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[数据结构]红黑树图解及其代码实现
一、红黑树的概念
红黑树，是一种二叉搜索树，但在每个结点上增加一个存储位表示结点的颜色，可以是Red或Black。 通过对任何一条从根到叶子的路径上各个结点着色方式的限制，红黑树确保没有一条路径会比其他路径长出俩倍，因而是接近平衡的。
二、红黑树的性质
- 每个结点不是红色就是黑色
- 根节点是黑色的
- 如果一个节点是红色的，则它的两个孩子结点是黑色的(没有连续的红色节点)
- 对于每个结点，从该结点到其所有后代叶结点的简单路径上，均包含相同数目的黑色结点(每条路径黑色节点的数量相同)
- 每个叶子结点都是黑色的(此处的叶子结点指的是空结点——NIL节点)
假设每条路径黑节点的数量是N，则：
N <= 任意路径长度 <= 2N
最短路径：全黑
最长路径：一黑一红最长路径不超过最短路径的2倍
AVL树：左右两边更均衡，高度更接近logN
满二多叉树：2^h - 1 = N
红黑树的两边没有那么均衡，整体高度：2 * logN假设红黑树中一条路径黑色节点的数量是X，高度为h，N为节点数量 则：X <= h <= 2X 2^X - 1 <= N <= 2^2X - 1 全黑满二叉树 ~ 一黑一红满二叉树- 1
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AVL严格平衡，效率logN，红黑树是近似平衡，效率是2 * logN
而AVL更加平衡是通过更多次旋转到达的
三、红黑树基本框架
enum Colour { RED, BLACK }; template <class K, class V> struct RBTreeNode { RBTreeNode<K, V>* _left; RBTreeNode<K, V>* _right; RBTreeNode<K, V>* _parent; pair<K, V> _kv; Colour _col; RBTreeNode(const pair<K, V>& kv) : _left(nullptr) , _right(nullptr) , _parent(nullptr) , _kv(kv) , _col(RED) {} }; template <class K, class V> struct RBTree { typedef RBTreeNode<K, V> Node; public: RBTree() : _root(nullptr) {} private: Node* _root; };- 1
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四、红黑树插入操作
前半部分插入按照线索二叉树的插入原则，比他大，插在右边，小就插在左边
在插入操作，我们是插入红结点还是黑节点？——红节点，因为插入了黑节点可能会影响其他路径了，问题比较严重，所以我们选择插入红色节点，然后再去进行调整
如图若22为新插入节点，并且是黑色，那么绿色路径有3个黑色节点，橙色路径有两个黑色节点，影响会更大一些bool Insert(const pair<K, V>& kv) { if (_root == nullptr) { _root = new Node(kv); _root->_col = BLACK; return true; } Node* parent = nullptr; Node* cur = _root; while (cur) { if (_root->_kv.first > kv.first) { parent = cur; cur = cur->_left; } else if (_root->_kv.first < kv.first) { parent = cur; cur = cur->_right; } else { return false; } } cur = new Node(kv); cur->_col = RED; if (parent->_kv.first > kv.first) { parent->_left = cur; } else { parent->_right = cur; } cur->_parent = parent; // 接下来控制平衡 }- 1
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在插入新节点后，要看红黑树性质是否被破坏
因为新节点的默认颜色是红色，因此：如果其双亲节点的颜色是黑色，没有违反红黑树任何性质，则不需要调整；但当新插入节点的双亲节点颜色为红色时，就违反了性质三不能有连在一起的红色节点，此时需要对红黑树分情况来讨论
约定:cur为当前节点，p为父节点，g为祖父节点，u为叔叔节点4.1 cur为红，p为红，g为黑，u存在且为红
bool Insert(const pair<K, V>& kv) { if (_root == nullptr) { _root = new Node(kv); _root->_col = BLACK; return true; } Node* parent = nullptr; Node* cur = _root; while (cur) { if (_root->_kv.first > kv.first) { parent = cur; cur = cur->_left; } else if (_root->_kv.first < kv.first) { parent = cur; cur = cur->_right; } else { return false; } } cur = new Node(kv); cur->_col = RED; if (parent->_kv.first > kv.first) { parent->_left = cur; } else { parent->_right = cur; } cur->_parent = parent; // 接下来控制平衡 while (parent && parent->_col == RED) { Node* grandFather = parent->_parent; if (parent == grandFather->_left) { Node* uncle = grandFather->_right; if (uncle && uncle->_col == RED) // 叔叔存在的情况 { parent->_col = uncle->_col = BLACK; grandFather->_col = RED; cur = grandFather; parent = cur->_parent; } } } _root->_col = BLACK; return true; }- 1
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4.2 cur为红，p为红，g为黑，u不存在/u存在且为黑 (单旋情况)
u情况说明：
如果u节点不存在，则cur一定是新插入节点，因为如果cur不是新插入节点，则cur和p一定有一个节点的颜色是黑色，这样子就不满足性质4（每条路径黑色节点个数要相同）
当u节点存在且为黑色的时候，cur节点的原先颜色一定是黑色，现在是红色的原因是子树在调整过程中，将cur节点由黑色改为了红色处理方法：
- 当左边高进行右旋，右边高进行左旋
- p和g进行变色，p变黑色、g变红色
4.2.1 左单旋
void RotateL(Node* parent) { Node* subR = parent->_right; Node* subRL = subR->_left; parent->_right = subRL; if (subRL) { subRL->_parent = parent; } Node* parentParent = parent->_parent; subR->_left = parent; parent->_parent = subR; if (_root == parent) { _root = subR; subR->_parent = nullptr; } else { if (parentParent->_left == parent) parentParent->_left = subR; else parentParent->_right = subR; subR->_parent = parentParent; } }- 1
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4.2.2 右单旋
void RotateR(Node* parent) { Node* subL = parent->_left; Node* subLR = subL->_right; parent->_left = subLR; if (subLR) subLR->_parent = parent; Node* parentParent = parent->_parent; subL->_right = parent; parent->_parent = subL; if (parent == _root) { _root = subL; _root->_parent = nullptr; } else { if (parentParent->_left == parent) parentParent->_left = subL; else parentParent->_right = subL; subL->_parent = parentParent; } }- 1
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4.3 cur为红，p为红，g为黑，u不存在/u存在且为黑 (双旋情况)
综上所述 我们代码可以进行完善bool Insert(const pair<K, V>& kv) { if (_root == nullptr) { _root = new Node(kv); _root->_col = BLACK; return true; } Node* parent = nullptr; Node* cur = _root; // 找到要插入的位置 while (cur) { if (_root->_kv.first > kv.first) { parent = cur; cur = cur->_left; } else if (_root->_kv.first < kv.first) { parent = cur; cur = cur->_right; } else { return false; } } // 找到该位置后进行插入 cur = new Node(kv); cur->_col = RED; if (parent->_kv.first > kv.first) { parent->_left = cur; } else { parent->_right = cur; } cur->_parent = parent; // 接下来控制平衡 // 只有插入的节点和双亲节点都是红色才需要去进行处理 while (parent && parent->_col == RED) { Node* grandFather = parent->_parent; if (parent == grandFather->_left) // parent在左侧的情况 { Node* uncle = grandFather->_right; // 情况1：叔叔存在且为红 if (uncle && uncle->_col == RED) { parent->_col = uncle->_col = BLACK; grandFather->_col = RED; cur = grandFather; parent = cur->_parent; } // 情况2：叔叔不存在/叔叔存在且为黑 else { /* 单旋 g p c 双旋 g p c */ if (cur == parent->_left) // 单旋 { RotateR(grandFather); parent->_col = BLACK; grandFather->_col = RED; } else // 双旋 { RotateL(parent); RotateR(grandFather); cur->_col = BLACK; grandFather->_col = RED; } break; // 旋转完后跳出循环 } } else // if (parent == grandFather->_left) // parent在右侧的情况 { Node* uncle = grandFather->_left; if (uncle && uncle->_col == RED) { // 变色+向上处理 parent->_col = uncle->_col = BLACK; grandFather->_col = RED; cur = grandFather; parent = cur->_parent; } else { /* 单旋 g p c 双旋 g p c */ if (cur == parent->_right) { RotateL(grandFather); parent->_col = BLACK; grandFather->_col = RED; } else { RotateR(parent); RotateL(grandFather); cur->_col = BLACK; grandFather->_col = RED; } break; } } } _root->_col = BLACK; return true; }- 1
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五、补充
5.1 求红黑树高度
int Height() { return _Height(_root); } int _Height(Node* root) { if (root == NULL) { return 0; } int leftHight = _Height(root->_left); int rightHight = _Height(root->_right); return leftHight > rightHight ? leftHight + 1 : rightHight + 1; }- 1
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5.2 判断一棵树是否为红黑树
bool IsBalance() { if (_root && _root->_col == RED) { cout << "根节点不是黑色" << endl; return false; } int banchmark = 0; // 基准值—计算一边的黑色节点数目 Node* left = _root; while (left) { if (left->_col == BLACK) ++banchmark; left = left->_left; } int blackNum = 0; return _IsBalance(_root, banchmark, blackNum); } bool _IsBalance(Node* root, int banchmark, int blackNum) { if (root == nullptr) { if (banchmark != blackNum) { cout << "黑色节点数量不相同" << endl; return false; } return true; } if (root->_col == RED && root->_parent->_col == RED) { cout << "存在连续的红色节点" << endl; return false; } if (root->_col == BLACK) blackNum++; return _IsBalance(root->_left, banchmark, blackNum) && _IsBalance(root->_right, banchmark, blackNum); }- 1
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六、红黑树与AVL数的比较
红黑树和AVL树都是高效的平衡二叉树，增删改查的时间复杂度都是O(logN)，红黑树不追求绝对平衡，其只需保证最长路径不超过最短路径的2倍，相对而言，降低了插入和旋转的次数，所以在经常进行增删的结构中性能比AVL树更优，而且红黑树实现比较简单，所以实际运用中红黑树更多
红黑树实际中应用有：
- map/set 、 multi_map/multi_set
- java库
- linux内核
- 其他库
七、完整代码
#pragma once #include#include #include using namespace std; enum Colour { RED, BLACK }; template <class K, class V> struct RBTreeNode { RBTreeNode<K, V>* _left; RBTreeNode<K, V>* _right; RBTreeNode<K, V>* _parent; pair<K, V> _kv; Colour _col; RBTreeNode(const pair<K, V>& kv) : _left(nullptr) , _right(nullptr) , _parent(nullptr) , _col(RED) , _kv(kv) {} }; template <class K, class V> struct RBTree { typedef RBTreeNode<K, V> Node; public: RBTree() : _root(nullptr) {} bool Insert(const pair<K, V>& kv) { if (_root == nullptr) { _root = new Node(kv); _root->_col = BLACK; return true; } Node* parent = nullptr; Node* cur = _root; // 找到要插入的位置 while (cur) { if (_root->_kv.first > kv.first) { parent = cur; cur = cur->_left; } else if (_root->_kv.first < kv.first) { parent = cur; cur = cur->_right; } else { return false; } } // 找到该位置后进行插入 cur = new Node(kv); cur->_col = RED; if (parent->_kv.first > kv.first) { parent->_left = cur; } else { parent->_right = cur; } cur->_parent = parent; // 接下来控制平衡 // 只有插入的节点和双亲节点都是红色才需要去进行处理 while (parent && parent->_col == RED) { Node* grandFather = parent->_parent; if (parent == grandFather->_left) // parent在左侧的情况 { Node* uncle = grandFather->_right; // 情况1：叔叔存在且为红 if (uncle && uncle->_col == RED) { parent->_col = uncle->_col = BLACK; grandFather->_col = RED; cur = grandFather; parent = cur->_parent; } // 情况2：叔叔不存在/叔叔存在且为黑 else { /* 单旋 g p c 双旋 g p c */ if (cur == parent->_left) // 单旋 { RotateR(grandFather); parent->_col = BLACK; grandFather->_col = RED; } else // 双旋 { RotateL(parent); RotateR(grandFather); cur->_col = BLACK; grandFather->_col = RED; } break; // 旋转完后跳出循环 } } else // if (parent == grandFather->_left) // parent在右侧的情况 { Node* uncle = grandFather->_left; if (uncle && uncle->_col == RED) { // 变色+向上处理 parent->_col = uncle->_col = BLACK; grandFather->_col = RED; cur = grandFather; parent = cur->_parent; } else { /* 单旋 g p c 双旋 g p c */ if (cur == parent->_right) { RotateL(grandFather); parent->_col = BLACK; grandFather->_col = RED; } else { RotateR(parent); RotateL(grandFather); cur->_col = BLACK; grandFather->_col = RED; } break; } } } _root->_col = BLACK; return true; } void RotateL(Node* parent) { Node* subR = parent->_right; Node* subRL = subR->_left; parent->_right = subRL; if (subRL) { subRL->_parent = parent; } Node* parentParent = parent->_parent; subR->_left = parent; parent->_parent = subR; if (_root == parent) { _root = subR; subR->_parent = nullptr; } else { if (parentParent->_left == parent) parentParent->_left = subR; else parentParent->_right = subR; subR->_parent = parentParent; } } void RotateR(Node * parent) { Node* subL = parent->_left; Node* subLR = subL->_right; parent->_left = subLR; if (subLR) subLR->_parent = parent; Node* parentParent = parent->_parent; subL->_right = parent; parent->_parent = subL; if (parent == _root) { _root = subL; _root->_parent = nullptr; } else { if (parentParent->_left == parent) parentParent->_left = subL; else parentParent->_right = subL; subL->_parent = parentParent; } } void InOrder() { _InOrder(_root); } void _InOrder(Node* root) { if (root == NULL) return; _InOrder(root->_left); cout << root->_kv.first << " -> " << root->_kv.second << endl; _InOrder(root->_right); } int Height() { return _Height(_root); } int _Height(Node* root) { if (root == NULL) { return 0; } int leftHight = _Height(root->_left); int rightHight = _Height(root->_right); return leftHight > rightHight ? leftHight + 1 : rightHight + 1; } // 判断是否为红黑树(防止只是单纯的二叉搜索树) bool IsBalance() { if (_root && _root->_col == RED) { cout << "根节点不是黑色" << endl; return false; } int banchmark = 0; // 基准值—计算一边的黑色节点数目 Node* left = _root; while (left) { if (left->_col == BLACK) ++banchmark; left = left->_left; } int blackNum = 0; return _IsBalance(_root, banchmark, blackNum); } bool _IsBalance(Node* root, int banchmark, int blackNum) { if (root == nullptr) { if (banchmark != blackNum) { cout << "黑色节点数量不相同" << endl; return false; } return true; } if (root->_col == RED && root->_parent->_col == RED) { cout << "存在连续的红色节点" << endl; return false; } if (root->_col == BLACK) blackNum++; return _IsBalance(root->_left, banchmark, blackNum) && _IsBalance(root->_right, banchmark, blackNum); } private: Node* _root; }; void TestRBTree() { RBTree<int, int> t; //int a[] = {5,4,3,2,1,0}; //int a[] = { 16, 3, 7, 11, 9, 26, 18, 14, 15 }; int a[] = { 4, 2, 6, 1, 3, 5, 15, 7, 16, 14 }; for (auto e : a) { t.Insert(make_pair(e, e)); /*if (!t.IsBalance()) { cout << "Insert" << e << endl; }*/ } t.InOrder(); cout << t.IsBalance() << endl; cout << "高度:" << t.Height() << endl; } - 1
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- 原文地址：https://blog.csdn.net/weixin_51304981/article/details/126343803
