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手把手实现红黑树Set和Map

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手把手实现红黑树Set和Map

好的，我们来手把手实现一个简化版的红黑树（RB-Tree），并基于此封装Set和Map容器。整个过程会分为几个步骤：

步骤1: 红黑树基础

红黑树是一种自平衡二叉搜索树，满足以下性质：

  1. 每个节点是红色或黑色
  2. 根节点是黑色
  3. 叶子节点（NIL节点）是黑色
  4. 红色节点的子节点必须是黑色（不能有连续红色节点）
  5. 从任一节点到其叶子节点的路径包含相同数量的黑色节点

步骤2: 节点结构设计

enum Color { RED, BLACK }; template<typename T> struct RBTreeNode { T data; // 节点存储的数据 Color color; RBTreeNode* parent; RBTreeNode* left; RBTreeNode* right; RBTreeNode(T val, Color c = RED) : data(val), color(c), parent(nullptr), left(nullptr), right(nullptr) {} };

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步骤3: 核心操作实现

3.1 左旋（Left Rotate）
template<typename T> void RBTree<T>::leftRotate(RBTreeNode<T>* x) { RBTreeNode<T>* y = x->right; x->right = y->left; if (y->left) y->left->parent = x; y->parent = x->parent; if (!x->parent) root = y; else if (x == x->parent->left) x->parent->left = y; else x->parent->right = y; y->left = x; x->parent = y; }
3.2 右旋（Right Rotate）
template<typename T> void RBTree<T>::rightRotate(RBTreeNode<T>* y) { RBTreeNode<T>* x = y->left; y->left = x->right; if (x->right) x->right->parent = y; x->parent = y->parent; if (!y->parent) root = x; else if (y == y->parent->left) y->parent->left = x; else y->parent->right = x; x->right = y; y->parent = x; }

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步骤4: 插入修复（Fixup）

插入后可能破坏红黑树性质，需通过颜色翻转和旋转修复：

template<typename T> void RBTree<T>::insertFixup(RBTreeNode<T>* z) { while (z->parent && z->parent->color == RED) { if (z->parent == z->parent->parent->left) { RBTreeNode<T>* uncle = z->parent->parent->right; if (uncle && uncle->color == RED) { // Case 1: 叔节点为红 z->parent->color = BLACK; uncle->color = BLACK; z->parent->parent->color = RED; z = z->parent->parent; } else { if (z == z->parent->right) { // Case 2: 三角型 z = z->parent; leftRotate(z); } // Case 3: 直线型 z->parent->color = BLACK; z->parent->parent->color = RED; rightRotate(z->parent->parent); } } else { // 对称操作（略） } } root->color = BLACK; // 根节点始终为黑 }

步骤5: 封装Set容器

template<typename Key> class Set { private: RBTree<Key> tree; // 红黑树实例 public: void insert(const Key& key) { tree.insert(key); } bool contains(const Key& key) { return tree.find(key) != nullptr; } // 迭代器、删除等操作（略） };

步骤6: 封装Map容器

通过存储键值对实现：

template<typename Key, typename Value> class Map { private: struct MapNode { Key key; Value value; bool operator<(const MapNode& other) const { return key < other.key; } }; RBTree<MapNode> tree; // 存储键值对 public: void insert(const Key& key, const Value& val) { tree.insert({key, val}); } Value& operator[](const Key& key) { auto node = tree.find({key, Value()}); if (!node) { insert(key, Value()); node = tree.find({key, Value()}); } return node->data.value; } };

总结

  1. 红黑树核心：旋转 + 颜色修复
  2. Set封装：直接存储单一类型数据
  3. Map封装：存储键值对，通过自定义比较规则排序

完整实现需补充删除操作、迭代器、边界检查等细节，但以上框架已覆盖核心逻辑。

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