linux/Documentation/rbtree.txt
<<
>>
Prefs
   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 each node.
 197This data can be used to augment some new functionality to rbtree.
 198Augmented rbtree is an optional feature built on top of basic rbtree
 199infrastructure. An rbtree user who wants this feature will have to call the
 200augmentation functions with the user provided augmentation callback
 201when inserting and erasing nodes.
 202
 203On insertion, the user must call rb_augment_insert() once the new node is in
 204place. This will cause the augmentation function callback to be called for
 205each node between the new node and the root which has been affected by the
 206insertion.
 207
 208When erasing a node, the user must call rb_augment_erase_begin() first to
 209retrieve the deepest node on the rebalance path. Then, after erasing the
 210original node, the user must call rb_augment_erase_end() with the deepest
 211node found earlier. This will cause the augmentation function to be called
 212for each affected node between the deepest node and the root.
 213
 214
 215Interval tree is an example of augmented rb tree. Reference -
 216"Introduction to Algorithms" by Cormen, Leiserson, Rivest and Stein.
 217More details about interval trees:
 218
 219Classical rbtree has a single key and it cannot be directly used to store
 220interval ranges like [lo:hi] and do a quick lookup for any overlap with a new
 221lo:hi or to find whether there is an exact match for a new lo:hi.
 222
 223However, rbtree can be augmented to store such interval ranges in a structured
 224way making it possible to do efficient lookup and exact match.
 225
 226This "extra information" stored in each node is the maximum hi
 227(max_hi) value among all the nodes that are its descendents. This
 228information can be maintained at each node just be looking at the node
 229and its immediate children. And this will be used in O(log n) lookup
 230for lowest match (lowest start address among all possible matches)
 231with something like:
 232
 233find_lowest_match(lo, hi, node)
 234{
 235        lowest_match = NULL;
 236        while (node) {
 237                if (max_hi(node->left) > lo) {
 238                        // Lowest overlap if any must be on left side
 239                        node = node->left;
 240                } else if (overlap(lo, hi, node)) {
 241                        lowest_match = node;
 242                        break;
 243                } else if (lo > node->lo) {
 244                        // Lowest overlap if any must be on right side
 245                        node = node->right;
 246                } else {
 247                        break;
 248                }
 249        }
 250        return lowest_match;
 251}
 252
 253Finding exact match will be to first find lowest match and then to follow
 254successor nodes looking for exact match, until the start of a node is beyond
 255the hi value we are looking for.
 256
lxr.linux.no kindly hosted by Redpill Linpro AS, provider of Linux consulting and operations services since 1995.