算法导论之红黑树

红黑树定义：是一棵二叉搜索树，它在每个结点上增加了一个存储位来表示结点的颜色，可以是红色也可以是黑色。通过对任意一条从根到叶子的简单路径上各个结点的颜色进行约束，红黑树确保没有一条路径会比其他路径长2倍，因而是近似于平衡的。

红黑树的性质：

1.每个节点或者是红色的，或者是黑色的。

2.根结点是黑色的。

3.每个叶子结点都是黑色的。

4.如果一个结点的颜色是红色的，则它的两个子结点都是黑色的。

5.对每个结点，从该结点到其他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。

C语言实现代码如下：

#include <stdio.h>struct RedBlackNode{int value;char color;struct RedBlackNode *right;struct RedBlackNode *left;struct RedBlackNode *parent;};typedef struct RedBlackNode RBNode;//定义全局变量当做叶子结点 RBNode *nil;//左旋转 RBNode *LEFT_ROTATE(RBNode *root,RBNode *x){RBNode *y=x->right;x->right=y->left;if(y->left!=nil){y->left->parent=x;}y->parent=x->parent;if(x->parent==nil){root=y;}else if(x==x->parent->left){x->parent->left=y;}else{x->parent->right=y;}y->left=x;x->parent=y;return root;}//右旋转 RBNode *RIGHT_ROTATE(RBNode *root,RBNode *x){RBNode *y=x->left;x->left=y->right;if(y->right!=nil){y->right->parent=x;}y->parent=x->parent;if(x->parent==nil){root=y;}else if(x==x->parent->right){x->parent->right=y;}else{x->parent->left=y;}y->right=x;x->parent=y;return root;}//用树v代替树u RBNode *RB_TRANSPLANT(RBNode *root,RBNode *u,RBNode *v){if(u->parent==nil){root=v;}else if(u==u->parent->left){u->parent->left=v;}else{u->parent->right=v;}v->parent=u->parent;return root;}//查找结点z下的最小值的结点 RBNode *TREE_MINIMUM(RBNode *z){RBNode *rs=z;while(rs->left!=nil){rs=rs->left;}return rs;}//删除后对红黑树进行调整 RBNode *RB_DELETE_FIXUP(RBNode *root,RBNode *x){RBNode *w = nil;while(x!=root && x->color=='B'){if(x==x->parent->left){w = x->parent->right;if(w->color=='R'){w->color='B';x->parent->color='R';root = LEFT_ROTATE(root,x->parent);w=x->parent->right;}if(w->left->color=='B' && w->right->color=='B'){w->color='R';x=x->parent;}else {if(w->right->color=='B'){w->left->color='B';w->color='R';root = RIGHT_ROTATE(root,w);w = x->parent->right;}w->color=x->parent->color;x->parent->color='B';w->right->color='B';root = LEFT_ROTATE(root,x->parent);x = root;}}else {RBNode *w = x->parent->left;if(w->color=='R'){w->color='B';x->parent->color='R';root = RIGHT_ROTATE(root,x->parent);w=x->parent->left;}if(w->right->color=='B' && w->left->color=='B'){w->color='R';x=x->parent;}else {if(w->left->color=='B'){w->right->color='B';w->color='R';root = LEFT_ROTATE(root,w);w = x->parent->left;}w->color=x->parent->color;x->parent->color='B';w->left->color='B';root = RIGHT_ROTATE(root,x->parent);x = root;}}}x->color='B';return root;}//删除红黑树中某一结点 RBNode *RB_DELETE(RBNode *root,RBNode *z){RBNode *y=z;RBNode *x=nil;char Y_ORIGINAL_COLOR=y->color;if(z->left==nil){x = z->right;root=RB_TRANSPLANT(root,z,z->right);}else if(z->right==nil){x = z->left;root = RB_TRANSPLANT(root,z,z->left);}else{y=TREE_MINIMUM(z->right);Y_ORIGINAL_COLOR=y->color;x=y->right;if(y->parent==z){x->parent=y;}else{root = RB_TRANSPLANT(root,y,y->right);y->right=z->right;y->right->parent=y;}root = RB_TRANSPLANT(root,z,y);y->left=z->left;y->left->parent=y;y->color=z->color;}if(Y_ORIGINAL_COLOR=='B'){root = RB_DELETE_FIXUP(root,x);}return root;}//插入某结点后的调整RBNode *RB_INSERT_FIXUP(RBNode *root,RBNode *z){while(z->parent->color=='R'){if(z->parent==z->parent->parent->left){RBNode *y=z->parent->parent->right;if(y->color=='R'){z->parent->color='B';y->color='B';z->parent->parent->color='R';z=z->parent->parent;}else{if(z==z->parent->right){z=z->parent;root = LEFT_ROTATE(root,z);}z->parent->color='B';z->parent->parent->color='R';root = RIGHT_ROTATE(root,z->parent->parent);}}else {RBNode *y=z->parent->parent->left;if(y->color=='R'){z->parent->color='B';y->color='B';z->parent->parent->color='R';z=z->parent->parent;}else{if(z==z->parent->left){z=z->parent;root = RIGHT_ROTATE(root,z);}z->parent->color='B';z->parent->parent->color='R';root = LEFT_ROTATE(root,z->parent->parent);}}}root->color='B';return root;}//插入一个节点 RBNode *RB_INSERT(RBNode *root,RBNode *z){RBNode *x=root,*y=nil;while(x!=nil){y=x;if(z->value<x->value){x=x->left;}else{x=x->right;}}z->parent=y;if(y==nil){root=z;}else if(z->value<y->value){y->left=z;}else{y->right=z;}z->left=nil;z->right=nil;z->color='R';root = RB_INSERT_FIXUP(root,z);return root;}//中序遍历树void inOrder(RBNode*tree) {if (tree != nil) {inOrder(tree->left);printf("%d,%c\n",tree->value,tree->color);inOrder(tree->right);}} //选择一个节点RBNode*selectNode(RBNode*root,int key){RBNode*temp = root;while (temp!=nil) {if (temp->value>key) {temp=temp->left;}else if (temp->value<key) {temp=temp->right;}else {return temp;}}return temp;} int main(int argc, char *argv[]){nil=(RBNode *)malloc(sizeof(RBNode));nil->color='B';RBNode *root = nil;int a[] = { 41, 38, 31, 12, 19, 8 };int i = 0,n=sizeof(a)/sizeof(int);for (; i < n; i++) {RBNode *z = (RBNode *)malloc(sizeof(RBNode));z->value = a[i];root = RB_INSERT(root, z);}inOrder(root);printf("\n测试删除根节点：\n");RBNode *deleteNode = selectNode(root,38);root = RB_DELETE(root,deleteNode);inOrder(root);return 0;}

Java语言实现代码如下：

package algorithms;public class RB_Tree {public rb_node T = null;private static String RED = "red";private static String BLACK = "black";private static rb_node T_nil = new rb_node(BLACK);public static class rb_node {String color;int key;rb_node left;rb_node right;rb_node p;public rb_node() {this.color = RED;this.key = Integer.MIN_VALUE;this.left = T_nil;this.right = T_nil;this.p = T_nil;}public rb_node(String color) {this.color = color;this.key = Integer.MIN_VALUE;this.left = null;this.right = null;this.p = null;}}private static rb_node LEFT_ROTATE(rb_node T, rb_node x) {rb_node y = x.right;x.right = y.left;if (y.left != T_nil) {y.left.p = x;}y.p = x.p;if (x.p == T_nil) {T = y;} else if (x == x.p.left) {x.p.left = y;} else {x.p.right = y;}y.left = x;x.p = y;return T;}private static rb_node RIGHT_ROTATE(rb_node T, rb_node x) {rb_node y = x.left;x.left = y.right;if (y.right != T_nil) {y.right.p = x;}y.p = x.p;if (x.p == T_nil) {T = y;} else if (x == x.p.left) {x.p.left = y;} else {x.p.right = y;}y.right = x;x.p = y;return T;}public rb_node RB_INSERT(rb_node T, rb_node z) {rb_node y = T_nil;rb_node x = T;while (x != T_nil) {y = x;if (z.key < x.key) {x = x.left;} else {x = x.right;}}z.p = y;if (y == T_nil) {T = z;} else if (z.key < y.key) {y.left = z;} else {y.right = z;}z.color = RED;T = RB_INSERT_FIXUP(T, z);return T;}private static rb_node RB_INSERT_FIXUP(rb_node T, rb_node z) {while (z.p.color == RED) {if (z.p == z.p.p.left) {rb_node y = z.p.p.right;if (y.color == RED) {z.p.color = BLACK;y.color = BLACK;z.p.p.color = RED;z = z.p.p;} else {if (z == z.p.right) {z = z.p;T = LEFT_ROTATE(T, z);}z.p.color = BLACK;z.p.p.color = RED;T = RIGHT_ROTATE(T, z.p.p);}} else {rb_node y = z.p.p.left;if (y.color == RED) {z.p.color = BLACK;y.color = BLACK;z.p.p.color = RED;z = z.p.p;} else {if (z == z.p.left) {z = z.p;T = RIGHT_ROTATE(T, z);}z.p.color = BLACK;z.p.p.color = RED;T = LEFT_ROTATE(T, z.p.p);}}}T.color = BLACK;return T;}public rb_node RB_DELETE(rb_node T, rb_node z) {rb_node y = z, x;String y_original_color = y.color;if (z.left == T_nil) {x = z.right;T = RB_TRANSPLANT(T, z, z.right);} else if (z.right == T_nil) {x = z.left;T = RB_TRANSPLANT(T, z, z.left);} else {y = TREE_MINIMUM(z.right);y_original_color = y.color;x = y.right;if (y.p == z) {x.p = y;} else {T = RB_TRANSPLANT(T, y, y.right);y.right = z.right;y.right.p = y;}T = RB_TRANSPLANT(T, z, y);y.left = z.left;y.left.p = y;y.color = z.color;}if (y_original_color == BLACK) {T = RB_DELETE_FIXUP(T, x);}return T;}private static rb_node RB_DELETE_FIXUP(rb_node T, rb_node x) {rb_node w = T_nil;while (x != T && x.color == BLACK) {if (x == x.p.left) {w = x.p.right;if (w.color == RED) {w.color = BLACK;x.p.color = RED;T = LEFT_ROTATE(T, x.p);w = x.p.right;}if (w.left.color == BLACK && w.right.color == BLACK) {w.color = RED;x = x.p;} else {if (w.right.color == BLACK) {w.left.color = BLACK;w.color = RED;T = RIGHT_ROTATE(T, w);w = x.p.right;}x.p.color = BLACK;w.color = RED;T = LEFT_ROTATE(T, x.p);x = T;}} else {w = x.p.left;if (w.color == RED) {w.color = BLACK;x.p.color = RED;T = RIGHT_ROTATE(T, x.p);w = x.p.left;}if (w.left.color == BLACK && w.right.color == BLACK) {w.color = RED;x = x.p;} else {if (w.left.color == BLACK) {w.right.color = BLACK;w.color = RED;T = LEFT_ROTATE(T, w);w = x.p.left;}x.p.color = BLACK;w.color = RED;T = RIGHT_ROTATE(T, x.p);x = T;}}}x.color = BLACK;return T;}private static rb_node TREE_MINIMUM(rb_node z) {while (z.left != T_nil) {z = z.left;}return z;}private static rb_node RB_TRANSPLANT(rb_node T, rb_node u, rb_node v) {if (u.p == T_nil) {T = v;} else if (u == u.p.left) {u.p.left = v;} else {u.p.right = v;}v.p = u.p;return T;}public rb_node selectNode(int key) {rb_node temp = this.T;while (temp != T_nil) {if (temp.key > key) {temp = temp.left;} else if (temp.key < key) {temp = temp.right;} else {return temp;}}return temp;}public RB_Tree() {T = T_nil;}public void preOrder(rb_node tree) {if (tree != T_nil) {System.out.print(tree.key + "->" + tree.color + "\t");preOrder(tree.left);preOrder(tree.right);}}public void inOrder(rb_node tree) {if (tree != T_nil) {inOrder(tree.left);System.out.print(tree.key + "->" + tree.color + "\t");inOrder(tree.right);}}public void postOrder(rb_node tree) {if (tree != T_nil) {postOrder(tree.left);postOrder(tree.right);System.out.print(tree.key + "," + tree.color);}}public static void main(String[] args) {String pre = "30 10 20 60 40 50 80 70 90 ";String in = "10 20 30 40 50 60 70 80 90 ";String post = "20 10 50 40 70 90 80 60 30 ";RB_Tree RB = new RB_Tree();// int a[] = { 10, 40, 30, 60, 90, 70, 20, 50, 80 };int a[] = { 41, 38, 31, 12, 19, 8 };for (int i = 0; i < a.length; i++) {rb_node z = new rb_node();z.key = a[i];RB.T = RB.RB_INSERT(RB.T, z);}RB.inOrder(RB.T);System.out.println();// 测试删除数据rb_node z = RB.selectNode(38);RB.T = RB.RB_DELETE(RB.T, z);RB.inOrder(RB.T);}}

在以上的两个版本中，都用了一个哨兵来标记叶子结点（即null结点）。

参考资料：
算法导论

备注：
转载请注明出处：http://blog.csdn.net/wsyw126/article/details/51448271
作者：WSYW12