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Based on kernel version 3.9. Page generated on 2013-05-02 23:13 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
197	each node, where the additional data for node N must be a function of
198	the contents of all nodes in the subtree rooted at N. This data can
199	be used to augment some new functionality to rbtree. Augmented rbtree
200	is an optional feature built on top of basic rbtree infrastructure.
201	An rbtree user who wants this feature will have to call the augmentation
202	functions with the user provided augmentation callback when inserting
203	and erasing nodes.
204	
205	C files implementing augmented rbtree manipulation must include
206	<linux/rbtree_augmented.h> instead of <linus/rbtree.h>. Note that
207	linux/rbtree_augmented.h exposes some rbtree implementations details
208	you are not expected to rely on; please stick to the documented APIs
209	there and do not include <linux/rbtree_augmented.h> from header files
210	either so as to minimize chances of your users accidentally relying on
211	such implementation details.
212	
213	On insertion, the user must update the augmented information on the path
214	leading to the inserted node, then call rb_link_node() as usual and
215	rb_augment_inserted() instead of the usual rb_insert_color() call.
216	If rb_augment_inserted() rebalances the rbtree, it will callback into
217	a user provided function to update the augmented information on the
218	affected subtrees.
219	
220	When erasing a node, the user must call rb_erase_augmented() instead of
221	rb_erase(). rb_erase_augmented() calls back into user provided functions
222	to updated the augmented information on affected subtrees.
223	
224	In both cases, the callbacks are provided through struct rb_augment_callbacks.
225	3 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	
238	The compiled code for rb_erase_augmented() may inline the propagation and
239	copy callbacks, which results in a large function, so each augmented rbtree
240	user should have a single rb_erase_augmented() call site in order to limit
241	compiled code size.
242	
243	
244	Sample usage:
245	
246	Interval tree is an example of augmented rb tree. Reference -
247	"Introduction to Algorithms" by Cormen, Leiserson, Rivest and Stein.
248	More details about interval trees:
249	
250	Classical rbtree has a single key and it cannot be directly used to store
251	interval ranges like [lo:hi] and do a quick lookup for any overlap with a new
252	lo:hi or to find whether there is an exact match for a new lo:hi.
253	
254	However, rbtree can be augmented to store such interval ranges in a structured
255	way making it possible to do efficient lookup and exact match.
256	
257	This "extra information" stored in each node is the maximum hi
258	(max_hi) value among all the nodes that are its descendents. This
259	information can be maintained at each node just be looking at the node
260	and its immediate children. And this will be used in O(log n) lookup
261	for lowest match (lowest start address among all possible matches)
262	with something like:
263	
264	struct interval_tree_node *
265	interval_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	
306	Insertion/removal are defined using the following augmented callbacks:
307	
308	static inline unsigned long
309	compute_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	
327	static 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	
340	static 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	
350	static 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	
361	static const struct rb_augment_callbacks augment_callbacks = {
362		augment_propagate, augment_copy, augment_rotate
363	};
364	
365	void 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	
388	void interval_tree_remove(struct interval_tree_node *node,
389				  struct rb_root *root)
390	{
391		rb_erase_augmented(&node->rb, root, &augment_callbacks);
392	}
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