { pParent->pRight = pNode; } pNode->pParent = pParent; balance(ppTree,pNode); return; } /* 查找後繼 */ Node * avl_successor( Node * pNode ) { Node * pSuccessor = NULL; if ( pNode->pRight != NULL ) { pSuccessor = pNode->pRight; while ( pSuccessor->pLeft != NULL ) { pSuccessor = pSuccessor->pLeft; } } else { pSuccessor = pNode->pParent; while ( pSuccessor != NULL ) { if ( pNode == pSuccessor->pLeft ) { break; } pNode = pSuccessor; pSuccessor = pSuccessor->pParent; } } return pSuccessor; } /* 刪除 */ Node * avl_delete( Node ** ppTree, Node * pNode ) { Node * pParent = NULL; if ( ppTree == NULL || *ppTree == NULL || pNode == NULL ) { return NULL; } pParent = pNode->pParent; if ( pNode->pLeft == NULL ) { if ( pParent != NULL ) { if ( pNode == pParent->pLeft ) { pParent->pLeft = pNode->pRight; pParent->nLHeight --; } else { pParent->pRight = pNode->pRight; pParent->nRHeight --; } } else { *ppTree = pNode->pRight; } if ( pNode->pRight != NULL ) { pNode->pRight->pParent = pParent; } } else if ( pNode->pRight == NULL ) { if ( pParent != NULL ) { if ( pNode == pParent->pLeft ) { pParent->pLeft = pNode->pLeft; pParent->nLHeight --; } else { pParent->pRight = pNode->pLeft; pParent->nRHeight --; } } else { *ppTree = pNode->pLeft; } if ( pNode->pLeft != NULL ) { pNode->pLeft->pParent = pParent; } } else { pParent = avl_successor( pNode ); pNode->nKey = pParent->nKey; return avl_delete( ppTree, pParent ); } if ( pParent != NULL ) { balance(ppTree,pParent); } return pNode; } /* 檢查性質 */ bool avl_check( Node * pTree ) { if ( pTree == NULL ) { return true; } /* 左右子樹高度差不超過1 */ if ( hdiff(pTree) < -1 || hdiff(pTree) > 1 ) { return false; } /* 左右高度檢查 */ if ( pTree->pLeft != NULL && hmax(pTree->pLeft)+1 != pTree->nLHeight ) { return false; } if ( pTree->pRight != NULL && hmax(pTree->pRight)+1 != pTree->nRHeight ) { return false; } if ( pTree->pRight == NULL && pTree->nRHeight != 0 ) { return false; } if ( pTree->pLeft == NULL && pTree->nLHeight != 0 ) { return false; } return avl_check(pTree->pRight) && avl_check(pTree->pLeft); } void avltree( int size ) { Node * pTree = NULL; srand( GetTickCount() ); /* 隨機插入 */ for ( int i = 0; i < size; i++ ) { Node * pNode = (Node *) new Node; if ( pNode != NULL ) { memset( pNode, 0, sizeof(Node) ); pNode->nKey = rand(); avl_insert( &pTree, pNode ); if ( !avl_check(pTree) ) { printf("Insert failed!/n"); return; } } } printf("Insert succeeded/n"); /* 刪除 */ for ( int i = 0; i < size; i++ ) { Node * pNode = pTree; delete avl_delete( &pTree, pNode ); if ( !avl_check(pTree) ) { printf("Delete failed!/n"); return; } } printf("Delete succeeded/n"); return; }

    main函數可以按照如下方式書寫：

#include "test.h" int main( int argc, char ** argv ) { avltree(1000); return 0; }

    AVL樹搜索、插入、刪除的時間複雜度爲O(lgn)。