1 =================================
2 Red-black Trees (rbtree) in Linux
3 =================================
4
5
6 :Date: January 18, 2007
7 :Author: Rob Landley <rob@landley.net>
8
9 What are red-black trees, and what are they fo
10 ----------------------------------------------
11
12 Red-black trees are a type of self-balancing b
13 storing sortable key/value data pairs. This d
14 are used to efficiently store sparse arrays an
15 to insert/access/delete nodes) and hash tables
16 be easily traversed in order, and must be tune
17 hash function where rbtrees scale gracefully s
18
19 Red-black trees are similar to AVL trees, but
20 worst case performance for insertion and delet
21 three rotations, respectively, to balance the
22 (but still O(log n)) lookup time.
23
24 To quote Linux Weekly News:
25
26 There are a number of red-black trees in u
27 The deadline and CFQ I/O schedulers employ
28 track requests; the packet CD/DVD driver d
29 The high-resolution timer code uses an rbt
30 timer requests. The ext3 filesystem track
31 red-black tree. Virtual memory areas (VMA
32 trees, as are epoll file descriptors, cryp
33 packets in the "hierarchical token bucket"
34
35 This document covers use of the Linux rbtree i
36 information on the nature and implementation o
37
38 Linux Weekly News article on red-black trees
39 https://lwn.net/Articles/184495/
40
41 Wikipedia entry on red-black trees
42 https://en.wikipedia.org/wiki/Red-black_tr
43
44 Linux implementation of red-black trees
45 ---------------------------------------
46
47 Linux's rbtree implementation lives in the fil
48 "#include <linux/rbtree.h>".
49
50 The Linux rbtree implementation is optimized f
51 less layer of indirection (and better cache lo
52 tree implementations. Instead of using pointe
53 structures, each instance of struct rb_node is
54 it organizes. And instead of using a comparis
55 users are expected to write their own tree sea
56 which call the provided rbtree functions. Loc
57 user of the rbtree code.
58
59 Creating a new rbtree
60 ---------------------
61
62 Data nodes in an rbtree tree are structures co
63
64 struct mytype {
65 struct rb_node node;
66 char *keystring;
67 };
68
69 When dealing with a pointer to the embedded st
70 structure may be accessed with the standard co
71 individual members may be accessed directly vi
72
73 At the root of each rbtree is an rb_root struc
74 empty via:
75
76 struct rb_root mytree = RB_ROOT;
77
78 Searching for a value in an rbtree
79 ----------------------------------
80
81 Writing a search function for your tree is fai
82 root, compare each value, and follow the left
83
84 Example::
85
86 struct mytype *my_search(struct rb_root *roo
87 {
88 struct rb_node *node = root->rb_node;
89
90 while (node) {
91 struct mytype *data = containe
92 int result;
93
94 result = strcmp(string, data->
95
96 if (result < 0)
97 node = node->rb_left;
98 else if (result > 0)
99 node = node->rb_right;
100 else
101 return data;
102 }
103 return NULL;
104 }
105
106 Inserting data into an rbtree
107 -----------------------------
108
109 Inserting data in the tree involves first sear
110 new node, then inserting the node and rebalanc
111
112 The search for insertion differs from the prev
113 location of the pointer on which to graft the
114 needs a link to its parent node for rebalancin
115
116 Example::
117
118 int my_insert(struct rb_root *root, struct m
119 {
120 struct rb_node **new = &(root->rb_node
121
122 /* Figure out where to put new node */
123 while (*new) {
124 struct mytype *this = containe
125 int result = strcmp(data->keys
126
127 parent = *new;
128 if (result < 0)
129 new = &((*new)->rb_lef
130 else if (result > 0)
131 new = &((*new)->rb_rig
132 else
133 return FALSE;
134 }
135
136 /* Add new node and rebalance tree. */
137 rb_link_node(&data->node, parent, new)
138 rb_insert_color(&data->node, root);
139
140 return TRUE;
141 }
142
143 Removing or replacing existing data in an rbtr
144 ----------------------------------------------
145
146 To remove an existing node from a tree, call::
147
148 void rb_erase(struct rb_node *victim, struct
149
150 Example::
151
152 struct mytype *data = mysearch(&mytree, "wal
153
154 if (data) {
155 rb_erase(&data->node, &mytree);
156 myfree(data);
157 }
158
159 To replace an existing node in a tree with a n
160
161 void rb_replace_node(struct rb_node *old, st
162 struct rb_root *tree);
163
164 Replacing a node this way does not re-sort the
165 have the same key as the old node, the rbtree
166
167 Iterating through the elements stored in an rb
168 ----------------------------------------------
169
170 Four functions are provided for iterating thro
171 sorted order. These work on arbitrary trees,
172 modified or wrapped (except for locking purpos
173
174 struct rb_node *rb_first(struct rb_root *tre
175 struct rb_node *rb_last(struct rb_root *tree
176 struct rb_node *rb_next(struct rb_node *node
177 struct rb_node *rb_prev(struct rb_node *node
178
179 To start iterating, call rb_first() or rb_last
180 of the tree, which will return a pointer to th
181 the first or last element in the tree. To con
182 node by calling rb_next() or rb_prev() on the
183 NULL when there are no more nodes left.
184
185 The iterator functions return a pointer to the
186 which the containing data structure may be acc
187 macro, and individual members may be accessed
188 rb_entry(node, type, member).
189
190 Example::
191
192 struct rb_node *node;
193 for (node = rb_first(&mytree); node; node =
194 printk("key=%s\n", rb_entry(node, stru
195
196 Cached rbtrees
197 --------------
198
199 Computing the leftmost (smallest) node is quit
200 search trees, such as for traversals or users
201 order for their own logic. To this end, users
202 to optimize O(logN) rb_first() calls to a simp
203 potentially expensive tree iterations. This is
204 overhead for maintenance; albeit larger memory
205
206 Similar to the rb_root structure, cached rbtre
207 empty via::
208
209 struct rb_root_cached mytree = RB_ROOT_CACHE
210
211 Cached rbtree is simply a regular rb_root with
212 leftmost node. This allows rb_root_cached to e
213 which permits augmented trees to be supported
214 interfaces::
215
216 struct rb_node *rb_first_cached(struct rb_ro
217 void rb_insert_color_cached(struct rb_node *
218 void rb_erase_cached(struct rb_node *node, s
219
220 Both insert and erase calls have their respect
221 trees::
222
223 void rb_insert_augmented_cached(struct rb_no
224 bool, struct
225 void rb_erase_augmented_cached(struct rb_nod
226 struct rb_aug
227
228
229 Support for Augmented rbtrees
230 -----------------------------
231
232 Augmented rbtree is an rbtree with "some" addi
233 each node, where the additional data for node
234 the contents of all nodes in the subtree roote
235 be used to augment some new functionality to r
236 is an optional feature built on top of basic r
237 An rbtree user who wants this feature will hav
238 functions with the user provided augmentation
239 and erasing nodes.
240
241 C files implementing augmented rbtree manipula
242 <linux/rbtree_augmented.h> instead of <linux/r
243 linux/rbtree_augmented.h exposes some rbtree i
244 you are not expected to rely on; please stick
245 there and do not include <linux/rbtree_augment
246 either so as to minimize chances of your users
247 such implementation details.
248
249 On insertion, the user must update the augment
250 leading to the inserted node, then call rb_lin
251 rb_augment_inserted() instead of the usual rb_
252 If rb_augment_inserted() rebalances the rbtree
253 a user provided function to update the augment
254 affected subtrees.
255
256 When erasing a node, the user must call rb_era
257 rb_erase(). rb_erase_augmented() calls back in
258 to updated the augmented information on affect
259
260 In both cases, the callbacks are provided thro
261 3 callbacks must be defined:
262
263 - A propagation callback, which updates the au
264 node and its ancestors, up to a given stop p
265 all the way to the root).
266
267 - A copy callback, which copies the augmented
268 to a newly assigned subtree root.
269
270 - A tree rotation callback, which copies the a
271 subtree to a newly assigned subtree root AND
272 information for the former subtree root.
273
274 The compiled code for rb_erase_augmented() may
275 copy callbacks, which results in a large funct
276 user should have a single rb_erase_augmented()
277 compiled code size.
278
279
280 Sample usage
281 ^^^^^^^^^^^^
282
283 Interval tree is an example of augmented rb tr
284 "Introduction to Algorithms" by Cormen, Leiser
285 More details about interval trees:
286
287 Classical rbtree has a single key and it canno
288 interval ranges like [lo:hi] and do a quick lo
289 lo:hi or to find whether there is an exact mat
290
291 However, rbtree can be augmented to store such
292 way making it possible to do efficient lookup
293
294 This "extra information" stored in each node i
295 (max_hi) value among all the nodes that are it
296 information can be maintained at each node jus
297 and its immediate children. And this will be u
298 for lowest match (lowest start address among a
299 with something like::
300
301 struct interval_tree_node *
302 interval_tree_first_match(struct rb_root *ro
303 unsigned long star
304 {
305 struct interval_tree_node *node;
306
307 if (!root->rb_node)
308 return NULL;
309 node = rb_entry(root->rb_node, struct
310
311 while (true) {
312 if (node->rb.rb_left) {
313 struct interval_tree_n
314 rb_entry(node-
315 struc
316 if (left->__subtree_la
317 /*
318 * Some nodes
319 * Iterate to
320 * If it also
321 * we are look
322 * matching in
323 * can't satis
324 */
325 node = left;
326 continue;
327 }
328 }
329 if (node->start <= last) {
330 if (node->last >= star
331 return node;
332 if (node->rb.rb_right)
333 node = rb_entr
334 struct
335 if (node->__su
336 contin
337 }
338 }
339 return NULL; /* No match */
340 }
341 }
342
343 Insertion/removal are defined using the follow
344
345 static inline unsigned long
346 compute_subtree_last(struct interval_tree_no
347 {
348 unsigned long max = node->last, subtre
349 if (node->rb.rb_left) {
350 subtree_last = rb_entry(node->
351 struct interval_tree_n
352 if (max < subtree_last)
353 max = subtree_last;
354 }
355 if (node->rb.rb_right) {
356 subtree_last = rb_entry(node->
357 struct interval_tree_n
358 if (max < subtree_last)
359 max = subtree_last;
360 }
361 return max;
362 }
363
364 static void augment_propagate(struct rb_node
365 {
366 while (rb != stop) {
367 struct interval_tree_node *nod
368 rb_entry(rb, struct in
369 unsigned long subtree_last = c
370 if (node->__subtree_last == su
371 break;
372 node->__subtree_last = subtree
373 rb = rb_parent(&node->rb);
374 }
375 }
376
377 static void augment_copy(struct rb_node *rb_
378 {
379 struct interval_tree_node *old =
380 rb_entry(rb_old, struct interv
381 struct interval_tree_node *new =
382 rb_entry(rb_new, struct interv
383
384 new->__subtree_last = old->__subtree_l
385 }
386
387 static void augment_rotate(struct rb_node *r
388 {
389 struct interval_tree_node *old =
390 rb_entry(rb_old, struct interv
391 struct interval_tree_node *new =
392 rb_entry(rb_new, struct interv
393
394 new->__subtree_last = old->__subtree_l
395 old->__subtree_last = compute_subtree_
396 }
397
398 static const struct rb_augment_callbacks aug
399 augment_propagate, augment_copy, augme
400 };
401
402 void interval_tree_insert(struct interval_tr
403 struct rb_root *ro
404 {
405 struct rb_node **link = &root->rb_node
406 unsigned long start = node->start, las
407 struct interval_tree_node *parent;
408
409 while (*link) {
410 rb_parent = *link;
411 parent = rb_entry(rb_parent, s
412 if (parent->__subtree_last < l
413 parent->__subtree_last
414 if (start < parent->start)
415 link = &parent->rb.rb_
416 else
417 link = &parent->rb.rb_
418 }
419
420 node->__subtree_last = last;
421 rb_link_node(&node->rb, rb_parent, lin
422 rb_insert_augmented(&node->rb, root, &
423 }
424
425 void interval_tree_remove(struct interval_tr
426 struct rb_root *ro
427 {
428 rb_erase_augmented(&node->rb, root, &a
429 }
Linux® is a registered trademark of Linus Torvalds in the United States and other countries.
TOMOYO® is a registered trademark of NTT DATA CORPORATION.