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 for?
10 ------------------------------------------------
11
12 Red-black trees are a type of self-balancing binary search tree, used for
13 storing sortable key/value data pairs. This differs from radix trees (which
14 are used to efficiently store sparse arrays and thus use long integer indexes
15 to insert/access/delete nodes) and hash tables (which are not kept sorted to
16 be easily traversed in order, and must be tuned for a specific size and
17 hash function where rbtrees scale gracefully storing arbitrary keys).
18
19 Red-black trees are similar to AVL trees, but provide faster real-time bounded
20 worst case performance for insertion and deletion (at most two rotations and
21 three rotations, respectively, to balance the tree), with slightly slower
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 use in the kernel.
27 The deadline and CFQ I/O schedulers employ rbtrees to
28 track requests; the packet CD/DVD driver does the same.
29 The high-resolution timer code uses an rbtree to organize outstanding
30 timer requests. The ext3 filesystem tracks directory entries in a
31 red-black tree. Virtual memory areas (VMAs) are tracked with red-black
32 trees, as are epoll file descriptors, cryptographic keys, and network
33 packets in the "hierarchical token bucket" scheduler.
34
35 This document covers use of the Linux rbtree implementation. For more
36 information on the nature and implementation of Red Black Trees, see:
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_tree
43
44 Linux implementation of red-black trees
45 ---------------------------------------
46
47 Linux's rbtree implementation lives in the file "lib/rbtree.c". To use it,
48 "#include <linux/rbtree.h>".
49
50 The Linux rbtree implementation is optimized for speed, and thus has one
51 less layer of indirection (and better cache locality) than more traditional
52 tree implementations. Instead of using pointers to separate rb_node and data
53 structures, each instance of struct rb_node is embedded in the data structure
54 it organizes. And instead of using a comparison callback function pointer,
55 users are expected to write their own tree search and insert functions
56 which call the provided rbtree functions. Locking is also left up to the
57 user of the rbtree code.
58
59 Creating a new rbtree
60 ---------------------
61
62 Data nodes in an rbtree tree are structures containing a struct rb_node member::
63
64 struct mytype {
65 struct rb_node node;
66 char *keystring;
67 };
68
69 When dealing with a pointer to the embedded struct rb_node, the containing data
70 structure may be accessed with the standard container_of() macro. In addition,
71 individual members may be accessed directly via rb_entry(node, type, member).
72
73 At the root of each rbtree is an rb_root structure, which is initialized to be
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 fairly straightforward: start at the
82 root, compare each value, and follow the left or right branch as necessary.
83
84 Example::
85
86 struct mytype *my_search(struct rb_root *root, char *string)
87 {
88 struct rb_node *node = root->rb_node;
89
90 while (node) {
91 struct mytype *data = container_of(node, struct mytype, node);
92 int result;
93
94 result = strcmp(string, data->keystring);
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 searching for the place to insert the
110 new node, then inserting the node and rebalancing ("recoloring") the tree.
111
112 The search for insertion differs from the previous search by finding the
113 location of the pointer on which to graft the new node. The new node also
114 needs a link to its parent node for rebalancing purposes.
115
116 Example::
117
118 int my_insert(struct rb_root *root, struct mytype *data)
119 {
120 struct rb_node **new = &(root->rb_node), *parent = NULL;
121
122 /* Figure out where to put new node */
123 while (*new) {
124 struct mytype *this = container_of(*new, struct mytype, node);
125 int result = strcmp(data->keystring, this->keystring);
126
127 parent = *new;
128 if (result < 0)
129 new = &((*new)->rb_left);
130 else if (result > 0)
131 new = &((*new)->rb_right);
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 rbtree
144 ------------------------------------------------
145
146 To remove an existing node from a tree, call::
147
148 void rb_erase(struct rb_node *victim, struct rb_root *tree);
149
150 Example::
151
152 struct mytype *data = mysearch(&mytree, "walrus");
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 new one with the same key, call::
160
161 void rb_replace_node(struct rb_node *old, struct rb_node *new,
162 struct rb_root *tree);
163
164 Replacing a node this way does not re-sort the tree: If the new node doesn't
165 have the same key as the old node, the rbtree will probably become corrupted.
166
167 Iterating through the elements stored in an rbtree (in sort order)
168 ------------------------------------------------------------------
169
170 Four functions are provided for iterating through an rbtree's contents in
171 sorted order. These work on arbitrary trees, and should not need to be
172 modified or wrapped (except for locking purposes)::
173
174 struct rb_node *rb_first(struct rb_root *tree);
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() with a pointer to the root
180 of the tree, which will return a pointer to the node structure contained in
181 the first or last element in the tree. To continue, fetch the next or previous
182 node by calling rb_next() or rb_prev() on the current node. This will return
183 NULL when there are no more nodes left.
184
185 The iterator functions return a pointer to the embedded struct rb_node, from
186 which the containing data structure may be accessed with the container_of()
187 macro, and individual members may be accessed directly via
188 rb_entry(node, type, member).
189
190 Example::
191
192 struct rb_node *node;
193 for (node = rb_first(&mytree); node; node = rb_next(node))
194 printk("key=%s\n", rb_entry(node, struct mytype, node)->keystring);
195
196 Cached rbtrees
197 --------------
198
199 Computing the leftmost (smallest) node is quite a common task for binary
200 search trees, such as for traversals or users relying on a the particular
201 order for their own logic. To this end, users can use 'struct rb_root_cached'
202 to optimize O(logN) rb_first() calls to a simple pointer fetch avoiding
203 potentially expensive tree iterations. This is done at negligible runtime
204 overhead for maintenance; albeit larger memory footprint.
205
206 Similar to the rb_root structure, cached rbtrees are initialized to be
207 empty via::
208
209 struct rb_root_cached mytree = RB_ROOT_CACHED;
210
211 Cached rbtree is simply a regular rb_root with an extra pointer to cache the
212 leftmost node. This allows rb_root_cached to exist wherever rb_root does,
213 which permits augmented trees to be supported as well as only a few extra
214 interfaces::
215
216 struct rb_node *rb_first_cached(struct rb_root_cached *tree);
217 void rb_insert_color_cached(struct rb_node *, struct rb_root_cached *, bool);
218 void rb_erase_cached(struct rb_node *node, struct rb_root_cached *);
219
220 Both insert and erase calls have their respective counterpart of augmented
221 trees::
222
223 void rb_insert_augmented_cached(struct rb_node *node, struct rb_root_cached *,
224 bool, struct rb_augment_callbacks *);
225 void rb_erase_augmented_cached(struct rb_node *, struct rb_root_cached *,
226 struct rb_augment_callbacks *);
227
228
229 Support for Augmented rbtrees
230 -----------------------------
231
232 Augmented rbtree is an rbtree with "some" additional data stored in
233 each node, where the additional data for node N must be a function of
234 the contents of all nodes in the subtree rooted at N. This data can
235 be used to augment some new functionality to rbtree. Augmented rbtree
236 is an optional feature built on top of basic rbtree infrastructure.
237 An rbtree user who wants this feature will have to call the augmentation
238 functions with the user provided augmentation callback when inserting
239 and erasing nodes.
240
241 C files implementing augmented rbtree manipulation must include
242 <linux/rbtree_augmented.h> instead of <linux/rbtree.h>. Note that
243 linux/rbtree_augmented.h exposes some rbtree implementations details
244 you are not expected to rely on; please stick to the documented APIs
245 there and do not include <linux/rbtree_augmented.h> from header files
246 either so as to minimize chances of your users accidentally relying on
247 such implementation details.
248
249 On insertion, the user must update the augmented information on the path
250 leading to the inserted node, then call rb_link_node() as usual and
251 rb_augment_inserted() instead of the usual rb_insert_color() call.
252 If rb_augment_inserted() rebalances the rbtree, it will callback into
253 a user provided function to update the augmented information on the
254 affected subtrees.
255
256 When erasing a node, the user must call rb_erase_augmented() instead of
257 rb_erase(). rb_erase_augmented() calls back into user provided functions
258 to updated the augmented information on affected subtrees.
259
260 In both cases, the callbacks are provided through struct rb_augment_callbacks.
261 3 callbacks must be defined:
262
263 - A propagation callback, which updates the augmented value for a given
264 node and its ancestors, up to a given stop point (or NULL to update
265 all the way to the root).
266
267 - A copy callback, which copies the augmented value for a given subtree
268 to a newly assigned subtree root.
269
270 - A tree rotation callback, which copies the augmented value for a given
271 subtree to a newly assigned subtree root AND recomputes the augmented
272 information for the former subtree root.
273
274 The compiled code for rb_erase_augmented() may inline the propagation and
275 copy callbacks, which results in a large function, so each augmented rbtree
276 user should have a single rb_erase_augmented() call site in order to limit
277 compiled code size.
278
279
280 Sample usage
281 ^^^^^^^^^^^^
282
283 Interval tree is an example of augmented rb tree. Reference -
284 "Introduction to Algorithms" by Cormen, Leiserson, Rivest and Stein.
285 More details about interval trees:
286
287 Classical rbtree has a single key and it cannot be directly used to store
288 interval ranges like [lo:hi] and do a quick lookup for any overlap with a new
289 lo:hi or to find whether there is an exact match for a new lo:hi.
290
291 However, rbtree can be augmented to store such interval ranges in a structured
292 way making it possible to do efficient lookup and exact match.
293
294 This "extra information" stored in each node is the maximum hi
295 (max_hi) value among all the nodes that are its descendants. This
296 information can be maintained at each node just be looking at the node
297 and its immediate children. And this will be used in O(log n) lookup
298 for lowest match (lowest start address among all possible matches)
299 with something like::
300
301 struct interval_tree_node *
302 interval_tree_first_match(struct rb_root *root,
303 unsigned long start, unsigned long last)
304 {
305 struct interval_tree_node *node;
306
307 if (!root->rb_node)
308 return NULL;
309 node = rb_entry(root->rb_node, struct interval_tree_node, rb);
310
311 while (true) {
312 if (node->rb.rb_left) {
313 struct interval_tree_node *left =
314 rb_entry(node->rb.rb_left,
315 struct interval_tree_node, rb);
316 if (left->__subtree_last >= start) {
317 /*
318 * Some nodes in left subtree satisfy Cond2.
319 * Iterate to find the leftmost such node N.
320 * If it also satisfies Cond1, that's the match
321 * we are looking for. Otherwise, there is no
322 * matching interval as nodes to the right of N
323 * can't satisfy Cond1 either.
324 */
325 node = left;
326 continue;
327 }
328 }
329 if (node->start <= last) { /* Cond1 */
330 if (node->last >= start) /* Cond2 */
331 return node; /* node is leftmost match */
332 if (node->rb.rb_right) {
333 node = rb_entry(node->rb.rb_right,
334 struct interval_tree_node, rb);
335 if (node->__subtree_last >= start)
336 continue;
337 }
338 }
339 return NULL; /* No match */
340 }
341 }
342
343 Insertion/removal are defined using the following augmented callbacks::
344
345 static inline unsigned long
346 compute_subtree_last(struct interval_tree_node *node)
347 {
348 unsigned long max = node->last, subtree_last;
349 if (node->rb.rb_left) {
350 subtree_last = rb_entry(node->rb.rb_left,
351 struct interval_tree_node, rb)->__subtree_last;
352 if (max < subtree_last)
353 max = subtree_last;
354 }
355 if (node->rb.rb_right) {
356 subtree_last = rb_entry(node->rb.rb_right,
357 struct interval_tree_node, rb)->__subtree_last;
358 if (max < subtree_last)
359 max = subtree_last;
360 }
361 return max;
362 }
363
364 static void augment_propagate(struct rb_node *rb, struct rb_node *stop)
365 {
366 while (rb != stop) {
367 struct interval_tree_node *node =
368 rb_entry(rb, struct interval_tree_node, rb);
369 unsigned long subtree_last = compute_subtree_last(node);
370 if (node->__subtree_last == subtree_last)
371 break;
372 node->__subtree_last = subtree_last;
373 rb = rb_parent(&node->rb);
374 }
375 }
376
377 static void augment_copy(struct rb_node *rb_old, struct rb_node *rb_new)
378 {
379 struct interval_tree_node *old =
380 rb_entry(rb_old, struct interval_tree_node, rb);
381 struct interval_tree_node *new =
382 rb_entry(rb_new, struct interval_tree_node, rb);
383
384 new->__subtree_last = old->__subtree_last;
385 }
386
387 static void augment_rotate(struct rb_node *rb_old, struct rb_node *rb_new)
388 {
389 struct interval_tree_node *old =
390 rb_entry(rb_old, struct interval_tree_node, rb);
391 struct interval_tree_node *new =
392 rb_entry(rb_new, struct interval_tree_node, rb);
393
394 new->__subtree_last = old->__subtree_last;
395 old->__subtree_last = compute_subtree_last(old);
396 }
397
398 static const struct rb_augment_callbacks augment_callbacks = {
399 augment_propagate, augment_copy, augment_rotate
400 };
401
402 void interval_tree_insert(struct interval_tree_node *node,
403 struct rb_root *root)
404 {
405 struct rb_node **link = &root->rb_node, *rb_parent = NULL;
406 unsigned long start = node->start, last = node->last;
407 struct interval_tree_node *parent;
408
409 while (*link) {
410 rb_parent = *link;
411 parent = rb_entry(rb_parent, struct interval_tree_node, rb);
412 if (parent->__subtree_last < last)
413 parent->__subtree_last = last;
414 if (start < parent->start)
415 link = &parent->rb.rb_left;
416 else
417 link = &parent->rb.rb_right;
418 }
419
420 node->__subtree_last = last;
421 rb_link_node(&node->rb, rb_parent, link);
422 rb_insert_augmented(&node->rb, root, &augment_callbacks);
423 }
424
425 void interval_tree_remove(struct interval_tree_node *node,
426 struct rb_root *root)
427 {
428 rb_erase_augmented(&node->rb, root, &augment_callbacks);
429 }
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