AVL Tree | Set 2 (Deletion)

95188 단어 데이터 구조
원본 링크:https://www.geeksforgeeks.org/avl-tree-set-2-deletion/
Steps to follow for deletion. To make sure that the given tree remains AVL after every deletion, we must augment the standard BST delete operation to perform some re-balancing. Following are two basic operations that can be performed to re-balance a BST without violating the BST property (keys(left) < key(root) < keys(right)).
  • Left Rotation
  • Right Rotation
  • T1, T2 and T3 are subtrees of the tree rooted with y (on left side)
    or x (on right side)
                    y                               x
                   / \     Right Rotation          /  \
                  x   T3   – --->        T1   y
                 / \       < - - - - - - -            / \
                T1  T2     Left Rotation            T2  T3
    Keys in both of the above trees follow the following order
          keys(T1) < key(x) < keys(T2) < key(y) < keys(T3)
    So BST property is not violated anywhere.
    

    c) Right Right Case
      z                                y
     /  \                            /   \ 
    T1   y     Left Rotate(z)       z      x
        /  \   - - - - - - - ->    / \    / \
       T2   x                     T1  T2 T3  T4
           / \
         T3  T4
    

    C implementation Following is the C implementation for AVL Tree Deletion. The following C implementation uses the recursive BST delete as basis. In the recursive BST delete, after deletion, we get pointers to all ancestors one by one in bottom up manner. So we don’t need parent pointer to travel up. The recursive code itself travels up and visits all the ancestors of the deleted node.
    우 리 는 재 귀적 인 프로그램 을 사용 하여 AVL 트 리 에서 노드 를 삭제 합 니 다. 재 귀적 인 프로그램 을 사용 할 때 노드 를 삭제 한 후에 이때 의 지침 은 아버 지 를 가리 키 고 자동 으로 재 귀적 으로 돌아 갈 때 아래 에서 위로 할아버지 의 노드 를 방문 합 니 다.
  • Perform the normal BST deleteon. 1) 정상 적 인 BST 삭 제 를 진행 합 니 다.
  • The current node must be one of the ancestors of the deleted node. Update the height of the current node. 현재 결점 은 결점 을 삭제 한 아버지 결점 을 가리 키 며 높이 를 업데이트 합 니 다.
  • Get the balance factor (left subtree height – right subtree height) of the current node. 균형 인 자 를 다시 계산 합 니 다.
  • If balance factor is greater than 1, then the current node is unbalanced and we are either in Left Left case or Left Right case. To check whether it is Left Left case or Left Right case, get the balance factor of left subtree. If balance factor of the left subtree is greater than or equal to 0, then it is Left Left case,else Left Right case. 균형 인자 가 1 보다 크 면 현재 결산 점 은 불 균형 적 입 니 다. left case or left right case 입 니 다. 도대체 어느 것 인지 확인 하기 위해 우 리 는 그의 왼쪽 나무의 균형 요 소 를 얻 을 수 있 습 니 다.왼쪽 트 리 의 균형 인자 > = 0 이면 left left case, 그렇지 않 으 면 left right case.
  • If balance factor is less than -1, then the current node is unbalanced and we are either in Right Right case or Right Left case. To check whether it is Right Right case or Right Left case, get the balance factor of right subtree. If the balance factor of the right subtree is smaller than or equal to 0, then it is Right Right case, else Right Left case.
  • #include
    #include
    using namespace std;
    class Node{
    public:
        int data;
        Node *left,*right;
        int height;
        Node():data(0),left(NULL),right(NULL),height(0){};
    };
    
    int getHeight(Node *node);
    int max(int a,int b);
    Node* newNode(int key);
    Node *rightRotate(Node *node);
    Node *leftRotate(Node *node);
    int getBalance(Node *node);
    Node *insert(Node *node,int key);
    Node *minValueNode(Node *node);
    Node *deletNode(Node *root,int key);
    void preOrder(Node *node);
    
    int main(){
        Node *root=NULL;
        root = insert(root, 9);
        root = insert(root, 11);
        root = insert(root, 10);
        root = insert(root, 6);
        root = insert(root, 5);
        root = insert(root, 0);
        root = insert(root, 2);
        root = insert(root, -1);
        root = insert(root, 1);
        preOrder(root);
        printf("
    "
    ); root = deletNode(root,10); preOrder(root); printf("
    "
    ); return 0; } int getHeight(Node *node){ if(node==NULL){ return 0; }return node->height; } // A utility function to get maximum of two integers int max(int a,int b){ return (a>b)?a:b; } Node* newNode(int key){ Node* node=new Node(); node->data=key; return node; } Node *rightRotate(Node *node){ Node *x=node->left; Node *T2=x->right; x->right=node; node->left=T2; node->height=max(getHeight(node->left),getHeight(node->right))+1; x->height=max(getHeight(x->left),getHeight(x->right))+1; return x; } Node *leftRotate(Node *node){ Node *y=node->right; Node *T2=y->left; y->left=node; node->right=T2; node->height=max(getHeight(node->left),getHeight(node->right))+1; y->height=max(getHeight(y->left),getHeight(y->right))+1; return y; } int getBalance(Node *node){ if(node==NULL){return 0;} return (getHeight(node->left)-getHeight(node->right)); } Node *insert(Node *node,int key){ if(node==NULL){return newNode(key);} else if(key>node->data){ node->right=insert(node->right,key); }else if(key<node->data){ node->left=insert(node->left,key); }else return node; node->height=max(getHeight(node->left),getHeight(node->right))+1; printf("node->data=%d,node->height=%d
    "
    ,node->data,node->height); int balfac=getBalance(node); // If this node becomes unbalanced,then there are 4 cases if(balfac>1&&key<node->left->data){ printf("rightRotate:%d
    "
    ,node->data); return rightRotate(node); } if(balfac>1&&key>node->left->data){ printf("001:leftRotate(%d)
    "
    ,node->left->data); node->left=leftRotate(node->left); printf("002:rightRotate(%d)
    "
    ,node->data); return rightRotate(node); } if(balfac<-1&&key>node->right->data){ cout<<"leftRotate:"<<node->data<<endl; return leftRotate(node); } if(balfac<-1&&key<node->right->data){ printf("001:rightRotate(%d)
    "
    ,node->right->data); node->right=rightRotate(node->right); printf("002:leftRotate(%d)
    "
    ,node->data); return leftRotate(node); } return node; /* return the (unchanged) node pointer */ } // Recursive function to delete a node with given key from subtree with // given root. It returns root of the modified subtree. Node *minValueNode(Node *node){ Node *current=node; while(current->left!=NULL){ current=current->left; } return current; } Node *deletNode(Node *root,int key){ // STEP 1: PERFORM STANDARD BST DELETE if(root==NULL){return root;} else if(key>root->data){ root->right=deletNode(root->right,key); }else if(key<root->data){ root->left=deletNode(root->left,key); }else{ // // node with only one child or no child if(root->left==NULL||root->right==NULL){ Node *tmp=root->left?root->left:root->right; // No child case if(tmp==NULL){ tmp=root; root=NULL; }else{ *root=*tmp; // Copy the contents of free(tmp); // the non-empty child } }else{ //node with two children: Get the inorder // successor (smallest in the right subtree) Node *tmp=minValueNode(root->right); // Copy the inorder successor's data to this node root->data=tmp->data; // Delete the inorder successor root->right=deletNode(root->right, tmp->data); } } // STEP 2: UPDATE HEIGHT OF THE CURRENT NODE root->height=1+max(getHeight(root->left),getHeight(root->right)); // STEP 3: GET THE BALANCE FACTOR OF THIS NODE (to check whether this // node became unbalanced) int balfac=getBalance(root); // If this node becomes unbalanced,then there are 4 cases(just like insert operation) // Left Left Case if(balfac>1&&getBalance(root->left)>=0){ return rightRotate(root); } // Left Right Case if(balfac>1&&getBalance(root->left)<0){ root->left=leftRotate(root->left); return rightRotate(root); } // Right Right Case if(balfac<-1&&getBalance(root->right)<=0){ return leftRotate(root); } if(balfac<-1&&getBalance(root->right)>0){ root->right=rightRotate(root->right); return leftRotate(root); } return root; } void preOrder(Node *node){ if(node==NULL){ return ; } printf("%d->",node->data); preOrder(node->left); preOrder(node->right); }
    
    // C++ program to delete a node from AVL Tree
    #include
    #include
    using namespace std;
    
    // An AVL tree node
    class Node
    {
        public:
        int key;
        Node *left;
        Node *right;
        int height;
    };
    
    // A utility function to get maximum
    // of two integers
    int max(int a, int b);
    
    // A utility function to get height
    // of the tree
    int height(Node *N)
    {
        if (N == NULL)
            return 0;
        return N->height;
    }
    
    // A utility function to get maximum
    // of two integers
    int max(int a, int b)
    {
        return (a > b)? a : b;
    }
    
    /* Helper function that allocates a
    new node with the given key and
    NULL left and right pointers. */
    Node* newNode(int key)
    {
        Node* node = new Node();
        node->key = key;
        node->left = NULL;
        node->right = NULL;
        node->height = 0; // new node is initially
                        // added at leaf
        return(node);
    }
    
    // A utility function to right
    // rotate subtree rooted with y
    // See the diagram given above.
    Node *rightRotate(Node *y)
    {   printf("rightRotate:%d
    "
    ,y->key); Node *x = y->left; Node *T2 = x->right; // Perform rotation x->right = y; y->left = T2; // Update heights y->height = max(height(y->left), height(y->right)) + 1; x->height = max(height(x->left), height(x->right)) + 1; // Return new root return x; } // A utility function to left // rotate subtree rooted with x // See the diagram given above. Node *leftRotate(Node *x) { printf("leftRotate:%d
    "
    ,x->key); Node *y = x->right; Node *T2 = y->left; // Perform rotation y->left = x; x->right = T2; // Update heights x->height = max(height(x->left), height(x->right)) + 1; y->height = max(height(y->left), height(y->right)) + 1; // Return new root return y; } // Get Balance factor of node N int getBalance(Node *N) { if (N == NULL) return 0; return height(N->left) - height(N->right); } Node* insert(Node* node, int key) { /* 1. Perform the normal BST rotation */ if (node == NULL) return(newNode(key)); if (key < node->key) node->left = insert(node->left, key); else if (key > node->key) node->right = insert(node->right, key); else // Equal keys not allowed return node; /* 2. Update height of this ancestor node */ node->height = 1 + max(height(node->left), height(node->right)); /* 3. Get the balance factor of this ancestor node to check whether this node became unbalanced */ int balance = getBalance(node); printf("node->data=%d,height=%d,balance=%d
    "
    ,node->key,node->height,balance); // If this node becomes unbalanced, // then there are 4 cases // Left Left Case if (balance > 1 && key < node->left->key){ return rightRotate(node);} // Right Right Case if (balance < -1 && key > node->right->key){ return leftRotate(node);} // Left Right Case if (balance > 1 && key > node->left->key) { node->left = leftRotate(node->left); return rightRotate(node); } // Right Left Case if (balance < -1 && key < node->right->key) { node->right = rightRotate(node->right); return leftRotate(node); } /* return the (unchanged) node pointer */ return node; // printf("root:%d
    ",node->key);
    } /* Given a non-empty binary search tree, return the node with minimum key value found in that tree. Note that the entire tree does not need to be searched. */ Node * minValueNode(Node* node) { Node* current = node; /* loop down to find the leftmost leaf */ while (current->left != NULL) current = current->left; return current; } // Recursive function to delete a node // with given key from subtree with // given root. It returns root of the // modified subtree. Node* deleteNode(Node* root, int key) { // STEP 1: PERFORM STANDARD BST DELETE if (root == NULL) return root; // If the key to be deleted is smaller // than the root's key, then it lies // in left subtree if ( key < root->key ) root->left = deleteNode(root->left, key); // If the key to be deleted is greater // than the root's key, then it lies // in right subtree else if( key > root->key ) root->right = deleteNode(root->right, key); // if key is same as root's key, then // This is the node to be deleted else { // node with only one child or no child if( (root->left == NULL) ||(root->right == NULL) ) { Node *temp = root->left ? root->left : root->right; // No child case if (temp == NULL) { temp = root; root = NULL; } else // One child case *root = *temp; // Copy the contents of // the non-empty child free(temp); } else { // node with two children: Get the inorder // successor (smallest in the right subtree) Node* temp = minValueNode(root->right); // Copy the inorder successor's // data to this node root->key = temp->key; // Delete the inorder successor root->right = deleteNode(root->right, temp->key); } } // If the tree had only one node // then return if (root == NULL) return root; // STEP 2: UPDATE HEIGHT OF THE CURRENT NODE root->height = 1 + max(height(root->left), height(root->right)); // STEP 3: GET THE BALANCE FACTOR OF // THIS NODE (to check whether this // node became unbalanced) int balance = getBalance(root); // If this node becomes unbalanced, // then there are 4 cases // Left Left Case if (balance > 1 && getBalance(root->left) >= 0) return rightRotate(root); // Left Right Case if (balance > 1 && getBalance(root->left) < 0) { root->left = leftRotate(root->left); return rightRotate(root); } // Right Right Case if (balance < -1 && getBalance(root->right) <= 0) return leftRotate(root); // Right Left Case if (balance < -1 && getBalance(root->right) > 0) { root->right = rightRotate(root->right); return leftRotate(root); } return root; } // A utility function to print preorder // traversal of the tree. // The function also prints height // of every node void preOrder(Node *root) { if(root != NULL) { cout << root->key << " "; preOrder(root->left); preOrder(root->right); } } // Driver Code int main() { Node *root = NULL; /* Constructing tree given in the above figure */ root = insert(root, 12); root = insert(root, 11); root = insert(root, 10); root = insert(root, 9); root = insert(root, 8); root = insert(root, -1); root = insert(root, 1); root = insert(root, 0); root = insert(root, 21); root = insert(root, 14); root = insert(root, 13); root = insert(root, 20); root = insert(root, 23); root = insert(root, 18); cout << "Preorder traversal of the " "constructed AVL tree is
    "
    ; preOrder(root); root = deleteNode(root, 20); cout << "
    Preorder traversal after"
    << " deletion of 10
    "
    ; preOrder(root); return 0; } // This code is contributed by rathbhupendra

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