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Linux/Documentation/rbtree.txt

  1 Red-black Trees (rbtree) in Linux
  2 January 18, 2007
  3 Rob Landley <rob@landley.net>
  4 =============================
  5 
  6 What are red-black trees, and what are they for?
  7 ------------------------------------------------
  8 
  9 Red-black trees are a type of self-balancing binary search tree, used for
 10 storing sortable key/value data pairs.  This differs from radix trees (which
 11 are used to efficiently store sparse arrays and thus use long integer indexes
 12 to insert/access/delete nodes) and hash tables (which are not kept sorted to
 13 be easily traversed in order, and must be tuned for a specific size and
 14 hash function where rbtrees scale gracefully storing arbitrary keys).
 15 
 16 Red-black trees are similar to AVL trees, but provide faster real-time bounded
 17 worst case performance for insertion and deletion (at most two rotations and
 18 three rotations, respectively, to balance the tree), with slightly slower
 19 (but still O(log n)) lookup time.
 20 
 21 To quote Linux Weekly News:
 22 
 23     There are a number of red-black trees in use in the kernel.
 24     The deadline and CFQ I/O schedulers employ rbtrees to
 25     track requests; the packet CD/DVD driver does the same.
 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 
 32 This document covers use of the Linux rbtree implementation.  For more
 33 information 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 
 41 Linux implementation of red-black trees
 42 ---------------------------------------
 43 
 44 Linux's rbtree implementation lives in the file "lib/rbtree.c".  To use it,
 45 "#include <linux/rbtree.h>".
 46 
 47 The Linux rbtree implementation is optimized for speed, and thus has one
 48 less layer of indirection (and better cache locality) than more traditional
 49 tree implementations.  Instead of using pointers to separate rb_node and data
 50 structures, each instance of struct rb_node is embedded in the data structure
 51 it organizes.  And instead of using a comparison callback function pointer,
 52 users are expected to write their own tree search and insert functions
 53 which call the provided rbtree functions.  Locking is also left up to the
 54 user of the rbtree code.
 55 
 56 Creating a new rbtree
 57 ---------------------
 58 
 59 Data 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 
 66 When dealing with a pointer to the embedded struct rb_node, the containing data
 67 structure may be accessed with the standard container_of() macro.  In addition,
 68 individual members may be accessed directly via rb_entry(node, type, member).
 69 
 70 At the root of each rbtree is an rb_root structure, which is initialized to be
 71 empty via:
 72 
 73   struct rb_root mytree = RB_ROOT;
 74 
 75 Searching for a value in an rbtree
 76 ----------------------------------
 77 
 78 Writing a search function for your tree is fairly straightforward: start at the
 79 root, compare each value, and follow the left or right branch as necessary.
 80 
 81 Example:
 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 
103 Inserting data into an rbtree
104 -----------------------------
105 
106 Inserting data in the tree involves first searching for the place to insert the
107 new node, then inserting the node and rebalancing ("recoloring") the tree.
108 
109 The search for insertion differs from the previous search by finding the
110 location of the pointer on which to graft the new node.  The new node also
111 needs a link to its parent node for rebalancing purposes.
112 
113 Example:
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. */
134         rb_link_node(&data->node, parent, new);
135         rb_insert_color(&data->node, root);
136 
137         return TRUE;
138   }
139 
140 Removing or replacing existing data in an rbtree
141 ------------------------------------------------
142 
143 To remove an existing node from a tree, call:
144 
145   void rb_erase(struct rb_node *victim, struct rb_root *tree);
146 
147 Example:
148 
149   struct mytype *data = mysearch(&mytree, "walrus");
150 
151   if (data) {
152         rb_erase(&data->node, &mytree);
153         myfree(data);
154   }
155 
156 To 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 
161 Replacing a node this way does not re-sort the tree: If the new node doesn't
162 have the same key as the old node, the rbtree will probably become corrupted.
163 
164 Iterating through the elements stored in an rbtree (in sort order)
165 ------------------------------------------------------------------
166 
167 Four functions are provided for iterating through an rbtree's contents in
168 sorted order.  These work on arbitrary trees, and should not need to be
169 modified 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 
176 To start iterating, call rb_first() or rb_last() with a pointer to the root
177 of the tree, which will return a pointer to the node structure contained in
178 the first or last element in the tree.  To continue, fetch the next or previous
179 node by calling rb_next() or rb_prev() on the current node.  This will return
180 NULL when there are no more nodes left.
181 
182 The iterator functions return a pointer to the embedded struct rb_node, from
183 which the containing data structure may be accessed with the container_of()
184 macro, and individual members may be accessed directly via
185 rb_entry(node, type, member).
186 
187 Example:
188 
189   struct rb_node *node;
190   for (node = rb_first(&mytree); node; node = rb_next(node))
191         printk("key=%s\n", rb_entry(node, struct mytype, node)->keystring);
192 
193 Support for Augmented rbtrees
194 -----------------------------
195 
196 Augmented rbtree is an rbtree with "some" additional data stored in
197 each node, where the additional data for node N must be a function of
198 the contents of all nodes in the subtree rooted at N. This data can
199 be used to augment some new functionality to rbtree. Augmented rbtree
200 is an optional feature built on top of basic rbtree infrastructure.
201 An rbtree user who wants this feature will have to call the augmentation
202 functions with the user provided augmentation callback when inserting
203 and erasing nodes.
204 
205 C files implementing augmented rbtree manipulation must include
206 <linux/rbtree_augmented.h> instead of <linux/rbtree.h>. Note that
207 linux/rbtree_augmented.h exposes some rbtree implementations details
208 you are not expected to rely on; please stick to the documented APIs
209 there and do not include <linux/rbtree_augmented.h> from header files
210 either so as to minimize chances of your users accidentally relying on
211 such implementation details.
212 
213 On insertion, the user must update the augmented information on the path
214 leading to the inserted node, then call rb_link_node() as usual and
215 rb_augment_inserted() instead of the usual rb_insert_color() call.
216 If rb_augment_inserted() rebalances the rbtree, it will callback into
217 a user provided function to update the augmented information on the
218 affected subtrees.
219 
220 When erasing a node, the user must call rb_erase_augmented() instead of
221 rb_erase(). rb_erase_augmented() calls back into user provided functions
222 to updated the augmented information on affected subtrees.
223 
224 In both cases, the callbacks are provided through struct rb_augment_callbacks.
225 3 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 
238 The compiled code for rb_erase_augmented() may inline the propagation and
239 copy callbacks, which results in a large function, so each augmented rbtree
240 user should have a single rb_erase_augmented() call site in order to limit
241 compiled code size.
242 
243 
244 Sample usage:
245 
246 Interval tree is an example of augmented rb tree. Reference -
247 "Introduction to Algorithms" by Cormen, Leiserson, Rivest and Stein.
248 More details about interval trees:
249 
250 Classical rbtree has a single key and it cannot be directly used to store
251 interval ranges like [lo:hi] and do a quick lookup for any overlap with a new
252 lo:hi or to find whether there is an exact match for a new lo:hi.
253 
254 However, rbtree can be augmented to store such interval ranges in a structured
255 way making it possible to do efficient lookup and exact match.
256 
257 This "extra information" stored in each node is the maximum hi
258 (max_hi) value among all the nodes that are its descendants. This
259 information can be maintained at each node just be looking at the node
260 and its immediate children. And this will be used in O(log n) lookup
261 for lowest match (lowest start address among all possible matches)
262 with something like:
263 
264 struct interval_tree_node *
265 interval_tree_first_match(struct rb_root *root,
266                           unsigned long start, unsigned long last)
267 {
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                         }
291                 }
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 
306 Insertion/removal are defined using the following augmented callbacks:
307 
308 static inline unsigned long
309 compute_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 
327 static 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 
340 static 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 
350 static 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 
361 static const struct rb_augment_callbacks augment_callbacks = {
362         augment_propagate, augment_copy, augment_rotate
363 };
364 
365 void 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;
381         }
382 
383         node->__subtree_last = last;
384         rb_link_node(&node->rb, rb_parent, link);
385         rb_insert_augmented(&node->rb, root, &augment_callbacks);
386 }
387 
388 void interval_tree_remove(struct interval_tree_node *node,
389                           struct rb_root *root)
390 {
391         rb_erase_augmented(&node->rb, root, &augment_callbacks);
392 }

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