Red-black Trees (rbtree) in Linux
  January 18, 2007
  Rob Landley <rob@landley.net>
  =============================
  
  What are red-black trees, and what are they for?
  ------------------------------------------------
  
  Red-black trees are a type of self-balancing binary search tree, used for
  storing sortable key/value data pairs.  This differs from radix trees (which
  are used to efficiently store sparse arrays and thus use long integer indexes
  to insert/access/delete nodes) and hash tables (which are not kept sorted to
  be easily traversed in order, and must be tuned for a specific size and
  hash function where rbtrees scale gracefully storing arbitrary keys).
  
  Red-black trees are similar to AVL trees, but provide faster real-time bounded
  worst case performance for insertion and deletion (at most two rotations and
  three rotations, respectively, to balance the tree), with slightly slower
  (but still O(log n)) lookup time.
  
  To quote Linux Weekly News:
  
      There are a number of red-black trees in use in the kernel.

      The deadline and CFQ I/O schedulers employ rbtrees to
      track requests; the packet CD/DVD driver does the same.

      The high-resolution timer code uses an rbtree to organize outstanding
      timer requests.  The ext3 filesystem tracks directory entries in a
      red-black tree.  Virtual memory areas (VMAs) are tracked with red-black
      trees, as are epoll file descriptors, cryptographic keys, and network
      packets in the "hierarchical token bucket" scheduler.
  
  This document covers use of the Linux rbtree implementation.  For more
  information on the nature and implementation of Red Black Trees,  see:
  
    Linux Weekly News article on red-black trees
      http://lwn.net/Articles/184495/
  
    Wikipedia entry on red-black trees
      http://en.wikipedia.org/wiki/Red-black_tree
  
  Linux implementation of red-black trees
  ---------------------------------------
  
  Linux's rbtree implementation lives in the file "lib/rbtree.c".  To use it,
  "#include <linux/rbtree.h>".
  
  The Linux rbtree implementation is optimized for speed, and thus has one
  less layer of indirection (and better cache locality) than more traditional
  tree implementations.  Instead of using pointers to separate rb_node and data
  structures, each instance of struct rb_node is embedded in the data structure
  it organizes.  And instead of using a comparison callback function pointer,
  users are expected to write their own tree search and insert functions
  which call the provided rbtree functions.  Locking is also left up to the
  user of the rbtree code.
  
  Creating a new rbtree
  ---------------------
  
  Data nodes in an rbtree tree are structures containing a struct rb_node member:
  
    struct mytype {
    	struct rb_node node;
    	char *keystring;
    };
  
  When dealing with a pointer to the embedded struct rb_node, the containing data
  structure may be accessed with the standard container_of() macro.  In addition,
  individual members may be accessed directly via rb_entry(node, type, member).
  
  At the root of each rbtree is an rb_root structure, which is initialized to be
  empty via:
  
    struct rb_root mytree = RB_ROOT;
  
  Searching for a value in an rbtree
  ----------------------------------
  
  Writing a search function for your tree is fairly straightforward: start at the
  root, compare each value, and follow the left or right branch as necessary.
  
  Example:
  
    struct mytype *my_search(struct rb_root *root, char *string)
    {
    	struct rb_node *node = root->rb_node;
  
    	while (node) {
    		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;
  		else
    			return data;
  	}
  	return NULL;
    }
  
  Inserting data into an rbtree
  -----------------------------
  
  Inserting data in the tree involves first searching for the place to insert the
  new node, then inserting the node and rebalancing ("recoloring") the tree.
  
  The search for insertion differs from the previous search by finding the
  location of the pointer on which to graft the new node.  The new node also
  needs a link to its parent node for rebalancing purposes.
  
  Example:
  
    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);
    		else
    			return FALSE;
    	}
  
    	/* Add new node and rebalance tree. */

    	rb_link_node(&data->node, parent, new);
    	rb_insert_color(&data->node, root);

  
  	return TRUE;
    }
  
  Removing or replacing existing data in an rbtree
  ------------------------------------------------
  
  To remove an existing node from a tree, call:
  
    void rb_erase(struct rb_node *victim, struct rb_root *tree);
  
  Example:

    struct mytype *data = mysearch(&mytree, "walrus");

  
    if (data) {

    	rb_erase(&data->node, &mytree);

    	myfree(data);
    }
  
  To replace an existing node in a tree with a new one with the same key, call:
  
    void rb_replace_node(struct rb_node *old, struct rb_node *new,
    			struct rb_root *tree);
  
  Replacing a node this way does not re-sort the tree: If the new node doesn't
  have the same key as the old node, the rbtree will probably become corrupted.
  
  Iterating through the elements stored in an rbtree (in sort order)
  ------------------------------------------------------------------
  
  Four functions are provided for iterating through an rbtree's contents in
  sorted order.  These work on arbitrary trees, and should not need to be
  modified or wrapped (except for locking purposes):
  
    struct rb_node *rb_first(struct rb_root *tree);
    struct rb_node *rb_last(struct rb_root *tree);
    struct rb_node *rb_next(struct rb_node *node);
    struct rb_node *rb_prev(struct rb_node *node);
  
  To start iterating, call rb_first() or rb_last() with a pointer to the root
  of the tree, which will return a pointer to the node structure contained in
  the first or last element in the tree.  To continue, fetch the next or previous
  node by calling rb_next() or rb_prev() on the current node.  This will return
  NULL when there are no more nodes left.
  
  The iterator functions return a pointer to the embedded struct rb_node, from
  which the containing data structure may be accessed with the container_of()
  macro, and individual members may be accessed directly via
  rb_entry(node, type, member).
  
  Example:
  
    struct rb_node *node;
    for (node = rb_first(&mytree); node; node = rb_next(node))

  	printk("key=%s
  ", rb_entry(node, struct mytype, node)->keystring);

  Support for Augmented rbtrees
  -----------------------------

  Augmented rbtree is an rbtree with "some" additional data stored in
  each node, where the additional data for node N must be a function of
  the contents of all nodes in the subtree rooted at N. This data can
  be used to augment some new functionality to rbtree. Augmented rbtree
  is an optional feature built on top of basic rbtree infrastructure.
  An rbtree user who wants this feature will have to call the augmentation
  functions with the user provided augmentation callback when inserting
  and erasing nodes.

  C files implementing augmented rbtree manipulation must include
  <linux/rbtree_augmented.h> instead of <linus/rbtree.h>. Note that
  linux/rbtree_augmented.h exposes some rbtree implementations details
  you are not expected to rely on; please stick to the documented APIs
  there and do not include <linux/rbtree_augmented.h> from header files
  either so as to minimize chances of your users accidentally relying on
  such implementation details.

  On insertion, the user must update the augmented information on the path
  leading to the inserted node, then call rb_link_node() as usual and
  rb_augment_inserted() instead of the usual rb_insert_color() call.
  If rb_augment_inserted() rebalances the rbtree, it will callback into
  a user provided function to update the augmented information on the
  affected subtrees.

  When erasing a node, the user must call rb_erase_augmented() instead of
  rb_erase(). rb_erase_augmented() calls back into user provided functions
  to updated the augmented information on affected subtrees.

  In both cases, the callbacks are provided through struct rb_augment_callbacks.
  3 callbacks must be defined:
  
  - A propagation callback, which updates the augmented value for a given
    node and its ancestors, up to a given stop point (or NULL to update
    all the way to the root).
  
  - A copy callback, which copies the augmented value for a given subtree
    to a newly assigned subtree root.
  
  - A tree rotation callback, which copies the augmented value for a given
    subtree to a newly assigned subtree root AND recomputes the augmented
    information for the former subtree root.

  The compiled code for rb_erase_augmented() may inline the propagation and
  copy callbacks, which results in a large function, so each augmented rbtree
  user should have a single rb_erase_augmented() call site in order to limit
  compiled code size.

  
  Sample usage:

  
  Interval tree is an example of augmented rb tree. Reference -
  "Introduction to Algorithms" by Cormen, Leiserson, Rivest and Stein.
  More details about interval trees:
  
  Classical rbtree has a single key and it cannot be directly used to store
  interval ranges like [lo:hi] and do a quick lookup for any overlap with a new
  lo:hi or to find whether there is an exact match for a new lo:hi.
  
  However, rbtree can be augmented to store such interval ranges in a structured
  way making it possible to do efficient lookup and exact match.
  
  This "extra information" stored in each node is the maximum hi

  (max_hi) value among all the nodes that are its descendants. This

  information can be maintained at each node just be looking at the node
  and its immediate children. And this will be used in O(log n) lookup
  for lowest match (lowest start address among all possible matches)
  with something like:

  struct interval_tree_node *
  interval_tree_first_match(struct rb_root *root,
  			  unsigned long start, unsigned long last)

  {

  	struct interval_tree_node *node;
  
  	if (!root->rb_node)
  		return NULL;
  	node = rb_entry(root->rb_node, struct interval_tree_node, rb);
  
  	while (true) {
  		if (node->rb.rb_left) {
  			struct interval_tree_node *left =
  				rb_entry(node->rb.rb_left,
  					 struct interval_tree_node, rb);
  			if (left->__subtree_last >= start) {
  				/*
  				 * Some nodes in left subtree satisfy Cond2.
  				 * Iterate to find the leftmost such node N.
  				 * If it also satisfies Cond1, that's the match
  				 * we are looking for. Otherwise, there is no
  				 * matching interval as nodes to the right of N
  				 * can't satisfy Cond1 either.
  				 */
  				node = left;
  				continue;
  			}

  		}

  		if (node->start <= last) {		/* Cond1 */
  			if (node->last >= start)	/* Cond2 */
  				return node;	/* node is leftmost match */
  			if (node->rb.rb_right) {
  				node = rb_entry(node->rb.rb_right,
  					struct interval_tree_node, rb);
  				if (node->__subtree_last >= start)
  					continue;
  			}
  		}
  		return NULL;	/* No match */
  	}
  }
  
  Insertion/removal are defined using the following augmented callbacks:
  
  static inline unsigned long
  compute_subtree_last(struct interval_tree_node *node)
  {
  	unsigned long max = node->last, subtree_last;
  	if (node->rb.rb_left) {
  		subtree_last = rb_entry(node->rb.rb_left,
  			struct interval_tree_node, rb)->__subtree_last;
  		if (max < subtree_last)
  			max = subtree_last;
  	}
  	if (node->rb.rb_right) {
  		subtree_last = rb_entry(node->rb.rb_right,
  			struct interval_tree_node, rb)->__subtree_last;
  		if (max < subtree_last)
  			max = subtree_last;
  	}
  	return max;
  }
  
  static void augment_propagate(struct rb_node *rb, struct rb_node *stop)
  {
  	while (rb != stop) {
  		struct interval_tree_node *node =
  			rb_entry(rb, struct interval_tree_node, rb);
  		unsigned long subtree_last = compute_subtree_last(node);
  		if (node->__subtree_last == subtree_last)
  			break;
  		node->__subtree_last = subtree_last;
  		rb = rb_parent(&node->rb);
  	}
  }
  
  static void augment_copy(struct rb_node *rb_old, struct rb_node *rb_new)
  {
  	struct interval_tree_node *old =
  		rb_entry(rb_old, struct interval_tree_node, rb);
  	struct interval_tree_node *new =
  		rb_entry(rb_new, struct interval_tree_node, rb);
  
  	new->__subtree_last = old->__subtree_last;
  }
  
  static void augment_rotate(struct rb_node *rb_old, struct rb_node *rb_new)
  {
  	struct interval_tree_node *old =
  		rb_entry(rb_old, struct interval_tree_node, rb);
  	struct interval_tree_node *new =
  		rb_entry(rb_new, struct interval_tree_node, rb);
  
  	new->__subtree_last = old->__subtree_last;
  	old->__subtree_last = compute_subtree_last(old);
  }
  
  static const struct rb_augment_callbacks augment_callbacks = {
  	augment_propagate, augment_copy, augment_rotate
  };
  
  void interval_tree_insert(struct interval_tree_node *node,
  			  struct rb_root *root)
  {
  	struct rb_node **link = &root->rb_node, *rb_parent = NULL;
  	unsigned long start = node->start, last = node->last;
  	struct interval_tree_node *parent;
  
  	while (*link) {
  		rb_parent = *link;
  		parent = rb_entry(rb_parent, struct interval_tree_node, rb);
  		if (parent->__subtree_last < last)
  			parent->__subtree_last = last;
  		if (start < parent->start)
  			link = &parent->rb.rb_left;
  		else
  			link = &parent->rb.rb_right;

  	}

  
  	node->__subtree_last = last;
  	rb_link_node(&node->rb, rb_parent, link);
  	rb_insert_augmented(&node->rb, root, &augment_callbacks);

  }

  void interval_tree_remove(struct interval_tree_node *node,
  			  struct rb_root *root)
  {
  	rb_erase_augmented(&node->rb, root, &augment_callbacks);
  }