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1Red-black Trees (rbtree) in Linux
2January 18, 2007
3Rob Landley <rob@landley.net>
4=============================
5
6What are red-black trees, and what are they for?
7------------------------------------------------
8
9Red-black trees are a type of self-balancing binary search tree, used for
10storing sortable key/value data pairs. This differs from radix trees (which
11are used to efficiently store sparse arrays and thus use long integer indexes
12to insert/access/delete nodes) and hash tables (which are not kept sorted to
13be easily traversed in order, and must be tuned for a specific size and
14hash function where rbtrees scale gracefully storing arbitrary keys).
15
16Red-black trees are similar to AVL trees, but provide faster real-time bounded
17worst case performance for insertion and deletion (at most two rotations and
18three rotations, respectively, to balance the tree), with slightly slower
19(but still O(log n)) lookup time.
20
21To quote Linux Weekly News:
22
23 There are a number of red-black trees in use in the kernel.
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24 The deadline and CFQ I/O schedulers employ rbtrees to
25 track requests; the packet CD/DVD driver does the same.
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26 The high-resolution timer code uses an rbtree to organize outstanding
27 timer requests. The ext3 filesystem tracks directory entries in a
28 red-black tree. Virtual memory areas (VMAs) are tracked with red-black
29 trees, as are epoll file descriptors, cryptographic keys, and network
30 packets in the "hierarchical token bucket" scheduler.
31
32This document covers use of the Linux rbtree implementation. For more
33information on the nature and implementation of Red Black Trees, see:
34
35 Linux Weekly News article on red-black trees
36 http://lwn.net/Articles/184495/
37
38 Wikipedia entry on red-black trees
39 http://en.wikipedia.org/wiki/Red-black_tree
40
41Linux implementation of red-black trees
42---------------------------------------
43
44Linux's rbtree implementation lives in the file "lib/rbtree.c". To use it,
45"#include <linux/rbtree.h>".
46
47The Linux rbtree implementation is optimized for speed, and thus has one
48less layer of indirection (and better cache locality) than more traditional
49tree implementations. Instead of using pointers to separate rb_node and data
50structures, each instance of struct rb_node is embedded in the data structure
51it organizes. And instead of using a comparison callback function pointer,
52users are expected to write their own tree search and insert functions
53which call the provided rbtree functions. Locking is also left up to the
54user of the rbtree code.
55
56Creating a new rbtree
57---------------------
58
59Data nodes in an rbtree tree are structures containing a struct rb_node member:
60
61 struct mytype {
62 struct rb_node node;
63 char *keystring;
64 };
65
66When dealing with a pointer to the embedded struct rb_node, the containing data
67structure may be accessed with the standard container_of() macro. In addition,
68individual members may be accessed directly via rb_entry(node, type, member).
69
70At the root of each rbtree is an rb_root structure, which is initialized to be
71empty via:
72
73 struct rb_root mytree = RB_ROOT;
74
75Searching for a value in an rbtree
76----------------------------------
77
78Writing a search function for your tree is fairly straightforward: start at the
79root, compare each value, and follow the left or right branch as necessary.
80
81Example:
82
83 struct mytype *my_search(struct rb_root *root, char *string)
84 {
85 struct rb_node *node = root->rb_node;
86
87 while (node) {
88 struct mytype *data = container_of(node, struct mytype, node);
89 int result;
90
91 result = strcmp(string, data->keystring);
92
93 if (result < 0)
94 node = node->rb_left;
95 else if (result > 0)
96 node = node->rb_right;
97 else
98 return data;
99 }
100 return NULL;
101 }
102
103Inserting data into an rbtree
104-----------------------------
105
106Inserting data in the tree involves first searching for the place to insert the
107new node, then inserting the node and rebalancing ("recoloring") the tree.
108
109The search for insertion differs from the previous search by finding the
110location of the pointer on which to graft the new node. The new node also
111needs a link to its parent node for rebalancing purposes.
112
113Example:
114
115 int my_insert(struct rb_root *root, struct mytype *data)
116 {
117 struct rb_node **new = &(root->rb_node), *parent = NULL;
118
119 /* Figure out where to put new node */
120 while (*new) {
121 struct mytype *this = container_of(*new, struct mytype, node);
122 int result = strcmp(data->keystring, this->keystring);
123
124 parent = *new;
125 if (result < 0)
126 new = &((*new)->rb_left);
127 else if (result > 0)
128 new = &((*new)->rb_right);
129 else
130 return FALSE;
131 }
132
133 /* Add new node and rebalance tree. */
27af1da4 134 rb_link_node(&data->node, parent, new);
135 rb_insert_color(&data->node, root);
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136
137 return TRUE;
138 }
139
140Removing or replacing existing data in an rbtree
141------------------------------------------------
142
143To remove an existing node from a tree, call:
144
145 void rb_erase(struct rb_node *victim, struct rb_root *tree);
146
147Example:
148
27af1da4 149 struct mytype *data = mysearch(&mytree, "walrus");
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150
151 if (data) {
27af1da4 152 rb_erase(&data->node, &mytree);
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153 myfree(data);
154 }
155
156To replace an existing node in a tree with a new one with the same key, call:
157
158 void rb_replace_node(struct rb_node *old, struct rb_node *new,
159 struct rb_root *tree);
160
161Replacing a node this way does not re-sort the tree: If the new node doesn't
162have the same key as the old node, the rbtree will probably become corrupted.
163
164Iterating through the elements stored in an rbtree (in sort order)
165------------------------------------------------------------------
166
167Four functions are provided for iterating through an rbtree's contents in
168sorted order. These work on arbitrary trees, and should not need to be
169modified or wrapped (except for locking purposes):
170
171 struct rb_node *rb_first(struct rb_root *tree);
172 struct rb_node *rb_last(struct rb_root *tree);
173 struct rb_node *rb_next(struct rb_node *node);
174 struct rb_node *rb_prev(struct rb_node *node);
175
176To start iterating, call rb_first() or rb_last() with a pointer to the root
177of the tree, which will return a pointer to the node structure contained in
178the first or last element in the tree. To continue, fetch the next or previous
179node by calling rb_next() or rb_prev() on the current node. This will return
180NULL when there are no more nodes left.
181
182The iterator functions return a pointer to the embedded struct rb_node, from
183which the containing data structure may be accessed with the container_of()
184macro, and individual members may be accessed directly via
185rb_entry(node, type, member).
186
187Example:
188
189 struct rb_node *node;
190 for (node = rb_first(&mytree); node; node = rb_next(node))
19034233 191 printk("key=%s\n", rb_entry(node, struct mytype, node)->keystring);
c742b531 192
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193Support for Augmented rbtrees
194-----------------------------
195
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196Augmented rbtree is an rbtree with "some" additional data stored in
197each node, where the additional data for node N must be a function of
198the contents of all nodes in the subtree rooted at N. This data can
199be used to augment some new functionality to rbtree. Augmented rbtree
200is an optional feature built on top of basic rbtree infrastructure.
201An rbtree user who wants this feature will have to call the augmentation
202functions with the user provided augmentation callback when inserting
203and erasing nodes.
2f175074 204
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205C files implementing augmented rbtree manipulation must include
206<linux/rbtree_augmented.h> instead of <linus/rbtree.h>. Note that
207linux/rbtree_augmented.h exposes some rbtree implementations details
208you are not expected to rely on; please stick to the documented APIs
209there and do not include <linux/rbtree_augmented.h> from header files
210either so as to minimize chances of your users accidentally relying on
211such implementation details.
212
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213On insertion, the user must update the augmented information on the path
214leading to the inserted node, then call rb_link_node() as usual and
215rb_augment_inserted() instead of the usual rb_insert_color() call.
216If rb_augment_inserted() rebalances the rbtree, it will callback into
217a user provided function to update the augmented information on the
218affected subtrees.
2f175074 219
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220When erasing a node, the user must call rb_erase_augmented() instead of
221rb_erase(). rb_erase_augmented() calls back into user provided functions
222to updated the augmented information on affected subtrees.
17d9ddc7 223
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224In both cases, the callbacks are provided through struct rb_augment_callbacks.
2253 callbacks must be defined:
226
227- A propagation callback, which updates the augmented value for a given
228 node and its ancestors, up to a given stop point (or NULL to update
229 all the way to the root).
230
231- A copy callback, which copies the augmented value for a given subtree
232 to a newly assigned subtree root.
233
234- A tree rotation callback, which copies the augmented value for a given
235 subtree to a newly assigned subtree root AND recomputes the augmented
236 information for the former subtree root.
237
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238The compiled code for rb_erase_augmented() may inline the propagation and
239copy callbacks, which results in a large function, so each augmented rbtree
240user should have a single rb_erase_augmented() call site in order to limit
241compiled code size.
242
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243
244Sample usage:
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245
246Interval tree is an example of augmented rb tree. Reference -
247"Introduction to Algorithms" by Cormen, Leiserson, Rivest and Stein.
248More details about interval trees:
249
250Classical rbtree has a single key and it cannot be directly used to store
251interval ranges like [lo:hi] and do a quick lookup for any overlap with a new
252lo:hi or to find whether there is an exact match for a new lo:hi.
253
254However, rbtree can be augmented to store such interval ranges in a structured
255way making it possible to do efficient lookup and exact match.
256
257This "extra information" stored in each node is the maximum hi
c98be0c9 258(max_hi) value among all the nodes that are its descendants. This
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259information can be maintained at each node just be looking at the node
260and its immediate children. And this will be used in O(log n) lookup
261for lowest match (lowest start address among all possible matches)
262with something like:
263
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264struct interval_tree_node *
265interval_tree_first_match(struct rb_root *root,
266 unsigned long start, unsigned long last)
17d9ddc7 267{
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268 struct interval_tree_node *node;
269
270 if (!root->rb_node)
271 return NULL;
272 node = rb_entry(root->rb_node, struct interval_tree_node, rb);
273
274 while (true) {
275 if (node->rb.rb_left) {
276 struct interval_tree_node *left =
277 rb_entry(node->rb.rb_left,
278 struct interval_tree_node, rb);
279 if (left->__subtree_last >= start) {
280 /*
281 * Some nodes in left subtree satisfy Cond2.
282 * Iterate to find the leftmost such node N.
283 * If it also satisfies Cond1, that's the match
284 * we are looking for. Otherwise, there is no
285 * matching interval as nodes to the right of N
286 * can't satisfy Cond1 either.
287 */
288 node = left;
289 continue;
290 }
17d9ddc7 291 }
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292 if (node->start <= last) { /* Cond1 */
293 if (node->last >= start) /* Cond2 */
294 return node; /* node is leftmost match */
295 if (node->rb.rb_right) {
296 node = rb_entry(node->rb.rb_right,
297 struct interval_tree_node, rb);
298 if (node->__subtree_last >= start)
299 continue;
300 }
301 }
302 return NULL; /* No match */
303 }
304}
305
306Insertion/removal are defined using the following augmented callbacks:
307
308static inline unsigned long
309compute_subtree_last(struct interval_tree_node *node)
310{
311 unsigned long max = node->last, subtree_last;
312 if (node->rb.rb_left) {
313 subtree_last = rb_entry(node->rb.rb_left,
314 struct interval_tree_node, rb)->__subtree_last;
315 if (max < subtree_last)
316 max = subtree_last;
317 }
318 if (node->rb.rb_right) {
319 subtree_last = rb_entry(node->rb.rb_right,
320 struct interval_tree_node, rb)->__subtree_last;
321 if (max < subtree_last)
322 max = subtree_last;
323 }
324 return max;
325}
326
327static void augment_propagate(struct rb_node *rb, struct rb_node *stop)
328{
329 while (rb != stop) {
330 struct interval_tree_node *node =
331 rb_entry(rb, struct interval_tree_node, rb);
332 unsigned long subtree_last = compute_subtree_last(node);
333 if (node->__subtree_last == subtree_last)
334 break;
335 node->__subtree_last = subtree_last;
336 rb = rb_parent(&node->rb);
337 }
338}
339
340static void augment_copy(struct rb_node *rb_old, struct rb_node *rb_new)
341{
342 struct interval_tree_node *old =
343 rb_entry(rb_old, struct interval_tree_node, rb);
344 struct interval_tree_node *new =
345 rb_entry(rb_new, struct interval_tree_node, rb);
346
347 new->__subtree_last = old->__subtree_last;
348}
349
350static void augment_rotate(struct rb_node *rb_old, struct rb_node *rb_new)
351{
352 struct interval_tree_node *old =
353 rb_entry(rb_old, struct interval_tree_node, rb);
354 struct interval_tree_node *new =
355 rb_entry(rb_new, struct interval_tree_node, rb);
356
357 new->__subtree_last = old->__subtree_last;
358 old->__subtree_last = compute_subtree_last(old);
359}
360
361static const struct rb_augment_callbacks augment_callbacks = {
362 augment_propagate, augment_copy, augment_rotate
363};
364
365void interval_tree_insert(struct interval_tree_node *node,
366 struct rb_root *root)
367{
368 struct rb_node **link = &root->rb_node, *rb_parent = NULL;
369 unsigned long start = node->start, last = node->last;
370 struct interval_tree_node *parent;
371
372 while (*link) {
373 rb_parent = *link;
374 parent = rb_entry(rb_parent, struct interval_tree_node, rb);
375 if (parent->__subtree_last < last)
376 parent->__subtree_last = last;
377 if (start < parent->start)
378 link = &parent->rb.rb_left;
379 else
380 link = &parent->rb.rb_right;
17d9ddc7 381 }
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382
383 node->__subtree_last = last;
384 rb_link_node(&node->rb, rb_parent, link);
385 rb_insert_augmented(&node->rb, root, &augment_callbacks);
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386}
387
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388void interval_tree_remove(struct interval_tree_node *node,
389 struct rb_root *root)
390{
391 rb_erase_augmented(&node->rb, root, &augment_callbacks);
392}