/* Copied from linux/lib/rbtree.c */
	/*
	  Red Black Trees
	  (C) 1999  Andrea Arcangeli <andrea@suse.de>
	  (C) 2002  David Woodhouse <dwmw2@infradead.org>
	  (C) 2012  Michel Lespinasse <walken@google.com>
	
	
	  linux/lib/rbtree.c
	*/
	
	#include "rbtree_augmented.h"
	
	/*
	 * red-black trees properties:  https://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.
	 */
	
	/*
	 * Notes on lockless lookups:
	 *
	 * All stores to the tree structure (rb_left and rb_right) must be done using
	 * WRITE_ONCE(). And we must not inadvertently cause (temporary) loops in the
	 * tree structure as seen in program order.
	 *
	 * These two requirements will allow lockless iteration of the tree -- not
	 * correct iteration mind you, tree rotations are not atomic so a lookup might
	 * miss entire subtrees.
	 *
	 * But they do guarantee that any such traversal will only see valid elements
	 * and that it will indeed complete -- does not get stuck in a loop.
	 *
	 * It also guarantees that if the lookup returns an element it is the 'correct'
	 * one. But not returning an element does _NOT_ mean it's not present.
	 *
	 * NOTE:
	 *
	 * Stores to __rb_parent_color are not important for simple lookups so those
	 * are left undone as of now. Nor did I check for loops involving parent
	 * pointers.
	 */
	
	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 (!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).
				 */
				rb_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.
			 */
			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 - 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.
					 */
					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 - 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;
					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;
			}
		}
	}
	
	/*
	 * 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
					 */
					tmp1 = sibling->rb_left;
					WRITE_ONCE(parent->rb_right, tmp1);
					WRITE_ONCE(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
					 *
					 * Note: p might be red, and then both
					 * p and sl are red after rotation(which
					 * breaks property 4). This is fixed in
					 * Case 4 (in __rb_rotate_set_parents()
					 *         which set sl the color of p
					 *         and set p RB_BLACK)
					 *
					 *   (p)            (sl)
					 *   / \            /  \
					 *  N   sl   -->   P    S
					 *       \        /      \
					 *        S      N        Sr
					 *         \
					 *          Sr
					 */
					tmp1 = tmp2->rb_right;
					WRITE_ONCE(sibling->rb_left, tmp1);
					WRITE_ONCE(tmp2->rb_right, sibling);
					WRITE_ONCE(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)
				 */
				tmp2 = sibling->rb_left;
				WRITE_ONCE(parent->rb_right, tmp2);
				WRITE_ONCE(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 */
					tmp1 = sibling->rb_right;
					WRITE_ONCE(parent->rb_left, tmp1);
					WRITE_ONCE(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 - left rotate at sibling */
					tmp1 = tmp2->rb_left;
					WRITE_ONCE(sibling->rb_right, tmp1);
					WRITE_ONCE(tmp2->rb_left, sibling);
					WRITE_ONCE(parent->rb_left, tmp2);
					if (tmp1)
						rb_set_parent_color(tmp1, sibling,
								    RB_BLACK);
					augment_rotate(sibling, tmp2);
					tmp1 = sibling;
					sibling = tmp2;
				}
				/* Case 4 - right rotate at parent + color flips */
				tmp2 = sibling->rb_right;
				WRITE_ONCE(parent->rb_left, tmp2);
				WRITE_ONCE(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 = {
		.propagate = dummy_propagate,
		.copy = dummy_copy,
		.rotate = 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);
	
		/* Copy the pointers/colour from the victim to the replacement */
		*new = *victim;
	
		/* Set the surrounding nodes to point to the replacement */
		if (victim->rb_left)
			rb_set_parent(victim->rb_left, new);
		if (victim->rb_right)
			rb_set_parent(victim->rb_right, new);
		__rb_change_child(victim, new, parent, root);
	}
	
	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);
	}