feat: add basic implementation and enhanced operations of red-black tree from Linux kernel source

4 files changed, 1115 insertions, 0 deletions

diff --git a/na/include/lume/rbtree.h b/na/include/lume/rbtree.h
new file mode 100644
index 0000000..331875c
--- /dev/null
+++ b/na/include/lume/rbtree.h
@@ -0,0 +1,219 @@
+/* SPDX-License-Identifier: GPL-2.0-only */
+/*
+ * Red-Black Trees — public API
+ * Adapted from Linux kernel include/linux/rbtree.h
+ *
+ * (C) 1999  Andrea Arcangeli <andrea@suse.de>
+ */
+
+#pragma once
+
+#include <lume/rbtree_types.h>
+
+/* ------------------------------------------------------------------ */
+/*  Linux-compat macros for freestanding kernel                       */
+/* ------------------------------------------------------------------ */
+
+#ifndef container_of
+#define container_of(ptr, type, member) \
+	reinterpret_cast<type *>( \
+		reinterpret_cast<char *>(ptr) - __builtin_offsetof(type, member))
+#endif
+
+#ifndef READ_ONCE
+#define READ_ONCE(x)  (*(volatile __typeof__(x) *)&(x))
+#endif
+
+/* ------------------------------------------------------------------ */
+/*  Core macros                                                       */
+/* ------------------------------------------------------------------ */
+
+#define rb_parent(r)   ((struct rb_node *)((r)->__rb_parent_color & ~3UL))
+
+#define rb_entry(ptr, type, member)  container_of(ptr, type, member)
+
+#define RB_EMPTY_ROOT(root)  (READ_ONCE((root)->rb_node) == nullptr)
+
+#define RB_EMPTY_NODE(node)  \
+	((node)->__rb_parent_color == (unsigned long)(node))
+#define RB_CLEAR_NODE(node)  \
+	((node)->__rb_parent_color = (unsigned long)(node))
+
+/* ------------------------------------------------------------------ */
+/*  Core API (implemented in lib/rbtree.cc)                           */
+/* ------------------------------------------------------------------ */
+
+#ifdef __cplusplus
+extern "C" {
+#endif
+
+void rb_insert_color(struct rb_node *, struct rb_root *);
+void rb_erase(struct rb_node *, struct rb_root *);
+
+struct rb_node *rb_next(const struct rb_node *);
+struct rb_node *rb_prev(const struct rb_node *);
+struct rb_node *rb_first(const struct rb_root *);
+struct rb_node *rb_last(const struct rb_root *);
+
+struct rb_node *rb_first_postorder(const struct rb_root *);
+struct rb_node *rb_next_postorder(const struct rb_node *);
+
+void rb_replace_node(struct rb_node *victim, struct rb_node *_new,
+		     struct rb_root *root);
+
+#ifdef __cplusplus
+}
+#endif
+
+/* ------------------------------------------------------------------ */
+/*  Inline helpers                                                    */
+/* ------------------------------------------------------------------ */
+
+static inline void rb_link_node(struct rb_node *node, struct rb_node *parent,
+				struct rb_node **rb_link)
+{
+	node->__rb_parent_color = (unsigned long)parent;
+	node->rb_left = node->rb_right = nullptr;
+	*rb_link = node;
+}
+
+#define rb_entry_safe(ptr, type, member) \
+	({ __typeof__(ptr) ____ptr = (ptr); \
+	   ____ptr ? rb_entry(____ptr, type, member) : nullptr; \
+	})
+
+/* ------------------------------------------------------------------ */
+/*  Cached (leftmost) tree helpers                                    */
+/* ------------------------------------------------------------------ */
+
+#define rb_first_cached(root) (root)->rb_leftmost
+
+static inline void rb_insert_color_cached(struct rb_node *node,
+					  struct rb_root_cached *root,
+					  bool leftmost)
+{
+	if (leftmost)
+		root->rb_leftmost = node;
+	rb_insert_color(node, &root->rb_root);
+}
+
+static inline void rb_erase_cached(struct rb_node *node,
+				   struct rb_root_cached *root)
+{
+	if (root->rb_leftmost == node)
+		root->rb_leftmost = rb_next(node);
+	rb_erase(node, &root->rb_root);
+}
+
+static inline void rb_replace_node_cached(struct rb_node *victim,
+					  struct rb_node *_new,
+					  struct rb_root_cached *root)
+{
+	if (root->rb_leftmost == victim)
+		root->rb_leftmost = _new;
+	rb_replace_node(victim, _new, &root->rb_root);
+}
+
+static inline void rb_erase_init(struct rb_node *n, struct rb_root *root)
+{
+	rb_erase(n, root);
+	RB_CLEAR_NODE(n);
+}
+
+/* ------------------------------------------------------------------ */
+/*  Generic insert / find helpers (inline, using function pointers)   */
+/* ------------------------------------------------------------------ */
+
+static inline void
+rb_add(struct rb_node *node, struct rb_root *tree,
+       bool (*less)(struct rb_node *, const struct rb_node *))
+{
+	struct rb_node **link = &tree->rb_node;
+	struct rb_node *parent = nullptr;
+
+	while (*link) {
+		parent = *link;
+		if (less(node, parent))
+			link = &parent->rb_left;
+		else
+			link = &parent->rb_right;
+	}
+
+	rb_link_node(node, parent, link);
+	rb_insert_color(node, tree);
+}
+
+static inline void
+rb_add_cached(struct rb_node *node, struct rb_root_cached *tree,
+	      bool (*less)(struct rb_node *, const struct rb_node *))
+{
+	struct rb_node **link = &tree->rb_root.rb_node;
+	struct rb_node *parent = nullptr;
+	bool leftmost = true;
+
+	while (*link) {
+		parent = *link;
+		if (less(node, parent)) {
+			link = &parent->rb_left;
+		} else {
+			link = &parent->rb_right;
+			leftmost = false;
+		}
+	}
+
+	rb_link_node(node, parent, link);
+	rb_insert_color_cached(node, tree, leftmost);
+}
+
+static inline struct rb_node *
+rb_find(const void *key, const struct rb_root *tree,
+	int (*cmp)(const void *key, const struct rb_node *))
+{
+	struct rb_node *node = tree->rb_node;
+
+	while (node) {
+		int c = cmp(key, node);
+		if (c < 0)
+			node = node->rb_left;
+		else if (c > 0)
+			node = node->rb_right;
+		else
+			return node;
+	}
+	return nullptr;
+}
+
+static inline struct rb_node *
+rb_find_add(struct rb_node *node, struct rb_root *tree,
+	    int (*cmp)(struct rb_node *, const struct rb_node *))
+{
+	struct rb_node **link = &tree->rb_node;
+	struct rb_node *parent = nullptr;
+	int c;
+
+	while (*link) {
+		parent = *link;
+		c = cmp(node, parent);
+		if (c < 0)
+			link = &parent->rb_left;
+		else if (c > 0)
+			link = &parent->rb_right;
+		else
+			return parent;
+	}
+
+	rb_link_node(node, parent, link);
+	rb_insert_color(node, tree);
+	return nullptr;
+}
+
+/* ------------------------------------------------------------------ */
+/*  Iteration macros                                                  */
+/* ------------------------------------------------------------------ */
+
+#define rbtree_postorder_for_each_entry_safe(pos, n, root, field) \
+	for (pos = rb_entry_safe(rb_first_postorder(root), __typeof__(*pos), field); \
+	     pos && ({ n = rb_entry_safe(rb_next_postorder(&pos->field), \
+			__typeof__(*pos), field); 1; }); \
+	     pos = n)
+
diff --git a/na/include/lume/rbtree_augmented.h b/na/include/lume/rbtree_augmented.h
new file mode 100644
index 0000000..07f367b
--- /dev/null
+++ b/na/include/lume/rbtree_augmented.h
@@ -0,0 +1,272 @@
+/* SPDX-License-Identifier: GPL-2.0-only */
+/*
+ * Red-Black Trees — augmented operations & internal helpers
+ * Adapted from Linux kernel include/linux/rbtree_augmented.h
+ *
+ * (C) 1999  Andrea Arcangeli <andrea@suse.de>
+ * (C) 2002  David Woodhouse <dwmw2@infradead.org>
+ * (C) 2012  Michel Lespinasse <walken@google.com>
+ */
+
+#pragma once
+
+#include <lume/rbtree.h>
+
+/* ------------------------------------------------------------------ */
+/*  Linux-compat macros for freestanding kernel                       */
+/* ------------------------------------------------------------------ */
+
+#ifndef WRITE_ONCE
+#define WRITE_ONCE(x, val)  do { \
+	__asm__ volatile("" ::: "memory"); \
+	(x) = (val); \
+	__asm__ volatile("" ::: "memory"); \
+} while (0)
+#endif
+
+#ifndef unlikely
+#define unlikely(x)   __builtin_expect(!!(x), 0)
+#endif
+
+#ifndef __always_inline
+#define __always_inline  inline __attribute__((always_inline))
+#endif
+
+/* ------------------------------------------------------------------ */
+/*  Augmented callback structure                                      */
+/* ------------------------------------------------------------------ */
+
+struct rb_augment_callbacks {
+	void (*propagate)(struct rb_node *node, struct rb_node *stop);
+	void (*copy)(struct rb_node *old, struct rb_node *_new);
+	void (*rotate)(struct rb_node *old, struct rb_node *_new);
+};
+
+extern "C" void __rb_insert_augmented(struct rb_node *node, struct rb_root *root,
+	void (*augment_rotate)(struct rb_node *old, struct rb_node *_new));
+
+extern "C" void __rb_erase_color(struct rb_node *parent, struct rb_root *root,
+	void (*augment_rotate)(struct rb_node *old, struct rb_node *_new));
+
+static inline void
+rb_insert_augmented(struct rb_node *node, struct rb_root *root,
+		    const struct rb_augment_callbacks *augment)
+{
+	__rb_insert_augmented(node, root, augment->rotate);
+}
+
+static inline void
+rb_insert_augmented_cached(struct rb_node *node,
+			   struct rb_root_cached *root, bool newleft,
+			   const struct rb_augment_callbacks *augment)
+{
+	if (newleft)
+		root->rb_leftmost = node;
+	rb_insert_augmented(node, &root->rb_root, augment);
+}
+
+/* ------------------------------------------------------------------ */
+/*  Color / parent encoding helpers                                   */
+/* ------------------------------------------------------------------ */
+
+#define RB_RED		0
+#define RB_BLACK	1
+
+#define __rb_parent(pc)    ((struct rb_node *)(pc & ~3UL))
+
+#define __rb_color(pc)     ((pc) & 1)
+#define __rb_is_black(pc)  __rb_color(pc)
+#define __rb_is_red(pc)    (!__rb_color(pc))
+#define rb_color(rb)       __rb_color((rb)->__rb_parent_color)
+#define rb_is_red(rb)      __rb_is_red((rb)->__rb_parent_color)
+#define rb_is_black(rb)    __rb_is_black((rb)->__rb_parent_color)
+
+static inline void rb_set_parent(struct rb_node *rb, struct rb_node *p)
+{
+	rb->__rb_parent_color = rb_color(rb) + (unsigned long)p;
+}
+
+static inline void rb_set_parent_color(struct rb_node *rb,
+				       struct rb_node *p, int color)
+{
+	rb->__rb_parent_color = (unsigned long)p + color;
+}
+
+static inline void
+__rb_change_child(struct rb_node *old, struct rb_node *_new,
+		  struct rb_node *parent, struct rb_root *root)
+{
+	if (parent) {
+		if (parent->rb_left == old)
+			WRITE_ONCE(parent->rb_left, _new);
+		else
+			WRITE_ONCE(parent->rb_right, _new);
+	} else
+		WRITE_ONCE(root->rb_node, _new);
+}
+
+/* ------------------------------------------------------------------ */
+/*  Augmented erase (inline)                                          */
+/* ------------------------------------------------------------------ */
+
+static __always_inline struct rb_node *
+__rb_erase_augmented(struct rb_node *node, struct rb_root *root,
+		     const struct rb_augment_callbacks *augment)
+{
+	struct rb_node *child = node->rb_right;
+	struct rb_node *tmp = node->rb_left;
+	struct rb_node *parent, *rebalance;
+	unsigned long pc;
+
+	if (!tmp) {
+		/*
+		 * Case 1: node to erase has no more than 1 child (easy!)
+		 */
+		pc = node->__rb_parent_color;
+		parent = __rb_parent(pc);
+		__rb_change_child(node, child, parent, root);
+		if (child) {
+			child->__rb_parent_color = pc;
+			rebalance = nullptr;
+		} else
+			rebalance = __rb_is_black(pc) ? parent : nullptr;
+		tmp = parent;
+	} else if (!child) {
+		/* Still case 1, but this time the child is node->rb_left */
+		tmp->__rb_parent_color = pc = node->__rb_parent_color;
+		parent = __rb_parent(pc);
+		__rb_change_child(node, tmp, parent, root);
+		rebalance = nullptr;
+		tmp = parent;
+	} else {
+		struct rb_node *successor = child, *child2;
+
+		tmp = child->rb_left;
+		if (!tmp) {
+			/*
+			 * Case 2: node's successor is its right child
+			 */
+			parent = successor;
+			child2 = successor->rb_right;
+
+			augment->copy(node, successor);
+		} else {
+			/*
+			 * Case 3: node's successor is leftmost under
+			 * node's right child subtree
+			 */
+			do {
+				parent = successor;
+				successor = tmp;
+				tmp = tmp->rb_left;
+			} while (tmp);
+			child2 = successor->rb_right;
+			WRITE_ONCE(parent->rb_left, child2);
+			WRITE_ONCE(successor->rb_right, child);
+			rb_set_parent(child, successor);
+
+			augment->copy(node, successor);
+			augment->propagate(parent, successor);
+		}
+
+		tmp = node->rb_left;
+		WRITE_ONCE(successor->rb_left, tmp);
+		rb_set_parent(tmp, successor);
+
+		pc = node->__rb_parent_color;
+		tmp = __rb_parent(pc);
+		__rb_change_child(node, successor, tmp, root);
+
+		if (child2) {
+			rb_set_parent_color(child2, parent, RB_BLACK);
+			rebalance = nullptr;
+		} else {
+			rebalance = rb_is_black(successor) ? parent : nullptr;
+		}
+		successor->__rb_parent_color = pc;
+		tmp = successor;
+	}
+
+	augment->propagate(tmp, nullptr);
+	return rebalance;
+}
+
+static __always_inline void
+rb_erase_augmented(struct rb_node *node, struct rb_root *root,
+		   const struct rb_augment_callbacks *augment)
+{
+	struct rb_node *rebalance = __rb_erase_augmented(node, root, augment);
+	if (rebalance)
+		__rb_erase_color(rebalance, root, augment->rotate);
+}
+
+static __always_inline void
+rb_erase_augmented_cached(struct rb_node *node, struct rb_root_cached *root,
+			  const struct rb_augment_callbacks *augment)
+{
+	if (root->rb_leftmost == node)
+		root->rb_leftmost = rb_next(node);
+	rb_erase_augmented(node, &root->rb_root, augment);
+}
+
+/* ------------------------------------------------------------------ */
+/*  Augmented callback declaration macros                             */
+/* ------------------------------------------------------------------ */
+
+#define RB_DECLARE_CALLBACKS(RBSTATIC, RBNAME,				\
+			     RBSTRUCT, RBFIELD, RBAUGMENTED, RBCOMPUTE)	\
+static inline void							\
+RBNAME ## _propagate(struct rb_node *rb, struct rb_node *stop)		\
+{									\
+	while (rb != stop) {						\
+		RBSTRUCT *node = rb_entry(rb, RBSTRUCT, RBFIELD);	\
+		if (RBCOMPUTE(node, true))				\
+			break;						\
+		rb = rb_parent(&node->RBFIELD);				\
+	}								\
+}									\
+static inline void							\
+RBNAME ## _copy(struct rb_node *rb_old, struct rb_node *rb_new)		\
+{									\
+	RBSTRUCT *old_ = rb_entry(rb_old, RBSTRUCT, RBFIELD);		\
+	RBSTRUCT *new_ = rb_entry(rb_new, RBSTRUCT, RBFIELD);		\
+	new_->RBAUGMENTED = old_->RBAUGMENTED;				\
+}									\
+static void								\
+RBNAME ## _rotate(struct rb_node *rb_old, struct rb_node *rb_new)	\
+{									\
+	RBSTRUCT *old_ = rb_entry(rb_old, RBSTRUCT, RBFIELD);		\
+	RBSTRUCT *new_ = rb_entry(rb_new, RBSTRUCT, RBFIELD);		\
+	new_->RBAUGMENTED = old_->RBAUGMENTED;				\
+	RBCOMPUTE(old_, false);						\
+}									\
+RBSTATIC const struct rb_augment_callbacks RBNAME = {			\
+	RBNAME ## _propagate,						\
+	RBNAME ## _copy,						\
+	RBNAME ## _rotate						\
+};
+
+#define RB_DECLARE_CALLBACKS_MAX(RBSTATIC, RBNAME, RBSTRUCT, RBFIELD,	      \
+				 RBTYPE, RBAUGMENTED, RBCOMPUTE)	      \
+static inline bool RBNAME ## _compute_max(RBSTRUCT *node, bool exit_early)    \
+{									      \
+	RBSTRUCT *child;						      \
+	RBTYPE max = RBCOMPUTE(node);					      \
+	if (node->RBFIELD.rb_left) {					      \
+		child = rb_entry(node->RBFIELD.rb_left, RBSTRUCT, RBFIELD);   \
+		if (child->RBAUGMENTED > max)				      \
+			max = child->RBAUGMENTED;			      \
+	}								      \
+	if (node->RBFIELD.rb_right) {					      \
+		child = rb_entry(node->RBFIELD.rb_right, RBSTRUCT, RBFIELD);  \
+		if (child->RBAUGMENTED > max)				      \
+			max = child->RBAUGMENTED;			      \
+	}								      \
+	if (exit_early && node->RBAUGMENTED == max)			      \
+		return true;						      \
+	node->RBAUGMENTED = max;					      \
+	return false;							      \
+}									      \
+RB_DECLARE_CALLBACKS(RBSTATIC, RBNAME,					      \
+		     RBSTRUCT, RBFIELD, RBAUGMENTED, RBNAME ## _compute_max)
+
diff --git a/na/include/lume/rbtree_types.h b/na/include/lume/rbtree_types.h
new file mode 100644
index 0000000..688ade4
--- /dev/null
+++ b/na/include/lume/rbtree_types.h
@@ -0,0 +1,28 @@
+/* SPDX-License-Identifier: GPL-2.0-only */
+/*
+ * Red-Black Tree type definitions
+ * Adapted from Linux kernel include/linux/rbtree_types.h
+ */
+
+#pragma once
+
+#include <lume/types.h>
+
+struct rb_node {
+	unsigned long  __rb_parent_color;
+	struct rb_node *rb_right;
+	struct rb_node *rb_left;
+} __attribute__((aligned(sizeof(long))));
+
+struct rb_root {
+	struct rb_node *rb_node;
+};
+
+struct rb_root_cached {
+	struct rb_root rb_root;
+	struct rb_node *rb_leftmost;
+};
+
+#define RB_ROOT		(struct rb_root) { nullptr, }
+#define RB_ROOT_CACHED	(struct rb_root_cached) { {nullptr, }, nullptr }
+
diff --git a/na/lib/rbtree.cc b/na/lib/rbtree.cc
new file mode 100644
index 0000000..6718a1a
--- /dev/null
+++ b/na/lib/rbtree.cc
@@ -0,0 +1,596 @@
+// SPDX-License-Identifier: GPL-2.0-only
+/*
+  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 <lume/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 (nullptr) 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 (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).
+			 */
+			rb_set_parent_color(node, nullptr, 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 = nullptr, *sibling, *tmp1, *tmp2;
+
+	while (true) {
+		/*
+		 * Loop invariants:
+		 * - node is black (or nullptr on first iteration)
+		 * - node is not the root (parent is not nullptr)
+		 * - 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);
+}
+
+struct rb_node *rb_next(const struct rb_node *node)
+{
+	struct rb_node *parent;
+
+	if (RB_EMPTY_NODE(node))
+		return nullptr;
+
+	/*
+	 * 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 nullptr;
+
+	/*
+	 * 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 nullptr;
+	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 nullptr;
+
+	return rb_left_deepest_node(root->rb_node);
+}
+
+struct rb_node *rb_first(const struct rb_root *root)
+{
+	struct rb_node	*n;
+
+	n = root->rb_node;
+	if (!n)
+		return nullptr;
+	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 nullptr;
+	while (n->rb_right)
+		n = n->rb_right;
+	return n;
+}