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392 lines
13 KiB
392 lines
13 KiB
Red-black Trees (rbtree) in Linux
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January 18, 2007
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Rob Landley <rob@landley.net>
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=============================
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What are red-black trees, and what are they for?
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------------------------------------------------
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Red-black trees are a type of self-balancing binary search tree, used for
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storing sortable key/value data pairs. This differs from radix trees (which
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are used to efficiently store sparse arrays and thus use long integer indexes
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to insert/access/delete nodes) and hash tables (which are not kept sorted to
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be easily traversed in order, and must be tuned for a specific size and
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hash function where rbtrees scale gracefully storing arbitrary keys).
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Red-black trees are similar to AVL trees, but provide faster real-time bounded
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worst case performance for insertion and deletion (at most two rotations and
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three rotations, respectively, to balance the tree), with slightly slower
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(but still O(log n)) lookup time.
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To quote Linux Weekly News:
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There are a number of red-black trees in use in the kernel.
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The deadline and CFQ I/O schedulers employ rbtrees to
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track requests; the packet CD/DVD driver does the same.
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The high-resolution timer code uses an rbtree to organize outstanding
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timer requests. The ext3 filesystem tracks directory entries in a
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red-black tree. Virtual memory areas (VMAs) are tracked with red-black
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trees, as are epoll file descriptors, cryptographic keys, and network
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packets in the "hierarchical token bucket" scheduler.
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This document covers use of the Linux rbtree implementation. For more
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information on the nature and implementation of Red Black Trees, see:
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Linux Weekly News article on red-black trees
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http://lwn.net/Articles/184495/
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Wikipedia entry on red-black trees
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http://en.wikipedia.org/wiki/Red-black_tree
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Linux implementation of red-black trees
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---------------------------------------
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Linux's rbtree implementation lives in the file "lib/rbtree.c". To use it,
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"#include <linux/rbtree.h>".
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The Linux rbtree implementation is optimized for speed, and thus has one
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less layer of indirection (and better cache locality) than more traditional
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tree implementations. Instead of using pointers to separate rb_node and data
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structures, each instance of struct rb_node is embedded in the data structure
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it organizes. And instead of using a comparison callback function pointer,
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users are expected to write their own tree search and insert functions
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which call the provided rbtree functions. Locking is also left up to the
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user of the rbtree code.
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Creating a new rbtree
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---------------------
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Data nodes in an rbtree tree are structures containing a struct rb_node member:
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struct mytype {
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struct rb_node node;
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char *keystring;
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};
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When dealing with a pointer to the embedded struct rb_node, the containing data
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structure may be accessed with the standard container_of() macro. In addition,
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individual members may be accessed directly via rb_entry(node, type, member).
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At the root of each rbtree is an rb_root structure, which is initialized to be
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empty via:
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struct rb_root mytree = RB_ROOT;
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Searching for a value in an rbtree
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----------------------------------
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Writing a search function for your tree is fairly straightforward: start at the
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root, compare each value, and follow the left or right branch as necessary.
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Example:
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struct mytype *my_search(struct rb_root *root, char *string)
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{
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struct rb_node *node = root->rb_node;
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while (node) {
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struct mytype *data = container_of(node, struct mytype, node);
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int result;
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result = strcmp(string, data->keystring);
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if (result < 0)
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node = node->rb_left;
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else if (result > 0)
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node = node->rb_right;
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else
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return data;
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}
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return NULL;
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}
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Inserting data into an rbtree
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-----------------------------
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Inserting data in the tree involves first searching for the place to insert the
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new node, then inserting the node and rebalancing ("recoloring") the tree.
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The search for insertion differs from the previous search by finding the
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location of the pointer on which to graft the new node. The new node also
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needs a link to its parent node for rebalancing purposes.
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Example:
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int my_insert(struct rb_root *root, struct mytype *data)
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{
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struct rb_node **new = &(root->rb_node), *parent = NULL;
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/* Figure out where to put new node */
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while (*new) {
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struct mytype *this = container_of(*new, struct mytype, node);
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int result = strcmp(data->keystring, this->keystring);
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parent = *new;
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if (result < 0)
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new = &((*new)->rb_left);
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else if (result > 0)
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new = &((*new)->rb_right);
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else
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return FALSE;
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}
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/* Add new node and rebalance tree. */
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rb_link_node(&data->node, parent, new);
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rb_insert_color(&data->node, root);
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return TRUE;
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}
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Removing or replacing existing data in an rbtree
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------------------------------------------------
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To remove an existing node from a tree, call:
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void rb_erase(struct rb_node *victim, struct rb_root *tree);
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Example:
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struct mytype *data = mysearch(&mytree, "walrus");
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if (data) {
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rb_erase(&data->node, &mytree);
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myfree(data);
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}
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To replace an existing node in a tree with a new one with the same key, call:
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void rb_replace_node(struct rb_node *old, struct rb_node *new,
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struct rb_root *tree);
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Replacing a node this way does not re-sort the tree: If the new node doesn't
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have the same key as the old node, the rbtree will probably become corrupted.
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Iterating through the elements stored in an rbtree (in sort order)
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------------------------------------------------------------------
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Four functions are provided for iterating through an rbtree's contents in
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sorted order. These work on arbitrary trees, and should not need to be
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modified or wrapped (except for locking purposes):
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struct rb_node *rb_first(struct rb_root *tree);
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struct rb_node *rb_last(struct rb_root *tree);
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struct rb_node *rb_next(struct rb_node *node);
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struct rb_node *rb_prev(struct rb_node *node);
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To start iterating, call rb_first() or rb_last() with a pointer to the root
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of the tree, which will return a pointer to the node structure contained in
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the first or last element in the tree. To continue, fetch the next or previous
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node by calling rb_next() or rb_prev() on the current node. This will return
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NULL when there are no more nodes left.
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The iterator functions return a pointer to the embedded struct rb_node, from
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which the containing data structure may be accessed with the container_of()
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macro, and individual members may be accessed directly via
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rb_entry(node, type, member).
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Example:
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struct rb_node *node;
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for (node = rb_first(&mytree); node; node = rb_next(node))
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printk("key=%s\n", rb_entry(node, struct mytype, node)->keystring);
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Support for Augmented rbtrees
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-----------------------------
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Augmented rbtree is an rbtree with "some" additional data stored in
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each node, where the additional data for node N must be a function of
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the contents of all nodes in the subtree rooted at N. This data can
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be used to augment some new functionality to rbtree. Augmented rbtree
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is an optional feature built on top of basic rbtree infrastructure.
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An rbtree user who wants this feature will have to call the augmentation
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functions with the user provided augmentation callback when inserting
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and erasing nodes.
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C files implementing augmented rbtree manipulation must include
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<linux/rbtree_augmented.h> instead of <linus/rbtree.h>. Note that
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linux/rbtree_augmented.h exposes some rbtree implementations details
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you are not expected to rely on; please stick to the documented APIs
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there and do not include <linux/rbtree_augmented.h> from header files
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either so as to minimize chances of your users accidentally relying on
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such implementation details.
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On insertion, the user must update the augmented information on the path
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leading to the inserted node, then call rb_link_node() as usual and
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rb_augment_inserted() instead of the usual rb_insert_color() call.
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If rb_augment_inserted() rebalances the rbtree, it will callback into
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a user provided function to update the augmented information on the
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affected subtrees.
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When erasing a node, the user must call rb_erase_augmented() instead of
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rb_erase(). rb_erase_augmented() calls back into user provided functions
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to updated the augmented information on affected subtrees.
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In both cases, the callbacks are provided through struct rb_augment_callbacks.
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3 callbacks must be defined:
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- A propagation callback, which updates the augmented value for a given
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node and its ancestors, up to a given stop point (or NULL to update
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all the way to the root).
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- A copy callback, which copies the augmented value for a given subtree
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to a newly assigned subtree root.
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- A tree rotation callback, which copies the augmented value for a given
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subtree to a newly assigned subtree root AND recomputes the augmented
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information for the former subtree root.
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The compiled code for rb_erase_augmented() may inline the propagation and
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copy callbacks, which results in a large function, so each augmented rbtree
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user should have a single rb_erase_augmented() call site in order to limit
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compiled code size.
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Sample usage:
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Interval tree is an example of augmented rb tree. Reference -
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"Introduction to Algorithms" by Cormen, Leiserson, Rivest and Stein.
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More details about interval trees:
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Classical rbtree has a single key and it cannot be directly used to store
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interval ranges like [lo:hi] and do a quick lookup for any overlap with a new
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lo:hi or to find whether there is an exact match for a new lo:hi.
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However, rbtree can be augmented to store such interval ranges in a structured
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way making it possible to do efficient lookup and exact match.
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This "extra information" stored in each node is the maximum hi
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(max_hi) value among all the nodes that are its descendents. This
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information can be maintained at each node just be looking at the node
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and its immediate children. And this will be used in O(log n) lookup
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for lowest match (lowest start address among all possible matches)
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with something like:
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struct interval_tree_node *
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interval_tree_first_match(struct rb_root *root,
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unsigned long start, unsigned long last)
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{
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struct interval_tree_node *node;
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if (!root->rb_node)
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return NULL;
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node = rb_entry(root->rb_node, struct interval_tree_node, rb);
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while (true) {
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if (node->rb.rb_left) {
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struct interval_tree_node *left =
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rb_entry(node->rb.rb_left,
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struct interval_tree_node, rb);
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if (left->__subtree_last >= start) {
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/*
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* Some nodes in left subtree satisfy Cond2.
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* Iterate to find the leftmost such node N.
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* If it also satisfies Cond1, that's the match
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* we are looking for. Otherwise, there is no
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* matching interval as nodes to the right of N
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* can't satisfy Cond1 either.
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*/
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node = left;
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continue;
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}
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}
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if (node->start <= last) { /* Cond1 */
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if (node->last >= start) /* Cond2 */
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return node; /* node is leftmost match */
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if (node->rb.rb_right) {
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node = rb_entry(node->rb.rb_right,
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struct interval_tree_node, rb);
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if (node->__subtree_last >= start)
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continue;
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}
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}
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return NULL; /* No match */
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}
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}
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Insertion/removal are defined using the following augmented callbacks:
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static inline unsigned long
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compute_subtree_last(struct interval_tree_node *node)
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{
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unsigned long max = node->last, subtree_last;
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if (node->rb.rb_left) {
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subtree_last = rb_entry(node->rb.rb_left,
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struct interval_tree_node, rb)->__subtree_last;
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if (max < subtree_last)
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max = subtree_last;
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}
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if (node->rb.rb_right) {
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subtree_last = rb_entry(node->rb.rb_right,
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struct interval_tree_node, rb)->__subtree_last;
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if (max < subtree_last)
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max = subtree_last;
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}
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return max;
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}
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static void augment_propagate(struct rb_node *rb, struct rb_node *stop)
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{
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while (rb != stop) {
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struct interval_tree_node *node =
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rb_entry(rb, struct interval_tree_node, rb);
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unsigned long subtree_last = compute_subtree_last(node);
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if (node->__subtree_last == subtree_last)
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break;
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node->__subtree_last = subtree_last;
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rb = rb_parent(&node->rb);
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}
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}
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static void augment_copy(struct rb_node *rb_old, struct rb_node *rb_new)
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{
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struct interval_tree_node *old =
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rb_entry(rb_old, struct interval_tree_node, rb);
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struct interval_tree_node *new =
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rb_entry(rb_new, struct interval_tree_node, rb);
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new->__subtree_last = old->__subtree_last;
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}
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static void augment_rotate(struct rb_node *rb_old, struct rb_node *rb_new)
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{
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struct interval_tree_node *old =
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rb_entry(rb_old, struct interval_tree_node, rb);
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struct interval_tree_node *new =
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rb_entry(rb_new, struct interval_tree_node, rb);
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new->__subtree_last = old->__subtree_last;
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old->__subtree_last = compute_subtree_last(old);
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}
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static const struct rb_augment_callbacks augment_callbacks = {
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augment_propagate, augment_copy, augment_rotate
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};
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void interval_tree_insert(struct interval_tree_node *node,
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struct rb_root *root)
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{
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struct rb_node **link = &root->rb_node, *rb_parent = NULL;
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unsigned long start = node->start, last = node->last;
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struct interval_tree_node *parent;
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while (*link) {
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rb_parent = *link;
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parent = rb_entry(rb_parent, struct interval_tree_node, rb);
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if (parent->__subtree_last < last)
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parent->__subtree_last = last;
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if (start < parent->start)
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link = &parent->rb.rb_left;
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else
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link = &parent->rb.rb_right;
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}
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node->__subtree_last = last;
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rb_link_node(&node->rb, rb_parent, link);
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rb_insert_augmented(&node->rb, root, &augment_callbacks);
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}
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void interval_tree_remove(struct interval_tree_node *node,
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struct rb_root *root)
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{
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rb_erase_augmented(&node->rb, root, &augment_callbacks);
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}
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