Documentation/rbtree.txt

   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     http://lwn.net/Articles/184495/
  40
  41   Wikipedia entry on red-black trees
  42     http://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 maintanence; 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   }