Root/Documentation/rbtree.txt

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

Archive Download this file



interactive