linux/Documentation/rbtree.txt
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   1Red-black Trees (rbtree) in Linux
   2January 18, 2007
   3Rob Landley <rob@landley.net>
   4=============================
   5
   6What are red-black trees, and what are they for?
   7------------------------------------------------
   8
   9Red-black trees are a type of self-balancing binary search tree, used for
  10storing sortable key/value data pairs.  This differs from radix trees (which
  11are used to efficiently store sparse arrays and thus use long integer indexes
  12to insert/access/delete nodes) and hash tables (which are not kept sorted to
  13be easily traversed in order, and must be tuned for a specific size and
  14hash function where rbtrees scale gracefully storing arbitrary keys).
  15
  16Red-black trees are similar to AVL trees, but provide faster real-time bounded
  17worst case performance for insertion and deletion (at most two rotations and
  18three rotations, respectively, to balance the tree), with slightly slower
  19(but still O(log n)) lookup time.
  20
  21To 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
  32This document covers use of the Linux rbtree implementation.  For more
  33information 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
  41Linux implementation of red-black trees
  42---------------------------------------
  43
  44Linux's rbtree implementation lives in the file "lib/rbtree.c".  To use it,
  45"#include <linux/rbtree.h>".
  46
  47The Linux rbtree implementation is optimized for speed, and thus has one
  48less layer of indirection (and better cache locality) than more traditional
  49tree implementations.  Instead of using pointers to separate rb_node and data
  50structures, each instance of struct rb_node is embedded in the data structure
  51it organizes.  And instead of using a comparison callback function pointer,
  52users are expected to write their own tree search and insert functions
  53which call the provided rbtree functions.  Locking is also left up to the
  54user of the rbtree code.
  55
  56Creating a new rbtree
  57---------------------
  58
  59Data 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
  66When dealing with a pointer to the embedded struct rb_node, the containing data
  67structure may be accessed with the standard container_of() macro.  In addition,
  68individual members may be accessed directly via rb_entry(node, type, member).
  69
  70At the root of each rbtree is an rb_root structure, which is initialized to be
  71empty via:
  72
  73  struct rb_root mytree = RB_ROOT;
  74
  75Searching for a value in an rbtree
  76----------------------------------
  77
  78Writing a search function for your tree is fairly straightforward: start at the
  79root, compare each value, and follow the left or right branch as necessary.
  80
  81Example:
  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
 103Inserting data into an rbtree
 104-----------------------------
 105
 106Inserting data in the tree involves first searching for the place to insert the
 107new node, then inserting the node and rebalancing ("recoloring") the tree.
 108
 109The search for insertion differs from the previous search by finding the
 110location of the pointer on which to graft the new node.  The new node also
 111needs a link to its parent node for rebalancing purposes.
 112
 113Example:
 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
 140Removing or replacing existing data in an rbtree
 141------------------------------------------------
 142
 143To remove an existing node from a tree, call:
 144
 145  void rb_erase(struct rb_node *victim, struct rb_root *tree);
 146
 147Example:
 148
 149  struct mytype *data = mysearch(&mytree, "walrus");
 150
 151  if (data) {
 152        rb_erase(&data->node, &mytree);
 153        myfree(data);
 154  }
 155
 156To 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
 161Replacing a node this way does not re-sort the tree: If the new node doesn't
 162have the same key as the old node, the rbtree will probably become corrupted.
 163
 164Iterating through the elements stored in an rbtree (in sort order)
 165------------------------------------------------------------------
 166
 167Four functions are provided for iterating through an rbtree's contents in
 168sorted order.  These work on arbitrary trees, and should not need to be
 169modified 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
 176To start iterating, call rb_first() or rb_last() with a pointer to the root
 177of the tree, which will return a pointer to the node structure contained in
 178the first or last element in the tree.  To continue, fetch the next or previous
 179node by calling rb_next() or rb_prev() on the current node.  This will return
 180NULL when there are no more nodes left.
 181
 182The iterator functions return a pointer to the embedded struct rb_node, from
 183which the containing data structure may be accessed with the container_of()
 184macro, and individual members may be accessed directly via
 185rb_entry(node, type, member).
 186
 187Example:
 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
 193Support for Augmented rbtrees
 194-----------------------------
 195
 196Augmented rbtree is an rbtree with "some" additional data stored in
 197each node, where the additional data for node N must be a function of
 198the contents of all nodes in the subtree rooted at N. This data can
 199be used to augment some new functionality to rbtree. Augmented rbtree
 200is an optional feature built on top of basic rbtree infrastructure.
 201An rbtree user who wants this feature will have to call the augmentation
 202functions with the user provided augmentation callback when inserting
 203and erasing nodes.
 204
 205C files implementing augmented rbtree manipulation must include
 206<linux/rbtree_augmented.h> instead of <linus/rbtree.h>. Note that
 207linux/rbtree_augmented.h exposes some rbtree implementations details
 208you are not expected to rely on; please stick to the documented APIs
 209there and do not include <linux/rbtree_augmented.h> from header files
 210either so as to minimize chances of your users accidentally relying on
 211such implementation details.
 212
 213On insertion, the user must update the augmented information on the path
 214leading to the inserted node, then call rb_link_node() as usual and
 215rb_augment_inserted() instead of the usual rb_insert_color() call.
 216If rb_augment_inserted() rebalances the rbtree, it will callback into
 217a user provided function to update the augmented information on the
 218affected subtrees.
 219
 220When erasing a node, the user must call rb_erase_augmented() instead of
 221rb_erase(). rb_erase_augmented() calls back into user provided functions
 222to updated the augmented information on affected subtrees.
 223
 224In both cases, the callbacks are provided through struct rb_augment_callbacks.
 2253 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
 238The compiled code for rb_erase_augmented() may inline the propagation and
 239copy callbacks, which results in a large function, so each augmented rbtree
 240user should have a single rb_erase_augmented() call site in order to limit
 241compiled code size.
 242
 243
 244Sample usage:
 245
 246Interval tree is an example of augmented rb tree. Reference -
 247"Introduction to Algorithms" by Cormen, Leiserson, Rivest and Stein.
 248More details about interval trees:
 249
 250Classical rbtree has a single key and it cannot be directly used to store
 251interval ranges like [lo:hi] and do a quick lookup for any overlap with a new
 252lo:hi or to find whether there is an exact match for a new lo:hi.
 253
 254However, rbtree can be augmented to store such interval ranges in a structured
 255way making it possible to do efficient lookup and exact match.
 256
 257This "extra information" stored in each node is the maximum hi
 258(max_hi) value among all the nodes that are its descendents. This
 259information can be maintained at each node just be looking at the node
 260and its immediate children. And this will be used in O(log n) lookup
 261for lowest match (lowest start address among all possible matches)
 262with something like:
 263
 264struct interval_tree_node *
 265interval_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
 306Insertion/removal are defined using the following augmented callbacks:
 307
 308static inline unsigned long
 309compute_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
 327static 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
 340static 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
 350static 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
 361static const struct rb_augment_callbacks augment_callbacks = {
 362        augment_propagate, augment_copy, augment_rotate
 363};
 364
 365void 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
 388void interval_tree_remove(struct interval_tree_node *node,
 389                          struct rb_root *root)
 390{
 391        rb_erase_augmented(&node->rb, root, &augment_callbacks);
 392}
 393
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