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Mauro Carvalho Chehabce0f95a2017-05-17 06:18:13 -03001=================================
Rob Landleyc742b532007-02-10 01:46:20 -08002Red-black Trees (rbtree) in Linux
Mauro Carvalho Chehabce0f95a2017-05-17 06:18:13 -03003=================================
4
5
6:Date: January 18, 2007
7:Author: Rob Landley <rob@landley.net>
Rob Landleyc742b532007-02-10 01:46:20 -08008
9What are red-black trees, and what are they for?
10------------------------------------------------
11
12Red-black trees are a type of self-balancing binary search tree, used for
13storing sortable key/value data pairs. This differs from radix trees (which
14are used to efficiently store sparse arrays and thus use long integer indexes
15to insert/access/delete nodes) and hash tables (which are not kept sorted to
16be easily traversed in order, and must be tuned for a specific size and
17hash function where rbtrees scale gracefully storing arbitrary keys).
18
19Red-black trees are similar to AVL trees, but provide faster real-time bounded
20worst case performance for insertion and deletion (at most two rotations and
21three rotations, respectively, to balance the tree), with slightly slower
22(but still O(log n)) lookup time.
23
24To quote Linux Weekly News:
25
26 There are a number of red-black trees in use in the kernel.
Randy Dunlap17a9e7b2010-11-11 12:09:59 +010027 The deadline and CFQ I/O schedulers employ rbtrees to
28 track requests; the packet CD/DVD driver does the same.
Rob Landleyc742b532007-02-10 01:46:20 -080029 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
35This document covers use of the Linux rbtree implementation. For more
36information on the nature and implementation of Red Black Trees, see:
37
38 Linux Weekly News article on red-black trees
39 http://lwn.net/Articles/184495/
40
41 Wikipedia entry on red-black trees
42 http://en.wikipedia.org/wiki/Red-black_tree
43
44Linux implementation of red-black trees
45---------------------------------------
46
47Linux's rbtree implementation lives in the file "lib/rbtree.c". To use it,
48"#include <linux/rbtree.h>".
49
50The Linux rbtree implementation is optimized for speed, and thus has one
51less layer of indirection (and better cache locality) than more traditional
52tree implementations. Instead of using pointers to separate rb_node and data
53structures, each instance of struct rb_node is embedded in the data structure
54it organizes. And instead of using a comparison callback function pointer,
55users are expected to write their own tree search and insert functions
56which call the provided rbtree functions. Locking is also left up to the
57user of the rbtree code.
58
59Creating a new rbtree
60---------------------
61
Mauro Carvalho Chehabce0f95a2017-05-17 06:18:13 -030062Data nodes in an rbtree tree are structures containing a struct rb_node member::
Rob Landleyc742b532007-02-10 01:46:20 -080063
64 struct mytype {
65 struct rb_node node;
66 char *keystring;
67 };
68
69When dealing with a pointer to the embedded struct rb_node, the containing data
70structure may be accessed with the standard container_of() macro. In addition,
71individual members may be accessed directly via rb_entry(node, type, member).
72
73At the root of each rbtree is an rb_root structure, which is initialized to be
74empty via:
75
76 struct rb_root mytree = RB_ROOT;
77
78Searching for a value in an rbtree
79----------------------------------
80
81Writing a search function for your tree is fairly straightforward: start at the
82root, compare each value, and follow the left or right branch as necessary.
83
Mauro Carvalho Chehabce0f95a2017-05-17 06:18:13 -030084Example::
Rob Landleyc742b532007-02-10 01:46:20 -080085
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
106Inserting data into an rbtree
107-----------------------------
108
109Inserting data in the tree involves first searching for the place to insert the
110new node, then inserting the node and rebalancing ("recoloring") the tree.
111
112The search for insertion differs from the previous search by finding the
113location of the pointer on which to graft the new node. The new node also
114needs a link to its parent node for rebalancing purposes.
115
Mauro Carvalho Chehabce0f95a2017-05-17 06:18:13 -0300116Example::
Rob Landleyc742b532007-02-10 01:46:20 -0800117
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. */
figo.zhang27af1da2009-04-17 10:58:48 +0800137 rb_link_node(&data->node, parent, new);
138 rb_insert_color(&data->node, root);
Rob Landleyc742b532007-02-10 01:46:20 -0800139
140 return TRUE;
141 }
142
143Removing or replacing existing data in an rbtree
144------------------------------------------------
145
Mauro Carvalho Chehabce0f95a2017-05-17 06:18:13 -0300146To remove an existing node from a tree, call::
Rob Landleyc742b532007-02-10 01:46:20 -0800147
148 void rb_erase(struct rb_node *victim, struct rb_root *tree);
149
Mauro Carvalho Chehabce0f95a2017-05-17 06:18:13 -0300150Example::
Rob Landleyc742b532007-02-10 01:46:20 -0800151
figo.zhang27af1da2009-04-17 10:58:48 +0800152 struct mytype *data = mysearch(&mytree, "walrus");
Rob Landleyc742b532007-02-10 01:46:20 -0800153
154 if (data) {
figo.zhang27af1da2009-04-17 10:58:48 +0800155 rb_erase(&data->node, &mytree);
Rob Landleyc742b532007-02-10 01:46:20 -0800156 myfree(data);
157 }
158
Mauro Carvalho Chehabce0f95a2017-05-17 06:18:13 -0300159To replace an existing node in a tree with a new one with the same key, call::
Rob Landleyc742b532007-02-10 01:46:20 -0800160
161 void rb_replace_node(struct rb_node *old, struct rb_node *new,
162 struct rb_root *tree);
163
164Replacing a node this way does not re-sort the tree: If the new node doesn't
165have the same key as the old node, the rbtree will probably become corrupted.
166
167Iterating through the elements stored in an rbtree (in sort order)
168------------------------------------------------------------------
169
170Four functions are provided for iterating through an rbtree's contents in
171sorted order. These work on arbitrary trees, and should not need to be
Mauro Carvalho Chehabce0f95a2017-05-17 06:18:13 -0300172modified or wrapped (except for locking purposes)::
Rob Landleyc742b532007-02-10 01:46:20 -0800173
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
179To start iterating, call rb_first() or rb_last() with a pointer to the root
180of the tree, which will return a pointer to the node structure contained in
181the first or last element in the tree. To continue, fetch the next or previous
182node by calling rb_next() or rb_prev() on the current node. This will return
183NULL when there are no more nodes left.
184
185The iterator functions return a pointer to the embedded struct rb_node, from
186which the containing data structure may be accessed with the container_of()
187macro, and individual members may be accessed directly via
188rb_entry(node, type, member).
189
Mauro Carvalho Chehabce0f95a2017-05-17 06:18:13 -0300190Example::
Rob Landleyc742b532007-02-10 01:46:20 -0800191
192 struct rb_node *node;
193 for (node = rb_first(&mytree); node; node = rb_next(node))
Wang Tinggong19034232009-05-14 11:00:20 +0200194 printk("key=%s\n", rb_entry(node, struct mytype, node)->keystring);
Rob Landleyc742b532007-02-10 01:46:20 -0800195
Davidlohr Buesocd9e61e2017-09-08 16:14:36 -0700196Cached rbtrees
197--------------
198
199Computing the leftmost (smallest) node is quite a common task for binary
200search trees, such as for traversals or users relying on a the particular
201order for their own logic. To this end, users can use 'struct rb_root_cached'
202to optimize O(logN) rb_first() calls to a simple pointer fetch avoiding
203potentially expensive tree iterations. This is done at negligible runtime
204overhead for maintanence; albeit larger memory footprint.
205
206Similar to the rb_root structure, cached rbtrees are initialized to be
207empty via:
208
209 struct rb_root_cached mytree = RB_ROOT_CACHED;
210
211Cached rbtree is simply a regular rb_root with an extra pointer to cache the
212leftmost node. This allows rb_root_cached to exist wherever rb_root does,
213which permits augmented trees to be supported as well as only a few extra
214interfaces:
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
220Both insert and erase calls have their respective counterpart of augmented
221trees:
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
Pallipadi, Venkatesh17d9ddc2010-02-10 15:23:44 -0800229Support for Augmented rbtrees
230-----------------------------
231
Michel Lespinasse14b94af2012-10-08 16:31:17 -0700232Augmented rbtree is an rbtree with "some" additional data stored in
233each node, where the additional data for node N must be a function of
234the contents of all nodes in the subtree rooted at N. This data can
235be used to augment some new functionality to rbtree. Augmented rbtree
236is an optional feature built on top of basic rbtree infrastructure.
237An rbtree user who wants this feature will have to call the augmentation
238functions with the user provided augmentation callback when inserting
239and erasing nodes.
Pallipadi, Venkatesh17d9ddc2010-02-10 15:23:44 -0800240
Michel Lespinasse9c079ad2012-10-08 16:31:33 -0700241C files implementing augmented rbtree manipulation must include
Alexey Klimov121e0242015-09-06 02:13:34 +0300242<linux/rbtree_augmented.h> instead of <linux/rbtree.h>. Note that
Michel Lespinasse9c079ad2012-10-08 16:31:33 -0700243linux/rbtree_augmented.h exposes some rbtree implementations details
244you are not expected to rely on; please stick to the documented APIs
245there and do not include <linux/rbtree_augmented.h> from header files
246either so as to minimize chances of your users accidentally relying on
247such implementation details.
248
Michel Lespinasse14b94af2012-10-08 16:31:17 -0700249On insertion, the user must update the augmented information on the path
250leading to the inserted node, then call rb_link_node() as usual and
251rb_augment_inserted() instead of the usual rb_insert_color() call.
252If rb_augment_inserted() rebalances the rbtree, it will callback into
253a user provided function to update the augmented information on the
254affected subtrees.
Sasha Levin2f175072011-07-24 11:23:20 +0300255
Michel Lespinasse14b94af2012-10-08 16:31:17 -0700256When erasing a node, the user must call rb_erase_augmented() instead of
257rb_erase(). rb_erase_augmented() calls back into user provided functions
258to updated the augmented information on affected subtrees.
Pallipadi, Venkatesh17d9ddc2010-02-10 15:23:44 -0800259
Michel Lespinasse14b94af2012-10-08 16:31:17 -0700260In both cases, the callbacks are provided through struct rb_augment_callbacks.
2613 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
Michel Lespinasse9c079ad2012-10-08 16:31:33 -0700274The compiled code for rb_erase_augmented() may inline the propagation and
275copy callbacks, which results in a large function, so each augmented rbtree
276user should have a single rb_erase_augmented() call site in order to limit
277compiled code size.
278
Michel Lespinasse14b94af2012-10-08 16:31:17 -0700279
Mauro Carvalho Chehabce0f95a2017-05-17 06:18:13 -0300280Sample usage
281^^^^^^^^^^^^
Pallipadi, Venkatesh17d9ddc2010-02-10 15:23:44 -0800282
283Interval tree is an example of augmented rb tree. Reference -
284"Introduction to Algorithms" by Cormen, Leiserson, Rivest and Stein.
285More details about interval trees:
286
287Classical rbtree has a single key and it cannot be directly used to store
288interval ranges like [lo:hi] and do a quick lookup for any overlap with a new
289lo:hi or to find whether there is an exact match for a new lo:hi.
290
291However, rbtree can be augmented to store such interval ranges in a structured
292way making it possible to do efficient lookup and exact match.
293
294This "extra information" stored in each node is the maximum hi
Carlos Garciac98be0c2014-04-04 22:31:00 -0400295(max_hi) value among all the nodes that are its descendants. This
Pallipadi, Venkatesh17d9ddc2010-02-10 15:23:44 -0800296information can be maintained at each node just be looking at the node
297and its immediate children. And this will be used in O(log n) lookup
298for lowest match (lowest start address among all possible matches)
Mauro Carvalho Chehabce0f95a2017-05-17 06:18:13 -0300299with something like::
Pallipadi, Venkatesh17d9ddc2010-02-10 15:23:44 -0800300
Mauro Carvalho Chehabce0f95a2017-05-17 06:18:13 -0300301 struct interval_tree_node *
302 interval_tree_first_match(struct rb_root *root,
303 unsigned long start, unsigned long last)
304 {
Michel Lespinasse14b94af2012-10-08 16:31:17 -0700305 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 }
Pallipadi, Venkatesh17d9ddc2010-02-10 15:23:44 -0800328 }
Michel Lespinasse14b94af2012-10-08 16:31:17 -0700329 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 */
Pallipadi, Venkatesh17d9ddc2010-02-10 15:23:44 -0800340 }
Mauro Carvalho Chehabce0f95a2017-05-17 06:18:13 -0300341 }
Pallipadi, Venkatesh17d9ddc2010-02-10 15:23:44 -0800342
Mauro Carvalho Chehabce0f95a2017-05-17 06:18:13 -0300343Insertion/removal are defined using the following augmented callbacks::
Michel Lespinasse14b94af2012-10-08 16:31:17 -0700344
Mauro Carvalho Chehabce0f95a2017-05-17 06:18:13 -0300345 static inline unsigned long
346 compute_subtree_last(struct interval_tree_node *node)
347 {
Michel Lespinasse14b94af2012-10-08 16:31:17 -0700348 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;
Mauro Carvalho Chehabce0f95a2017-05-17 06:18:13 -0300362 }
Michel Lespinasse14b94af2012-10-08 16:31:17 -0700363
Mauro Carvalho Chehabce0f95a2017-05-17 06:18:13 -0300364 static void augment_propagate(struct rb_node *rb, struct rb_node *stop)
365 {
Michel Lespinasse14b94af2012-10-08 16:31:17 -0700366 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 }
Mauro Carvalho Chehabce0f95a2017-05-17 06:18:13 -0300375 }
Michel Lespinasse14b94af2012-10-08 16:31:17 -0700376
Mauro Carvalho Chehabce0f95a2017-05-17 06:18:13 -0300377 static void augment_copy(struct rb_node *rb_old, struct rb_node *rb_new)
378 {
Michel Lespinasse14b94af2012-10-08 16:31:17 -0700379 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;
Mauro Carvalho Chehabce0f95a2017-05-17 06:18:13 -0300385 }
Michel Lespinasse14b94af2012-10-08 16:31:17 -0700386
Mauro Carvalho Chehabce0f95a2017-05-17 06:18:13 -0300387 static void augment_rotate(struct rb_node *rb_old, struct rb_node *rb_new)
388 {
Michel Lespinasse14b94af2012-10-08 16:31:17 -0700389 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);
Mauro Carvalho Chehabce0f95a2017-05-17 06:18:13 -0300396 }
Michel Lespinasse14b94af2012-10-08 16:31:17 -0700397
Mauro Carvalho Chehabce0f95a2017-05-17 06:18:13 -0300398 static const struct rb_augment_callbacks augment_callbacks = {
Michel Lespinasse14b94af2012-10-08 16:31:17 -0700399 augment_propagate, augment_copy, augment_rotate
Mauro Carvalho Chehabce0f95a2017-05-17 06:18:13 -0300400 };
Michel Lespinasse14b94af2012-10-08 16:31:17 -0700401
Mauro Carvalho Chehabce0f95a2017-05-17 06:18:13 -0300402 void interval_tree_insert(struct interval_tree_node *node,
403 struct rb_root *root)
404 {
Michel Lespinasse14b94af2012-10-08 16:31:17 -0700405 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);
Mauro Carvalho Chehabce0f95a2017-05-17 06:18:13 -0300423 }
Michel Lespinasse14b94af2012-10-08 16:31:17 -0700424
Mauro Carvalho Chehabce0f95a2017-05-17 06:18:13 -0300425 void interval_tree_remove(struct interval_tree_node *node,
426 struct rb_root *root)
427 {
Michel Lespinasse14b94af2012-10-08 16:31:17 -0700428 rb_erase_augmented(&node->rb, root, &augment_callbacks);
Mauro Carvalho Chehabce0f95a2017-05-17 06:18:13 -0300429 }