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。
红黑树的性质1. 每个结点不是黑色就是红色
2. 根结点是黑色的
3. 如果一个节点是红色的,那么它的两个孩子结点是黑色的
4. 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,均包含数目相同的黑色结点
5. 每个叶子节点都是黑色的。(此处叶子结点指的是空结点)
红黑树能保证最长路径中结点个数不会超过最短路径结点个数的两倍
红黑树的实现enum Color { RED, BLACK }; templatestruct RBTreeNode { RBTreeNode * _left; RBTreeNode * _right; RBTreeNode * _parent; pair _kv; Color _col; RBTreeNode(const pair & kv = pair ()) :_kv(kv) , _left(nullptr) , _right(nullptr) , _parent(nullptr) , _col(RED) {} };
结点
树
二叉树的插入1. 按照二叉搜索的树规则插入新节点
public: bool Insert(const pair检查新结点插入后,红黑树的性质是否遭到破坏& kv) { if (_root == nullptr) { _root = new Node(kv); _root->_col = BLACK; return true; } Node* parent = nullptr; Node* cur = _root; while (cur) { if (cur->_kv.first < kv.first) { parent = cur; cur = cur->_right; } else if (cur->_kv.first > kv.first) { parent = cur; cur = cur->left; } else { return false; } } cur = new Node(kv); cur->_col = RED; if (parent->_kv.first < kv.first) { parent->_right = cur; } else { parent->_left = cur; } cur->_parent = parent; while (parent && parent->_col == RED) { Node* grandfather = parent->_parent; assert(grandfather); if (grandfather->_left == parent) { Node* uncle = grandfather->_right; //uncle exist and uncle is red if (uncle && uncle->_col == RED) { parent->_col = uncle->_col = BLACK; grandfather->_col = RED; cur = grandfather; parent = cur->_parent; } else // uncle does not exist or uncle is black { if (cur == parent->_left) { RotateR(grandfather); parent->_col = BLACK; grandfather->_col = RED; } else { RotateL(parent); RotateR(grandfather); cur->_col = BLACK; grandfather->_col = RED; } break; } } else { Node* uncle = grandfather->_left; //uncle exist and uncle is red if (uncle && uncle->_col == RED) { parent->_col = uncle->_col = BLACK; grandfather->_col = RED; cur = grandfather; parent = cur->_parent; } else //uncle does not exist or uncle is black { 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; }
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
情况一:cur为红,p为红,g为黑,u存在且为红
p u 变红,g变黑
情况二: cur为红,p为红,g为黑,u不存在/u存在且为黑
p为g的左孩子,cur为p的左孩子,则进行右单旋转;相反,
p为g的右孩子,cur为p的右孩子,则进行左单旋转
p、g变色–p变黑,g变红
情况三:cur为红,p为红,g为黑,u不存在/u存在且为黑
p为g的左孩子,cur为p的右孩子,则针对p做左单旋转;相反,
p为g的右孩子,cur为p的左孩子,则针对p做右单旋转
则转换成了情况2