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Based on kernel version 3.4. Page generated on 2012-05-21 22:09 EST.

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 each node.
197	This data can be used to augment some new functionality to rbtree.
198	Augmented rbtree is an optional feature built on top of basic rbtree
199	infrastructure. An rbtree user who wants this feature will have to call the
200	augmentation functions with the user provided augmentation callback
201	when inserting and erasing nodes.
202	
203	On insertion, the user must call rb_augment_insert() once the new node is in
204	place. This will cause the augmentation function callback to be called for
205	each node between the new node and the root which has been affected by the
206	insertion.
207	
208	When erasing a node, the user must call rb_augment_erase_begin() first to
209	retrieve the deepest node on the rebalance path. Then, after erasing the
210	original node, the user must call rb_augment_erase_end() with the deepest
211	node found earlier. This will cause the augmentation function to be called
212	for each affected node between the deepest node and the root.
213	
214	
215	Interval tree is an example of augmented rb tree. Reference -
216	"Introduction to Algorithms" by Cormen, Leiserson, Rivest and Stein.
217	More details about interval trees:
218	
219	Classical rbtree has a single key and it cannot be directly used to store
220	interval ranges like [lo:hi] and do a quick lookup for any overlap with a new
221	lo:hi or to find whether there is an exact match for a new lo:hi.
222	
223	However, rbtree can be augmented to store such interval ranges in a structured
224	way making it possible to do efficient lookup and exact match.
225	
226	This "extra information" stored in each node is the maximum hi
227	(max_hi) value among all the nodes that are its descendents. This
228	information can be maintained at each node just be looking at the node
229	and its immediate children. And this will be used in O(log n) lookup
230	for lowest match (lowest start address among all possible matches)
231	with something like:
232	
233	find_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	
253	Finding exact match will be to first find lowest match and then to follow
254	successor nodes looking for exact match, until the start of a node is beyond
255	the hi value we are looking for.
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