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+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\n", 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 descendents. 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);
+}
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