linux内核的红黑树

很多博文都介绍过红黑树，linux内核对红黑树的实现无疑是经典中的经典，那么就来看看这经典中的经典是怎么样的。

1. 回忆树的基本知识

什么是树，有很多博文中会罗列几条规则，满足那几条规则的数据结构就称为树，这种定义方式是很数学化的表达方式。这里抓出来两条最重要的规则：

1)树的节点有且仅有一个父节点；
2)任何非页节点都可以分为若干不相交的子树。
其实这两条规则都是表述同一个含义，就是树是不相交的，节点之间不能构成回环。
常用到的是二叉树，二叉树对节点孩子的个数进行了约束，即节点的孩子个数不超过两个。二叉树的一个重要应用是排序与查找，所以就有了排序二叉树或者说查找二叉树。排序二叉树的左子树的节点的值都小于根节点，右子树的的节点的值都大于根节点。在红黑树的左旋和右旋操作中要用到该性质，下图所示，即为一个左旋转。问题是为什么D可以作为A的右孩子，因为D总是比A大的，这就是二叉树性质的运用。

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旋转前

旋转后

图1 二叉树性质的运用
二叉树在不幸的时候会退化为链表结构，为了这个有了平衡二叉树，红黑树就是我们常用的平衡二叉树之一。是的，今天的主角就是红黑树。

2. 什么是红黑树

linux源码(linux 5.13)中对红黑树的定义是如下的描述，那么黑节点的孩子可以为黑吗？这是完全有可能的。

1) A node is either red or black
2) The root is black //根节点为黑
3) All leaves (NULL) are black //所有的叶节点为黑
4) Both children of every red node are black //所有红节点的孩子为黑
5) Every simple path from root to leaves contains the same numberof black nodes.

3. linux源码中红黑树的实现

我们从使用者的角度来看红黑树是如何实现的。在linux中，双向链表、哈希链表、红黑树的实现都有一个共同点，那就是数据结构只用于连接，而不存放数据。需要使用这些连接的数据结构，其中必须包含相应连接的成员。不论是链表还是树，都有根节点。红黑树的根节点，就是一个指向节点的指针。

struct rb_root {struct rb_node *rb_node;
};
struct rb_node {unsigned long  __rb_parent_color;struct rb_node *rb_right;struct rb_node *rb_left;
} __attribute__((aligned(sizeof(long))));//最低的4位始终为0，可作其它用途

定义一个树的根节点使用宏RB_ROOT，rb_node指针被初始化为NULL。RB_ROOT返回的是一个rb_root实例，由于one_tree是数组，所以右边需要加花括号。one_tree是数组名，是用于赋值给指针。root_tree就是指向红黑树根节点的指针，当有很多红黑树时，只需把root_tree当做一个数组看待。

struct rb_root one_tree[1] = { RB_ROOT };
struct rb_root *root_tree = one_tree;

在红黑树中查找的样例如下所示，linux内核并没有采用提供回调函数的方式让使用者直接使用，而是需要开发者自己定义。在如下的流程中，开发者可以改变比较大小的方式，还可以根据需要增加一些其它的处理流程。红黑树的查找与普通的二叉排序树并没有区别。

struct mytype *my_search(struct rb_root *root, char *string){struct rb_node *node = root->rb_node; //rb_node是一个指针while (node) {//参数的第三个参数node是结构体mytype中的成员名称struct mytype *data = container_of(node, struct mytype, node);int result;result = strcmp(string, data->keystring);if (result < 0)node = node->rb_left;else if (result > 0)node = node->rb_right;elsereturn data;}return NULL;}

有差别的是在红黑树中插入节点。与查找不同，一开始定义的指针new是一个指向根节点的指针的指针。*new的含义即是指向节点的指针。为什么要用指向指针的指针呢？目的是为了在通过函数传参时能修改指针的值。rb_link_node将data->node中的数据成员进行了修改，__rb_parent_color的颜色修改为其父节点的颜色，左孩子和右孩子都设置为NULL，并将new指针指向node指针。那么new指针的含义是什么，它可能是左孩子，也可能是右孩子，更确切的含义是当前节点要插入的位置。

int my_insert(struct rb_root *root, struct mytype *data){struct rb_node **new = &(root->rb_node), *parent = NULL;/* Figure out where to put new node */while (*new) {struct mytype *this = container_of(*new, struct mytype, node);int result = strcmp(data->keystring, this->keystring);parent = *new;if (result < 0)new = &((*new)->rb_left);else if (result > 0)new = &((*new)->rb_right);elsereturn FALSE;}/* Add new node and rebalance tree. */rb_link_node(&data->node, parent, new);rb_insert_color(&data->node, root);return TRUE;}

rb_insert_color函数是红黑树相对于普通二叉排序树最大的差别。这个函数中平衡二叉树的方式主要有3种：a)改变颜色，0表示red，1表示black。对于对齐的节点，低4位总是0，所以默认的情况下是红色的节点。那么对于父节点是black的情况，直接插入即可，无需改变颜色。如果父节点是红色则需要改变颜色。b)左旋转，将一个节点的右孩子变为父节点，节点本身变为左孩子。c)右旋转，将一个节点的做孩子变为父节点，节点本身变为右孩子。

static __always_inline void
__rb_insert(struct rb_node *node, struct rb_root *root,void (*augment_rotate)(struct rb_node *old, struct rb_node *new))
{// 节点在没有上色之前__rb_parent_color表示指向父节点的指针，且低位未被标记struct rb_node *parent = rb_red_parent(node), *gparent, *tmp;while (true) {/** Loop invariant: node is red.*/if (unlikely(!parent)) {/** The inserted node is root. Either this is the* first node, or we recursed at Case 1 below and* are no longer violating 4).*/// 根节点直接设置为黑色，1表示黑色，__rb_parent_color被设置为1.// 因为根节点没有父节点，所以第二个参数为NULLrb_set_parent_color(node, NULL, RB_BLACK);break;}/** If there is a black parent, we are done.* Otherwise, take some corrective action as,* per 4), we don't want a red root or two* consecutive red nodes.*/// 父节点为黑色，无需做任何处理，颜色呢？if(rb_is_black(parent))break;// 父节点为红色才能走到此处为红色意味着低位为0.gparent = rb_red_parent(parent);tmp = gparent->rb_right;if (parent != tmp) {    /* parent == gparent->rb_left */if (tmp && rb_is_red(tmp)) {// 父节点存在兄弟/** Case 1 - node's uncle is red (color flips).**       G            g*      / \          / \*     p   u  -->   P   U*    /            /*   n            n** However, since g's parent might be red, and* 4) does not allow this, we need to recurse* at g.*/// 父节点及其兄弟的parent都设置为黑，parent和兄弟不能直接访问其父节点了rb_set_parent_color(tmp, gparent, RB_BLACK);rb_set_parent_color(parent, gparent, RB_BLACK);node = gparent;parent = rb_parent(node);//最低两位被设为0，parent已经被设置为祖父的parentrb_set_parent_color(node, parent, RB_RED);continue;}tmp = parent->rb_right;if (node == tmp) {/** Case 2 - node's uncle is black and node is* the parent's right child (left rotate at parent).**      G             G*     / \           / \*    p   U  -->    n   U*     \           /*      n         p** This still leaves us in violation of 4), the* continuation into Case 3 will fix that.*/tmp = node->rb_left;WRITE_ONCE(parent->rb_right, tmp);WRITE_ONCE(node->rb_left, parent);if (tmp)rb_set_parent_color(tmp, parent,RB_BLACK);rb_set_parent_color(parent, node, RB_RED);augment_rotate(parent, node);parent = node;tmp = node->rb_right;}/** Case 3 - node's uncle is black and node is* the parent's left child (right rotate at gparent).**        G           P*       / \         / \*      p   U  -->  n   g*     /                 \*    n                   U*/WRITE_ONCE(gparent->rb_left, tmp); /* == parent->rb_right */WRITE_ONCE(parent->rb_right, gparent);if (tmp)rb_set_parent_color(tmp, gparent, RB_BLACK);__rb_rotate_set_parents(gparent, parent, root, RB_RED);augment_rotate(gparent, parent);break;} else {tmp = gparent->rb_left;if (tmp && rb_is_red(tmp)) {/* Case 1 - color flips */rb_set_parent_color(tmp, gparent, RB_BLACK);rb_set_parent_color(parent, gparent, RB_BLACK);node = gparent;parent = rb_parent(node);rb_set_parent_color(node, parent, RB_RED);continue;}tmp = parent->rb_left;if (node == tmp) {/* Case 2 - right rotate at parent */tmp = node->rb_right;// 非锁的方式读时确保不是致命的WRITE_ONCE(parent->rb_left, tmp);WRITE_ONCE(node->rb_right, parent);if (tmp)rb_set_parent_color(tmp, parent,RB_BLACK);rb_set_parent_color(parent, node, RB_RED);augment_rotate(parent, node);parent = node;tmp = node->rb_left;}/* Case 3 - left rotate at gparent */WRITE_ONCE(gparent->rb_right, tmp); /* == parent->rb_left */WRITE_ONCE(parent->rb_left, gparent);if (tmp)rb_set_parent_color(tmp, gparent, RB_BLACK);__rb_rotate_set_parents(gparent, parent, root, RB_RED);augment_rotate(gparent, parent);break;}}
}