tools/lib/rbtree.c - linux - Git at Google

 /*
   Red Black Trees
   (C) 1999  Andrea Arcangeli <andrea@suse.de>
   (C) 2002  David Woodhouse <dwmw2@infradead.org>
   (C) 2012  Michel Lespinasse <walken@google.com>

   This program is free software; you can redistribute it and/or modify
   it under the terms of the GNU General Public License as published by
   the Free Software Foundation; either version 2 of the License, or
   (at your option) any later version.

   This program is distributed in the hope that it will be useful,
   but WITHOUT ANY WARRANTY; without even the implied warranty of
   MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
   GNU General Public License for more details.

   You should have received a copy of the GNU General Public License
   along with this program; if not, write to the Free Software
   Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA  02111-1307  USA

   linux/lib/rbtree.c
 */

 #include <linux/rbtree_augmented.h>

 /*
  * red-black trees properties:  http://en.wikipedia.org/wiki/Rbtree
  *
  *  1) A node is either red or black
  *  2) The root is black
  *  3) All leaves (NULL) are black
  *  4) Both children of every red node are black
  *  5) Every simple path from root to leaves contains the same number
  *     of black nodes.
  *
  *  4 and 5 give the O(log n) guarantee, since 4 implies you cannot have two
  *  consecutive red nodes in a path and every red node is therefore followed by
  *  a black. So if B is the number of black nodes on every simple path (as per
  *  5), then the longest possible path due to 4 is 2B.
  *
  *  We shall indicate color with case, where black nodes are uppercase and red
  *  nodes will be lowercase. Unknown color nodes shall be drawn as red within
  *  parentheses and have some accompanying text comment.
  */

 static inline void rb_set_black(struct rb_node *rb)
 {
 	rb->__rb_parent_color |= RB_BLACK;
 }

 static inline struct rb_node *rb_red_parent(struct rb_node *red)
 {
 	return (struct rb_node *)red->__rb_parent_color;
 }

 /*
  * Helper function for rotations:
  * - old's parent and color get assigned to new
  * - old gets assigned new as a parent and 'color' as a color.
  */
 static inline void
 __rb_rotate_set_parents(struct rb_node *old, struct rb_node *new,
 			struct rb_root *root, int color)
 {
 	struct rb_node *parent = rb_parent(old);
 	new->__rb_parent_color = old->__rb_parent_color;
 	rb_set_parent_color(old, new, color);
 	__rb_change_child(old, new, parent, root);
 }

 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))
 {
 	struct rb_node *parent = rb_red_parent(node), *gparent, *tmp;

 	while (true) {
 		/*
 		 * Loop invariant: node is red
 		 *
 		 * If there is a black parent, we are done.
 		 * Otherwise, take some corrective action as we don't
 		 * want a red root or two consecutive red nodes.
 		 */
 		if (!parent) {
 			rb_set_parent_color(node, NULL, RB_BLACK);
 			break;
 		} else if (rb_is_black(parent))
 			break;

 		gparent = rb_red_parent(parent);

 		tmp = gparent->rb_right;
 		if (parent != tmp) {	/* parent == gparent->rb_left */
 			if (tmp && rb_is_red(tmp)) {
 				/*
 				 * Case 1 - 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.
 				 */
 				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_right;
 			if (node == tmp) {
 				/*
 				 * Case 2 - 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.
 				 */
 				parent->rb_right = tmp = node->rb_left;
 				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 - right rotate at gparent
 			 *
 			 *        G           P
 			 *       / \         / \
 			 *      p   U  -->  n   g
 			 *     /                 \
 			 *    n                   U
 			 */
 			gparent->rb_left = tmp;  /* == parent->rb_right */
 			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 */
 				parent->rb_left = tmp = node->rb_right;
 				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 */
 			gparent->rb_right = tmp;  /* == parent->rb_left */
 			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;
 		}
 	}
 }

 /*
  * Inline version for rb_erase() use - we want to be able to inline
  * and eliminate the dummy_rotate callback there
  */
 static __always_inline void
 ____rb_erase_color(struct rb_node *parent, struct rb_root *root,
 	void (*augment_rotate)(struct rb_node *old, struct rb_node *new))
 {
 	struct rb_node *node = NULL, *sibling, *tmp1, *tmp2;

 	while (true) {
 		/*
 		 * Loop invariants:
 		 * - node is black (or NULL on first iteration)
 		 * - node is not the root (parent is not NULL)
 		 * - All leaf paths going through parent and node have a
 		 *   black node count that is 1 lower than other leaf paths.
 		 */
 		sibling = parent->rb_right;
 		if (node != sibling) {	/* node == parent->rb_left */
 			if (rb_is_red(sibling)) {
 				/*
 				 * Case 1 - left rotate at parent
 				 *
 				 *     P               S
 				 *    / \             / \
 				 *   N   s    -->    p   Sr
 				 *      / \         / \
 				 *     Sl  Sr      N   Sl
 				 */
 				parent->rb_right = tmp1 = sibling->rb_left;
 				sibling->rb_left = parent;
 				rb_set_parent_color(tmp1, parent, RB_BLACK);
 				__rb_rotate_set_parents(parent, sibling, root,
 							RB_RED);
 				augment_rotate(parent, sibling);
 				sibling = tmp1;
 			}
 			tmp1 = sibling->rb_right;
 			if (!tmp1 || rb_is_black(tmp1)) {
 				tmp2 = sibling->rb_left;
 				if (!tmp2 || rb_is_black(tmp2)) {
 					/*
 					 * Case 2 - sibling color flip
 					 * (p could be either color here)
 					 *
 					 *    (p)           (p)
 					 *    / \           / \
 					 *   N   S    -->  N   s
 					 *      / \           / \
 					 *     Sl  Sr        Sl  Sr
 					 *
 					 * This leaves us violating 5) which
 					 * can be fixed by flipping p to black
 					 * if it was red, or by recursing at p.
 					 * p is red when coming from Case 1.
 					 */
 					rb_set_parent_color(sibling, parent,
 							    RB_RED);
 					if (rb_is_red(parent))
 						rb_set_black(parent);
 					else {
 						node = parent;
 						parent = rb_parent(node);
 						if (parent)
 							continue;
 					}
 					break;
 				}
 				/*
 				 * Case 3 - right rotate at sibling
 				 * (p could be either color here)
 				 *
 				 *   (p)           (p)
 				 *   / \           / \
 				 *  N   S    -->  N   Sl
 				 *     / \             \
 				 *    sl  Sr            s
 				 *                       \
 				 *                        Sr
 				 */
 				sibling->rb_left = tmp1 = tmp2->rb_right;
 				tmp2->rb_right = sibling;
 				parent->rb_right = tmp2;
 				if (tmp1)
 					rb_set_parent_color(tmp1, sibling,
 							    RB_BLACK);
 				augment_rotate(sibling, tmp2);
 				tmp1 = sibling;
 				sibling = tmp2;
 			}
 			/*
 			 * Case 4 - left rotate at parent + color flips
 			 * (p and sl could be either color here.
 			 *  After rotation, p becomes black, s acquires
 			 *  p's color, and sl keeps its color)
 			 *
 			 *      (p)             (s)
 			 *      / \             / \
 			 *     N   S     -->   P   Sr
 			 *        / \         / \
 			 *      (sl) sr      N  (sl)
 			 */
 			parent->rb_right = tmp2 = sibling->rb_left;
 			sibling->rb_left = parent;
 			rb_set_parent_color(tmp1, sibling, RB_BLACK);
 			if (tmp2)
 				rb_set_parent(tmp2, parent);
 			__rb_rotate_set_parents(parent, sibling, root,
 						RB_BLACK);
 			augment_rotate(parent, sibling);
 			break;
 		} else {
 			sibling = parent->rb_left;
 			if (rb_is_red(sibling)) {
 				/* Case 1 - right rotate at parent */
 				parent->rb_left = tmp1 = sibling->rb_right;
 				sibling->rb_right = parent;
 				rb_set_parent_color(tmp1, parent, RB_BLACK);
 				__rb_rotate_set_parents(parent, sibling, root,
 							RB_RED);
 				augment_rotate(parent, sibling);
 				sibling = tmp1;
 			}
 			tmp1 = sibling->rb_left;
 			if (!tmp1 || rb_is_black(tmp1)) {
 				tmp2 = sibling->rb_right;
 				if (!tmp2 || rb_is_black(tmp2)) {
 					/* Case 2 - sibling color flip */
 					rb_set_parent_color(sibling, parent,
 							    RB_RED);
 					if (rb_is_red(parent))
 						rb_set_black(parent);
 					else {
 						node = parent;
 						parent = rb_parent(node);
 						if (parent)
 							continue;
 					}
 					break;
 				}
 				/* Case 3 - right rotate at sibling */
 				sibling->rb_right = tmp1 = tmp2->rb_left;
 				tmp2->rb_left = sibling;
 				parent->rb_left = tmp2;
 				if (tmp1)
 					rb_set_parent_color(tmp1, sibling,
 							    RB_BLACK);
 				augment_rotate(sibling, tmp2);
 				tmp1 = sibling;
 				sibling = tmp2;
 			}
 			/* Case 4 - left rotate at parent + color flips */
 			parent->rb_left = tmp2 = sibling->rb_right;
 			sibling->rb_right = parent;
 			rb_set_parent_color(tmp1, sibling, RB_BLACK);
 			if (tmp2)
 				rb_set_parent(tmp2, parent);
 			__rb_rotate_set_parents(parent, sibling, root,
 						RB_BLACK);
 			augment_rotate(parent, sibling);
 			break;
 		}
 	}
 }

 /* Non-inline version for rb_erase_augmented() use */
 void __rb_erase_color(struct rb_node *parent, struct rb_root *root,
 	void (*augment_rotate)(struct rb_node *old, struct rb_node *new))
 {
 	____rb_erase_color(parent, root, augment_rotate);
 }

 /*
  * Non-augmented rbtree manipulation functions.
  *
  * We use dummy augmented callbacks here, and have the compiler optimize them
  * out of the rb_insert_color() and rb_erase() function definitions.
  */

 static inline void dummy_propagate(struct rb_node *node, struct rb_node *stop) {}
 static inline void dummy_copy(struct rb_node *old, struct rb_node *new) {}
 static inline void dummy_rotate(struct rb_node *old, struct rb_node *new) {}

 static const struct rb_augment_callbacks dummy_callbacks = {
 	dummy_propagate, dummy_copy, dummy_rotate
 };

 void rb_insert_color(struct rb_node *node, struct rb_root *root)
 {
 	__rb_insert(node, root, dummy_rotate);
 }

 void rb_erase(struct rb_node *node, struct rb_root *root)
 {
 	struct rb_node *rebalance;
 	rebalance = __rb_erase_augmented(node, root, &dummy_callbacks);
 	if (rebalance)
 		____rb_erase_color(rebalance, root, dummy_rotate);
 }

 /*
  * Augmented rbtree manipulation functions.
  *
  * This instantiates the same __always_inline functions as in the non-augmented
  * case, but this time with user-defined callbacks.
  */

 void __rb_insert_augmented(struct rb_node *node, struct rb_root *root,
 	void (*augment_rotate)(struct rb_node *old, struct rb_node *new))
 {
 	__rb_insert(node, root, augment_rotate);
 }

 /*
  * This function returns the first node (in sort order) of the tree.
  */
 struct rb_node *rb_first(const struct rb_root *root)
 {
 	struct rb_node	*n;

 	n = root->rb_node;
 	if (!n)
 		return NULL;
 	while (n->rb_left)
 		n = n->rb_left;
 	return n;
 }

 struct rb_node *rb_last(const struct rb_root *root)
 {
 	struct rb_node	*n;

 	n = root->rb_node;
 	if (!n)
 		return NULL;
 	while (n->rb_right)
 		n = n->rb_right;
 	return n;
 }

 struct rb_node *rb_next(const struct rb_node *node)
 {
 	struct rb_node *parent;

 	if (RB_EMPTY_NODE(node))
 		return NULL;

 	/*
 	 * If we have a right-hand child, go down and then left as far
 	 * as we can.
 	 */
 	if (node->rb_right) {
 		node = node->rb_right;
 		while (node->rb_left)
 			node=node->rb_left;
 		return (struct rb_node *)node;
 	}

 	/*
 	 * No right-hand children. Everything down and left is smaller than us,
 	 * so any 'next' node must be in the general direction of our parent.
 	 * Go up the tree; any time the ancestor is a right-hand child of its
 	 * parent, keep going up. First time it's a left-hand child of its
 	 * parent, said parent is our 'next' node.
 	 */
 	while ((parent = rb_parent(node)) && node == parent->rb_right)
 		node = parent;

 	return parent;
 }

 struct rb_node *rb_prev(const struct rb_node *node)
 {
 	struct rb_node *parent;

 	if (RB_EMPTY_NODE(node))
 		return NULL;

 	/*
 	 * If we have a left-hand child, go down and then right as far
 	 * as we can.
 	 */
 	if (node->rb_left) {
 		node = node->rb_left;
 		while (node->rb_right)
 			node=node->rb_right;
 		return (struct rb_node *)node;
 	}

 	/*
 	 * No left-hand children. Go up till we find an ancestor which
 	 * is a right-hand child of its parent.
 	 */
 	while ((parent = rb_parent(node)) && node == parent->rb_left)
 		node = parent;

 	return parent;
 }

 void rb_replace_node(struct rb_node *victim, struct rb_node *new,
 		     struct rb_root *root)
 {
 	struct rb_node *parent = rb_parent(victim);

 	/* Set the surrounding nodes to point to the replacement */
 	__rb_change_child(victim, new, parent, root);
 	if (victim->rb_left)
 		rb_set_parent(victim->rb_left, new);
 	if (victim->rb_right)
 		rb_set_parent(victim->rb_right, new);

 	/* Copy the pointers/colour from the victim to the replacement */
 	*new = *victim;
 }

 static struct rb_node *rb_left_deepest_node(const struct rb_node *node)
 {
 	for (;;) {
 		if (node->rb_left)
 			node = node->rb_left;
 		else if (node->rb_right)
 			node = node->rb_right;
 		else
 			return (struct rb_node *)node;
 	}
 }

 struct rb_node *rb_next_postorder(const struct rb_node *node)
 {
 	const struct rb_node *parent;
 	if (!node)
 		return NULL;
 	parent = rb_parent(node);

 	/* If we're sitting on node, we've already seen our children */
 	if (parent && node == parent->rb_left && parent->rb_right) {
 		/* If we are the parent's left node, go to the parent's right
 		 * node then all the way down to the left */
 		return rb_left_deepest_node(parent->rb_right);
 	} else
 		/* Otherwise we are the parent's right node, and the parent
 		 * should be next */
 		return (struct rb_node *)parent;
 }

 struct rb_node *rb_first_postorder(const struct rb_root *root)
 {
 	if (!root->rb_node)
 		return NULL;

 	return rb_left_deepest_node(root->rb_node);
 }
	/*
	Red Black Trees
	(C) 1999 Andrea Arcangeli <andrea@suse.de>
	(C) 2002 David Woodhouse <dwmw2@infradead.org>
	(C) 2012 Michel Lespinasse <walken@google.com>

	This program is free software; you can redistribute it and/or modify
	it under the terms of the GNU General Public License as published by
	the Free Software Foundation; either version 2 of the License, or
	(at your option) any later version.

	This program is distributed in the hope that it will be useful,
	but WITHOUT ANY WARRANTY; without even the implied warranty of
	MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
	GNU General Public License for more details.

	You should have received a copy of the GNU General Public License
	along with this program; if not, write to the Free Software
	Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA

	linux/lib/rbtree.c
	*/

	#include <linux/rbtree_augmented.h>

	/*
	* red-black trees properties: http://en.wikipedia.org/wiki/Rbtree
	*
	* 1) A node is either red or black
	* 2) The root is black
	* 3) All leaves (NULL) are black
	* 4) Both children of every red node are black
	* 5) Every simple path from root to leaves contains the same number
	* of black nodes.
	*
	* 4 and 5 give the O(log n) guarantee, since 4 implies you cannot have two
	* consecutive red nodes in a path and every red node is therefore followed by
	* a black. So if B is the number of black nodes on every simple path (as per
	* 5), then the longest possible path due to 4 is 2B.
	*
	* We shall indicate color with case, where black nodes are uppercase and red
	* nodes will be lowercase. Unknown color nodes shall be drawn as red within
	* parentheses and have some accompanying text comment.
	*/

	static inline void rb_set_black(struct rb_node *rb)
	{
	rb->__rb_parent_color \|= RB_BLACK;
	}

	static inline struct rb_node rb_red_parent(struct rb_node red)
	{
	return (struct rb_node *)red->__rb_parent_color;
	}

	/*
	* Helper function for rotations:
	* - old's parent and color get assigned to new
	* - old gets assigned new as a parent and 'color' as a color.
	*/
	static inline void
	__rb_rotate_set_parents(struct rb_node old, struct rb_node new,
	struct rb_root *root, int color)
	{
	struct rb_node *parent = rb_parent(old);
	new->__rb_parent_color = old->__rb_parent_color;
	rb_set_parent_color(old, new, color);
	__rb_change_child(old, new, parent, root);
	}

	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))
	{
	struct rb_node parent = rb_red_parent(node), gparent, *tmp;

	while (true) {
	/*
	* Loop invariant: node is red
	*
	* If there is a black parent, we are done.
	* Otherwise, take some corrective action as we don't
	* want a red root or two consecutive red nodes.
	*/
	if (!parent) {
	rb_set_parent_color(node, NULL, RB_BLACK);
	break;
	} else if (rb_is_black(parent))
	break;

	gparent = rb_red_parent(parent);

	tmp = gparent->rb_right;
	if (parent != tmp) { /* parent == gparent->rb_left */
	if (tmp && rb_is_red(tmp)) {
	/*
	* Case 1 - 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.
	*/
	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_right;
	if (node == tmp) {
	/*
	* Case 2 - 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.
	*/
	parent->rb_right = tmp = node->rb_left;
	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 - right rotate at gparent
	*
	* G P
	* / \ / \
	* p U --> n g
	* / \
	* n U
	*/
	gparent->rb_left = tmp; /* == parent->rb_right */
	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 */
	parent->rb_left = tmp = node->rb_right;
	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 */
	gparent->rb_right = tmp; /* == parent->rb_left */
	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;
	}
	}
	}

	/*
	* Inline version for rb_erase() use - we want to be able to inline
	* and eliminate the dummy_rotate callback there
	*/
	static __always_inline void
	____rb_erase_color(struct rb_node parent, struct rb_root root,
	void (augment_rotate)(struct rb_node old, struct rb_node *new))
	{
	struct rb_node node = NULL, sibling, tmp1, tmp2;

	while (true) {
	/*
	* Loop invariants:
	* - node is black (or NULL on first iteration)
	* - node is not the root (parent is not NULL)
	* - All leaf paths going through parent and node have a
	* black node count that is 1 lower than other leaf paths.
	*/
	sibling = parent->rb_right;
	if (node != sibling) { /* node == parent->rb_left */
	if (rb_is_red(sibling)) {
	/*
	* Case 1 - left rotate at parent
	*
	* P S
	* / \ / \
	* N s --> p Sr
	* / \ / \
	* Sl Sr N Sl
	*/
	parent->rb_right = tmp1 = sibling->rb_left;
	sibling->rb_left = parent;
	rb_set_parent_color(tmp1, parent, RB_BLACK);
	__rb_rotate_set_parents(parent, sibling, root,
	RB_RED);
	augment_rotate(parent, sibling);
	sibling = tmp1;
	}
	tmp1 = sibling->rb_right;
	if (!tmp1 \|\| rb_is_black(tmp1)) {
	tmp2 = sibling->rb_left;
	if (!tmp2 \|\| rb_is_black(tmp2)) {
	/*
	* Case 2 - sibling color flip
	* (p could be either color here)
	*
	* (p) (p)
	* / \ / \
	* N S --> N s
	* / \ / \
	* Sl Sr Sl Sr
	*
	* This leaves us violating 5) which
	* can be fixed by flipping p to black
	* if it was red, or by recursing at p.
	* p is red when coming from Case 1.
	*/
	rb_set_parent_color(sibling, parent,
	RB_RED);
	if (rb_is_red(parent))
	rb_set_black(parent);
	else {
	node = parent;
	parent = rb_parent(node);
	if (parent)
	continue;
	}
	break;
	}
	/*
	* Case 3 - right rotate at sibling
	* (p could be either color here)
	*
	* (p) (p)
	* / \ / \
	* N S --> N Sl
	* / \ \
	* sl Sr s
	* \
	* Sr
	*/
	sibling->rb_left = tmp1 = tmp2->rb_right;
	tmp2->rb_right = sibling;
	parent->rb_right = tmp2;
	if (tmp1)
	rb_set_parent_color(tmp1, sibling,
	RB_BLACK);
	augment_rotate(sibling, tmp2);
	tmp1 = sibling;
	sibling = tmp2;
	}
	/*
	* Case 4 - left rotate at parent + color flips
	* (p and sl could be either color here.
	* After rotation, p becomes black, s acquires
	* p's color, and sl keeps its color)
	*
	* (p) (s)
	* / \ / \
	* N S --> P Sr
	* / \ / \
	* (sl) sr N (sl)
	*/
	parent->rb_right = tmp2 = sibling->rb_left;
	sibling->rb_left = parent;
	rb_set_parent_color(tmp1, sibling, RB_BLACK);
	if (tmp2)
	rb_set_parent(tmp2, parent);
	__rb_rotate_set_parents(parent, sibling, root,
	RB_BLACK);
	augment_rotate(parent, sibling);
	break;
	} else {
	sibling = parent->rb_left;
	if (rb_is_red(sibling)) {
	/* Case 1 - right rotate at parent */
	parent->rb_left = tmp1 = sibling->rb_right;
	sibling->rb_right = parent;
	rb_set_parent_color(tmp1, parent, RB_BLACK);
	__rb_rotate_set_parents(parent, sibling, root,
	RB_RED);
	augment_rotate(parent, sibling);
	sibling = tmp1;
	}
	tmp1 = sibling->rb_left;
	if (!tmp1 \|\| rb_is_black(tmp1)) {
	tmp2 = sibling->rb_right;
	if (!tmp2 \|\| rb_is_black(tmp2)) {
	/* Case 2 - sibling color flip */
	rb_set_parent_color(sibling, parent,
	RB_RED);
	if (rb_is_red(parent))
	rb_set_black(parent);
	else {
	node = parent;
	parent = rb_parent(node);
	if (parent)
	continue;
	}
	break;
	}
	/* Case 3 - right rotate at sibling */
	sibling->rb_right = tmp1 = tmp2->rb_left;
	tmp2->rb_left = sibling;
	parent->rb_left = tmp2;
	if (tmp1)
	rb_set_parent_color(tmp1, sibling,
	RB_BLACK);
	augment_rotate(sibling, tmp2);
	tmp1 = sibling;
	sibling = tmp2;
	}
	/* Case 4 - left rotate at parent + color flips */
	parent->rb_left = tmp2 = sibling->rb_right;
	sibling->rb_right = parent;
	rb_set_parent_color(tmp1, sibling, RB_BLACK);
	if (tmp2)
	rb_set_parent(tmp2, parent);
	__rb_rotate_set_parents(parent, sibling, root,
	RB_BLACK);
	augment_rotate(parent, sibling);
	break;
	}
	}
	}

	/* Non-inline version for rb_erase_augmented() use */
	void __rb_erase_color(struct rb_node parent, struct rb_root root,
	void (augment_rotate)(struct rb_node old, struct rb_node *new))
	{
	____rb_erase_color(parent, root, augment_rotate);
	}

	/*
	* Non-augmented rbtree manipulation functions.
	*
	* We use dummy augmented callbacks here, and have the compiler optimize them
	* out of the rb_insert_color() and rb_erase() function definitions.
	*/

	static inline void dummy_propagate(struct rb_node node, struct rb_node stop) {}
	static inline void dummy_copy(struct rb_node old, struct rb_node new) {}
	static inline void dummy_rotate(struct rb_node old, struct rb_node new) {}

	static const struct rb_augment_callbacks dummy_callbacks = {
	dummy_propagate, dummy_copy, dummy_rotate
	};

	void rb_insert_color(struct rb_node node, struct rb_root root)
	{
	__rb_insert(node, root, dummy_rotate);
	}

	void rb_erase(struct rb_node node, struct rb_root root)
	{
	struct rb_node *rebalance;
	rebalance = __rb_erase_augmented(node, root, &dummy_callbacks);
	if (rebalance)
	____rb_erase_color(rebalance, root, dummy_rotate);
	}

	/*
	* Augmented rbtree manipulation functions.
	*
	* This instantiates the same __always_inline functions as in the non-augmented
	* case, but this time with user-defined callbacks.
	*/

	void __rb_insert_augmented(struct rb_node node, struct rb_root root,
	void (augment_rotate)(struct rb_node old, struct rb_node *new))
	{
	__rb_insert(node, root, augment_rotate);
	}

	/*
	* This function returns the first node (in sort order) of the tree.
	*/
	struct rb_node rb_first(const struct rb_root root)
	{
	struct rb_node *n;

	n = root->rb_node;
	if (!n)
	return NULL;
	while (n->rb_left)
	n = n->rb_left;
	return n;
	}

	struct rb_node rb_last(const struct rb_root root)
	{
	struct rb_node *n;

	n = root->rb_node;
	if (!n)
	return NULL;
	while (n->rb_right)
	n = n->rb_right;
	return n;
	}

	struct rb_node rb_next(const struct rb_node node)
	{
	struct rb_node *parent;

	if (RB_EMPTY_NODE(node))
	return NULL;

	/*
	* If we have a right-hand child, go down and then left as far
	* as we can.
	*/
	if (node->rb_right) {
	node = node->rb_right;
	while (node->rb_left)
	node=node->rb_left;
	return (struct rb_node *)node;
	}

	/*
	* No right-hand children. Everything down and left is smaller than us,
	* so any 'next' node must be in the general direction of our parent.
	* Go up the tree; any time the ancestor is a right-hand child of its
	* parent, keep going up. First time it's a left-hand child of its
	* parent, said parent is our 'next' node.
	*/
	while ((parent = rb_parent(node)) && node == parent->rb_right)
	node = parent;

	return parent;
	}

	struct rb_node rb_prev(const struct rb_node node)
	{
	struct rb_node *parent;

	if (RB_EMPTY_NODE(node))
	return NULL;

	/*
	* If we have a left-hand child, go down and then right as far
	* as we can.
	*/
	if (node->rb_left) {
	node = node->rb_left;
	while (node->rb_right)
	node=node->rb_right;
	return (struct rb_node *)node;
	}

	/*
	* No left-hand children. Go up till we find an ancestor which
	* is a right-hand child of its parent.
	*/
	while ((parent = rb_parent(node)) && node == parent->rb_left)
	node = parent;

	return parent;
	}

	void rb_replace_node(struct rb_node victim, struct rb_node new,
	struct rb_root *root)
	{
	struct rb_node *parent = rb_parent(victim);

	/* Set the surrounding nodes to point to the replacement */
	__rb_change_child(victim, new, parent, root);
	if (victim->rb_left)
	rb_set_parent(victim->rb_left, new);
	if (victim->rb_right)
	rb_set_parent(victim->rb_right, new);

	/* Copy the pointers/colour from the victim to the replacement */
	new = victim;
	}

	static struct rb_node rb_left_deepest_node(const struct rb_node node)
	{
	for (;;) {
	if (node->rb_left)
	node = node->rb_left;
	else if (node->rb_right)
	node = node->rb_right;
	else
	return (struct rb_node *)node;
	}
	}

	struct rb_node rb_next_postorder(const struct rb_node node)
	{
	const struct rb_node *parent;
	if (!node)
	return NULL;
	parent = rb_parent(node);

	/* If we're sitting on node, we've already seen our children */
	if (parent && node == parent->rb_left && parent->rb_right) {
	/* If we are the parent's left node, go to the parent's right
	* node then all the way down to the left */
	return rb_left_deepest_node(parent->rb_right);
	} else
	/* Otherwise we are the parent's right node, and the parent
	* should be next */
	return (struct rb_node *)parent;
	}

	struct rb_node rb_first_postorder(const struct rb_root root)
	{
	if (!root->rb_node)
	return NULL;

	return rb_left_deepest_node(root->rb_node);
	}