242 lines
8.1 KiB
C++
242 lines
8.1 KiB
C++
#include <iostream>
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#include <vector>
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using namespace std;
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template <class T>
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struct Node {
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bool leaf; // 是否为叶子节点
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vector<T> keys; // 存储关键字
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vector<Node*> children; // 孩子节点指针
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Node(bool _leaf) : leaf(_leaf) {}
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};
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template <class T>
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class BTree {
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public:
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BTree(int _t) : t(_t), root(nullptr) {}
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void insert(const T& key); // 插入关键字
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void remove(const T& key); // 删除关键字
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void traverse() { traverse(root); cout << endl; } // 中序遍历
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private:
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Node<T>* root;
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int t; // 最小度数
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// ----------- 插入相关辅助函数 -----------
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void insertNonFull(Node<T>* node, const T& key); // 在非满节点插入
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void splitChild(Node<T>* parent, int idx); // 分裂子节点
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// ----------- 删除相关辅助函数 -----------
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void remove(Node<T>* node, const T& key); // 递归删除
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void removeFromLeaf(Node<T>* node, int idx); // 从叶子节点删除
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void removeFromNonLeaf(Node<T>* node, int idx); // 从非叶子节点删除
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T getPredecessor(Node<T>* node, int idx); // 找前驱
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T getSuccessor(Node<T>* node, int idx); // 找后继
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void fill(Node<T>* node, int idx); // 保证孩子节点至少 t-1 个关键字
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void borrowFromPrev(Node<T>* node, int idx); // 从左兄弟借一个关键字
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void borrowFromNext(Node<T>* node, int idx); // 从右兄弟借一个关键字
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void merge(Node<T>* node, int idx); // 与右兄弟合并
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// ----------- 遍历辅助函数 -----------
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void traverse(Node<T>* node);
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};
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// ---------------------- 插入相关 ----------------------
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template<class T>
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void BTree<T>::insert(const T& key) {
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if (!root) { // 树空
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root = new Node<T>(true);
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root->keys.push_back(key);
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} else {
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// 根节点满，先分裂根节点
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if (root->keys.size() == 2*t - 1) {
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Node<T>* s = new Node<T>(false);
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s->children.push_back(root);
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splitChild(s, 0);
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root = s;
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}
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insertNonFull(root, key); // 插入非满节点
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}
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}
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template<class T>
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void BTree<T>::insertNonFull(Node<T>* node, const T& key) {
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int i = node->keys.size() - 1;
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if (node->leaf) { // 叶子节点直接插入
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node->keys.push_back(key);
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while (i >= 0 && node->keys[i] > key) {
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node->keys[i+1] = node->keys[i];
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i--;
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}
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node->keys[i+1] = key;
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} else { // 内部节点
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while (i >= 0 && node->keys[i] > key) i--;
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i++;
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if (node->children[i]->keys.size() == 2*t-1) {
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splitChild(node, i); // 分裂满子节点
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if (node->keys[i] < key) i++;
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}
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insertNonFull(node->children[i], key);
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}
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}
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template<class T>
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void BTree<T>::splitChild(Node<T>* parent, int idx) {
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Node<T>* y = parent->children[idx];
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Node<T>* z = new Node<T>(y->leaf);
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// 把 y 的后 t-1 个关键字移动到 z
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for (int j = 0; j < t-1; j++) z->keys.push_back(y->keys[j+t]);
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if (!y->leaf) {
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for (int j = 0; j < t; j++) z->children.push_back(y->children[j+t]);
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y->children.resize(t);
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}
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y->keys.resize(t-1); // 保留前 t-1 个关键字
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parent->children.insert(parent->children.begin()+idx+1, z);
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parent->keys.insert(parent->keys.begin()+idx, y->keys[t-1]); // 中间关键字上移
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}
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// ---------------------- 删除相关 ----------------------
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template<class T>
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void BTree<T>::remove(const T& key) {
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if (!root) return;
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remove(root, key);
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// 如果根节点被删除后没有关键字，调整 root
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if (root->keys.empty()) {
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Node<T>* oldRoot = root;
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if (root->leaf) root = nullptr;
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else root = root->children[0];
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delete oldRoot;
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}
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}
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template<class T>
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void BTree<T>::remove(Node<T>* node, const T& key) {
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int idx = 0;
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while (idx < node->keys.size() && node->keys[idx] < key) idx++;
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if (idx < node->keys.size() && node->keys[idx] == key) {
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if (node->leaf) removeFromLeaf(node, idx); // 情况1: 叶子节点直接删除
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else removeFromNonLeaf(node, idx); // 情况2: 内部节点
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} else {
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if (node->leaf) return; // key不存在
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bool flag = (idx == node->keys.size());
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// 如果孩子节点关键字不足 t 个，需要调整
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if (node->children[idx]->keys.size() < t) fill(node, idx);
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if (flag && idx > node->keys.size()) remove(node->children[idx-1], key);
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else remove(node->children[idx], key);
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}
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}
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template<class T>
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void BTree<T>::removeFromLeaf(Node<T>* node, int idx) {
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// 叶子节点直接删除关键字
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node->keys.erase(node->keys.begin() + idx);
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}
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template<class T>
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void BTree<T>::removeFromNonLeaf(Node<T>* node, int idx) {
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T k = node->keys[idx];
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// 如果左孩子关键字 >= t，用前驱替换
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if (node->children[idx]->keys.size() >= t) {
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T pred = getPredecessor(node, idx);
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node->keys[idx] = pred;
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remove(node->children[idx], pred);
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}
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// 如果右孩子关键字 >= t，用后继替换
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else if (node->children[idx+1]->keys.size() >= t) {
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T succ = getSuccessor(node, idx);
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node->keys[idx] = succ;
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remove(node->children[idx+1], succ);
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}
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// 左右孩子都不足 t，则合并孩子
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else {
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merge(node, idx);
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remove(node->children[idx], k);
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}
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}
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template<class T>
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T BTree<T>::getPredecessor(Node<T>* node, int idx) {
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Node<T>* cur = node->children[idx];
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while (!cur->leaf) cur = cur->children.back();
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return cur->keys.back();
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}
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template<class T>
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T BTree<T>::getSuccessor(Node<T>* node, int idx) {
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Node<T>* cur = node->children[idx+1];
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while (!cur->leaf) cur = cur->children[0];
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return cur->keys[0];
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}
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template<class T>
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void BTree<T>::fill(Node<T>* node, int idx) {
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// 尝试从兄弟借关键字，如果不够则合并
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if (idx != 0 && node->children[idx-1]->keys.size() >= t)
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borrowFromPrev(node, idx);
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else if (idx != node->keys.size() && node->children[idx+1]->keys.size() >= t)
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borrowFromNext(node, idx);
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else {
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if (idx != node->keys.size()) merge(node, idx);
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else merge(node, idx-1);
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}
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}
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template<class T>
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void BTree<T>::borrowFromPrev(Node<T>* node, int idx) {
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Node<T>* child = node->children[idx];
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Node<T>* sibling = node->children[idx-1];
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// 将父节点关键字下移到 child 前面
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child->keys.insert(child->keys.begin(), node->keys[idx-1]);
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if (!child->leaf) child->children.insert(child->children.begin(), sibling->children.back());
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node->keys[idx-1] = sibling->keys.back();
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sibling->keys.pop_back();
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if (!sibling->leaf) sibling->children.pop_back();
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}
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template<class T>
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void BTree<T>::borrowFromNext(Node<T>* node, int idx) {
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Node<T>* child = node->children[idx];
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Node<T>* sibling = node->children[idx+1];
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// 将父节点关键字下移到 child 后面
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child->keys.push_back(node->keys[idx]);
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if (!child->leaf) child->children.push_back(sibling->children[0]);
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node->keys[idx] = sibling->keys[0];
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sibling->keys.erase(sibling->keys.begin());
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if (!sibling->leaf) sibling->children.erase(sibling->children.begin());
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}
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template<class T>
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void BTree<T>::merge(Node<T>* node, int idx) {
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Node<T>* child = node->children[idx];
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Node<T>* sibling = node->children[idx+1];
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// 将父节点关键字和 sibling 的关键字都移入 child
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child->keys.push_back(node->keys[idx]);
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child->keys.insert(child->keys.end(), sibling->keys.begin(), sibling->keys.end());
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if (!child->leaf) child->children.insert(child->children.end(), sibling->children.begin(), sibling->children.end());
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// 删除父节点中的关键字和 sibling 指针
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node->keys.erase(node->keys.begin() + idx);
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node->children.erase(node->children.begin() + idx + 1);
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delete sibling;
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}
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// ---------------------- 遍历 ----------------------
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template<class T>
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void BTree<T>::traverse(Node<T>* node) {
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if (!node) return;
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int i;
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for (i = 0; i < node->keys.size(); i++) {
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if (!node->leaf) traverse(node->children[i]);
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cout << node->keys[i] << " ";
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}
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if (!node->leaf) traverse(node->children[i]);
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}
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