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Rob Landleyc742b532007-02-10 01:46:20 -08001Red-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.
Randy Dunlap17a9e7b2010-11-11 12:09:59 +010024 The deadline and CFQ I/O schedulers employ rbtrees to
25 track requests; the packet CD/DVD driver does the same.
Rob Landleyc742b532007-02-10 01:46:20 -080026 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. */
figo.zhang27af1da2009-04-17 10:58:48 +0800134 rb_link_node(&data->node, parent, new);
135 rb_insert_color(&data->node, root);
Rob Landleyc742b532007-02-10 01:46:20 -0800136
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
figo.zhang27af1da2009-04-17 10:58:48 +0800149 struct mytype *data = mysearch(&mytree, "walrus");
Rob Landleyc742b532007-02-10 01:46:20 -0800150
151 if (data) {
figo.zhang27af1da2009-04-17 10:58:48 +0800152 rb_erase(&data->node, &mytree);
Rob Landleyc742b532007-02-10 01:46:20 -0800153 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))
Wang Tinggong19034232009-05-14 11:00:20 +0200191 printk("key=%s\n", rb_entry(node, struct mytype, node)->keystring);
Rob Landleyc742b532007-02-10 01:46:20 -0800192
Pallipadi, Venkatesh17d9ddc2010-02-10 15:23:44 -0800193Support 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. rbtree user who wants this feature will have an augment
200callback function in rb_root initialized.
201
202This callback function will be called from rbtree core routines whenever
203a node has a change in one or both of its children. It is the responsibility
204of the callback function to recalculate the additional data that is in the
205rb node using new children information. Note that if this new additional
206data affects the parent node's additional data, then callback function has
207to handle it and do the recursive updates.
208
209
210Interval tree is an example of augmented rb tree. Reference -
211"Introduction to Algorithms" by Cormen, Leiserson, Rivest and Stein.
212More details about interval trees:
213
214Classical rbtree has a single key and it cannot be directly used to store
215interval ranges like [lo:hi] and do a quick lookup for any overlap with a new
216lo:hi or to find whether there is an exact match for a new lo:hi.
217
218However, rbtree can be augmented to store such interval ranges in a structured
219way making it possible to do efficient lookup and exact match.
220
221This "extra information" stored in each node is the maximum hi
222(max_hi) value among all the nodes that are its descendents. This
223information can be maintained at each node just be looking at the node
224and its immediate children. And this will be used in O(log n) lookup
225for lowest match (lowest start address among all possible matches)
226with something like:
227
228find_lowest_match(lo, hi, node)
229{
230 lowest_match = NULL;
231 while (node) {
232 if (max_hi(node->left) > lo) {
233 // Lowest overlap if any must be on left side
234 node = node->left;
235 } else if (overlap(lo, hi, node)) {
236 lowest_match = node;
237 break;
238 } else if (lo > node->lo) {
239 // Lowest overlap if any must be on right side
240 node = node->right;
241 } else {
242 break;
243 }
244 }
245 return lowest_match;
246}
247
248Finding exact match will be to first find lowest match and then to follow
249successor nodes looking for exact match, until the start of a node is beyond
250the hi value we are looking for.