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 }