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 On insertion, the user must update the augmented information on the path
 206 leading to the inserted node, then call rb_link_node() as usual and
 207 rb_augment_inserted() instead of the usual rb_insert_color() call.
 208 If rb_augment_inserted() rebalances the rbtree, it will callback into
 209 a user provided function to update the augmented information on the
 210 affected subtrees.
 211
 212 When erasing a node, the user must call rb_erase_augmented() instead of
 213 rb_erase(). rb_erase_augmented() calls back into user provided functions
 214 to updated the augmented information on affected subtrees.
 215
 216 In both cases, the callbacks are provided through struct rb_augment_callbacks.
 217 3 callbacks must be defined:
 218
 219 - A propagation callback, which updates the augmented value for a given
 220   node and its ancestors, up to a given stop point (or NULL to update
 221   all the way to the root).
 222
 223 - A copy callback, which copies the augmented value for a given subtree
 224   to a newly assigned subtree root.
 225
 226 - A tree rotation callback, which copies the augmented value for a given
 227   subtree to a newly assigned subtree root AND recomputes the augmented
 228   information for the former subtree root.
 229
 230
 231 Sample usage:
 232
 233 Interval tree is an example of augmented rb tree. Reference -
 234 "Introduction to Algorithms" by Cormen, Leiserson, Rivest and Stein.
 235 More details about interval trees:
 236
 237 Classical rbtree has a single key and it cannot be directly used to store
 238 interval ranges like [lo:hi] and do a quick lookup for any overlap with a new
 239 lo:hi or to find whether there is an exact match for a new lo:hi.
 240
 241 However, rbtree can be augmented to store such interval ranges in a structured
 242 way making it possible to do efficient lookup and exact match.
 243
 244 This "extra information" stored in each node is the maximum hi
 245 (max_hi) value among all the nodes that are its descendents. This
 246 information can be maintained at each node just be looking at the node
 247 and its immediate children. And this will be used in O(log n) lookup
 248 for lowest match (lowest start address among all possible matches)
 249 with something like:
 250
 251 struct interval_tree_node *
 252 interval_tree_first_match(struct rb_root *root,
 253                           unsigned long start, unsigned long last)
 254 {
 255         struct interval_tree_node *node;
 256
 257         if (!root->rb_node)
 258                 return NULL;
 259         node = rb_entry(root->rb_node, struct interval_tree_node, rb);
 260
 261         while (true) {
 262                 if (node->rb.rb_left) {
 263                         struct interval_tree_node *left =
 264                                 rb_entry(node->rb.rb_left,
 265                                          struct interval_tree_node, rb);
 266                         if (left->__subtree_last >= start) {
 267                                 /*
 268                                  * Some nodes in left subtree satisfy Cond2.
 269                                  * Iterate to find the leftmost such node N.
 270                                  * If it also satisfies Cond1, that's the match
 271                                  * we are looking for. Otherwise, there is no
 272                                  * matching interval as nodes to the right of N
 273                                  * can't satisfy Cond1 either.
 274                                  */
 275                                 node = left;
 276                                 continue;
 277                         }
 278                 }
 279                 if (node->start <= last) {              /* Cond1 */
 280                         if (node->last >= start)        /* Cond2 */
 281                                 return node;    /* node is leftmost match */
 282                         if (node->rb.rb_right) {
 283                                 node = rb_entry(node->rb.rb_right,
 284                                         struct interval_tree_node, rb);
 285                                 if (node->__subtree_last >= start)
 286                                         continue;
 287                         }
 288                 }
 289                 return NULL;    /* No match */
 290         }
 291 }
 292
 293 Insertion/removal are defined using the following augmented callbacks:
 294
 295 static inline unsigned long
 296 compute_subtree_last(struct interval_tree_node *node)
 297 {
 298         unsigned long max = node->last, subtree_last;
 299         if (node->rb.rb_left) {
 300                 subtree_last = rb_entry(node->rb.rb_left,
 301                         struct interval_tree_node, rb)->__subtree_last;
 302                 if (max < subtree_last)
 303                         max = subtree_last;
 304         }
 305         if (node->rb.rb_right) {
 306                 subtree_last = rb_entry(node->rb.rb_right,
 307                         struct interval_tree_node, rb)->__subtree_last;
 308                 if (max < subtree_last)
 309                         max = subtree_last;
 310         }
 311         return max;
 312 }
 313
 314 static void augment_propagate(struct rb_node *rb, struct rb_node *stop)
 315 {
 316         while (rb != stop) {
 317                 struct interval_tree_node *node =
 318                         rb_entry(rb, struct interval_tree_node, rb);
 319                 unsigned long subtree_last = compute_subtree_last(node);
 320                 if (node->__subtree_last == subtree_last)
 321                         break;
 322                 node->__subtree_last = subtree_last;
 323                 rb = rb_parent(&node->rb);
 324         }
 325 }
 326
 327 static void augment_copy(struct rb_node *rb_old, struct rb_node *rb_new)
 328 {
 329         struct interval_tree_node *old =
 330                 rb_entry(rb_old, struct interval_tree_node, rb);
 331         struct interval_tree_node *new =
 332                 rb_entry(rb_new, struct interval_tree_node, rb);
 333
 334         new->__subtree_last = old->__subtree_last;
 335 }
 336
 337 static void augment_rotate(struct rb_node *rb_old, struct rb_node *rb_new)
 338 {
 339         struct interval_tree_node *old =
 340                 rb_entry(rb_old, struct interval_tree_node, rb);
 341         struct interval_tree_node *new =
 342                 rb_entry(rb_new, struct interval_tree_node, rb);
 343
 344         new->__subtree_last = old->__subtree_last;
 345         old->__subtree_last = compute_subtree_last(old);
 346 }
 347
 348 static const struct rb_augment_callbacks augment_callbacks = {
 349         augment_propagate, augment_copy, augment_rotate
 350 };
 351
 352 void interval_tree_insert(struct interval_tree_node *node,
 353                           struct rb_root *root)
 354 {
 355         struct rb_node **link = &root->rb_node, *rb_parent = NULL;
 356         unsigned long start = node->start, last = node->last;
 357         struct interval_tree_node *parent;
 358
 359         while (*link) {
 360                 rb_parent = *link;
 361                 parent = rb_entry(rb_parent, struct interval_tree_node, rb);
 362                 if (parent->__subtree_last < last)
 363                         parent->__subtree_last = last;
 364                 if (start < parent->start)
 365                         link = &parent->rb.rb_left;
 366                 else
 367                         link = &parent->rb.rb_right;
 368         }
 369
 370         node->__subtree_last = last;
 371         rb_link_node(&node->rb, rb_parent, link);
 372         rb_insert_augmented(&node->rb, root, &augment_callbacks);
 373 }
 374
 375 void interval_tree_remove(struct interval_tree_node *node,
 376                           struct rb_root *root)
 377 {
 378         rb_erase_augmented(&node->rb, root, &augment_callbacks);
 379 }