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Based on kernel version 4.13.3. Page generated on 2017-09-23 13:55 EST.

1	=================================
2	Red-black Trees (rbtree) in Linux
3	=================================
4	
5	
6	:Date: January 18, 2007
7	:Author: Rob Landley <rob@landley.net>
8	
9	What are red-black trees, and what are they for?
10	------------------------------------------------
11	
12	Red-black trees are a type of self-balancing binary search tree, used for
13	storing sortable key/value data pairs.  This differs from radix trees (which
14	are used to efficiently store sparse arrays and thus use long integer indexes
15	to insert/access/delete nodes) and hash tables (which are not kept sorted to
16	be easily traversed in order, and must be tuned for a specific size and
17	hash function where rbtrees scale gracefully storing arbitrary keys).
18	
19	Red-black trees are similar to AVL trees, but provide faster real-time bounded
20	worst case performance for insertion and deletion (at most two rotations and
21	three rotations, respectively, to balance the tree), with slightly slower
22	(but still O(log n)) lookup time.
23	
24	To quote Linux Weekly News:
25	
26	    There are a number of red-black trees in use in the kernel.
27	    The deadline and CFQ I/O schedulers employ rbtrees to
28	    track requests; the packet CD/DVD driver does the same.
29	    The high-resolution timer code uses an rbtree to organize outstanding
30	    timer requests.  The ext3 filesystem tracks directory entries in a
31	    red-black tree.  Virtual memory areas (VMAs) are tracked with red-black
32	    trees, as are epoll file descriptors, cryptographic keys, and network
33	    packets in the "hierarchical token bucket" scheduler.
34	
35	This document covers use of the Linux rbtree implementation.  For more
36	information on the nature and implementation of Red Black Trees,  see:
37	
38	  Linux Weekly News article on red-black trees
39	    http://lwn.net/Articles/184495/
40	
41	  Wikipedia entry on red-black trees
42	    http://en.wikipedia.org/wiki/Red-black_tree
43	
44	Linux implementation of red-black trees
45	---------------------------------------
46	
47	Linux's rbtree implementation lives in the file "lib/rbtree.c".  To use it,
48	"#include <linux/rbtree.h>".
49	
50	The Linux rbtree implementation is optimized for speed, and thus has one
51	less layer of indirection (and better cache locality) than more traditional
52	tree implementations.  Instead of using pointers to separate rb_node and data
53	structures, each instance of struct rb_node is embedded in the data structure
54	it organizes.  And instead of using a comparison callback function pointer,
55	users are expected to write their own tree search and insert functions
56	which call the provided rbtree functions.  Locking is also left up to the
57	user of the rbtree code.
58	
59	Creating a new rbtree
60	---------------------
61	
62	Data nodes in an rbtree tree are structures containing a struct rb_node member::
63	
64	  struct mytype {
65	  	struct rb_node node;
66	  	char *keystring;
67	  };
68	
69	When dealing with a pointer to the embedded struct rb_node, the containing data
70	structure may be accessed with the standard container_of() macro.  In addition,
71	individual members may be accessed directly via rb_entry(node, type, member).
72	
73	At the root of each rbtree is an rb_root structure, which is initialized to be
74	empty via:
75	
76	  struct rb_root mytree = RB_ROOT;
77	
78	Searching for a value in an rbtree
79	----------------------------------
80	
81	Writing a search function for your tree is fairly straightforward: start at the
82	root, compare each value, and follow the left or right branch as necessary.
83	
84	Example::
85	
86	  struct mytype *my_search(struct rb_root *root, char *string)
87	  {
88	  	struct rb_node *node = root->rb_node;
89	
90	  	while (node) {
91	  		struct mytype *data = container_of(node, struct mytype, node);
92			int result;
93	
94			result = strcmp(string, data->keystring);
95	
96			if (result < 0)
97	  			node = node->rb_left;
98			else if (result > 0)
99	  			node = node->rb_right;
100			else
101	  			return data;
102		}
103		return NULL;
104	  }
105	
106	Inserting data into an rbtree
107	-----------------------------
108	
109	Inserting data in the tree involves first searching for the place to insert the
110	new node, then inserting the node and rebalancing ("recoloring") the tree.
111	
112	The search for insertion differs from the previous search by finding the
113	location of the pointer on which to graft the new node.  The new node also
114	needs a link to its parent node for rebalancing purposes.
115	
116	Example::
117	
118	  int my_insert(struct rb_root *root, struct mytype *data)
119	  {
120	  	struct rb_node **new = &(root->rb_node), *parent = NULL;
121	
122	  	/* Figure out where to put new node */
123	  	while (*new) {
124	  		struct mytype *this = container_of(*new, struct mytype, node);
125	  		int result = strcmp(data->keystring, this->keystring);
126	
127			parent = *new;
128	  		if (result < 0)
129	  			new = &((*new)->rb_left);
130	  		else if (result > 0)
131	  			new = &((*new)->rb_right);
132	  		else
133	  			return FALSE;
134	  	}
135	
136	  	/* Add new node and rebalance tree. */
137	  	rb_link_node(&data->node, parent, new);
138	  	rb_insert_color(&data->node, root);
139	
140		return TRUE;
141	  }
142	
143	Removing or replacing existing data in an rbtree
144	------------------------------------------------
145	
146	To remove an existing node from a tree, call::
147	
148	  void rb_erase(struct rb_node *victim, struct rb_root *tree);
149	
150	Example::
151	
152	  struct mytype *data = mysearch(&mytree, "walrus");
153	
154	  if (data) {
155	  	rb_erase(&data->node, &mytree);
156	  	myfree(data);
157	  }
158	
159	To replace an existing node in a tree with a new one with the same key, call::
160	
161	  void rb_replace_node(struct rb_node *old, struct rb_node *new,
162	  			struct rb_root *tree);
163	
164	Replacing a node this way does not re-sort the tree: If the new node doesn't
165	have the same key as the old node, the rbtree will probably become corrupted.
166	
167	Iterating through the elements stored in an rbtree (in sort order)
168	------------------------------------------------------------------
169	
170	Four functions are provided for iterating through an rbtree's contents in
171	sorted order.  These work on arbitrary trees, and should not need to be
172	modified or wrapped (except for locking purposes)::
173	
174	  struct rb_node *rb_first(struct rb_root *tree);
175	  struct rb_node *rb_last(struct rb_root *tree);
176	  struct rb_node *rb_next(struct rb_node *node);
177	  struct rb_node *rb_prev(struct rb_node *node);
178	
179	To start iterating, call rb_first() or rb_last() with a pointer to the root
180	of the tree, which will return a pointer to the node structure contained in
181	the first or last element in the tree.  To continue, fetch the next or previous
182	node by calling rb_next() or rb_prev() on the current node.  This will return
183	NULL when there are no more nodes left.
184	
185	The iterator functions return a pointer to the embedded struct rb_node, from
186	which the containing data structure may be accessed with the container_of()
187	macro, and individual members may be accessed directly via
188	rb_entry(node, type, member).
189	
190	Example::
191	
192	  struct rb_node *node;
193	  for (node = rb_first(&mytree); node; node = rb_next(node))
194		printk("key=%s\n", rb_entry(node, struct mytype, node)->keystring);
195	
196	Support for Augmented rbtrees
197	-----------------------------
198	
199	Augmented rbtree is an rbtree with "some" additional data stored in
200	each node, where the additional data for node N must be a function of
201	the contents of all nodes in the subtree rooted at N. This data can
202	be used to augment some new functionality to rbtree. Augmented rbtree
203	is an optional feature built on top of basic rbtree infrastructure.
204	An rbtree user who wants this feature will have to call the augmentation
205	functions with the user provided augmentation callback when inserting
206	and erasing nodes.
207	
208	C files implementing augmented rbtree manipulation must include
209	<linux/rbtree_augmented.h> instead of <linux/rbtree.h>. Note that
210	linux/rbtree_augmented.h exposes some rbtree implementations details
211	you are not expected to rely on; please stick to the documented APIs
212	there and do not include <linux/rbtree_augmented.h> from header files
213	either so as to minimize chances of your users accidentally relying on
214	such implementation details.
215	
216	On insertion, the user must update the augmented information on the path
217	leading to the inserted node, then call rb_link_node() as usual and
218	rb_augment_inserted() instead of the usual rb_insert_color() call.
219	If rb_augment_inserted() rebalances the rbtree, it will callback into
220	a user provided function to update the augmented information on the
221	affected subtrees.
222	
223	When erasing a node, the user must call rb_erase_augmented() instead of
224	rb_erase(). rb_erase_augmented() calls back into user provided functions
225	to updated the augmented information on affected subtrees.
226	
227	In both cases, the callbacks are provided through struct rb_augment_callbacks.
228	3 callbacks must be defined:
229	
230	- A propagation callback, which updates the augmented value for a given
231	  node and its ancestors, up to a given stop point (or NULL to update
232	  all the way to the root).
233	
234	- A copy callback, which copies the augmented value for a given subtree
235	  to a newly assigned subtree root.
236	
237	- A tree rotation callback, which copies the augmented value for a given
238	  subtree to a newly assigned subtree root AND recomputes the augmented
239	  information for the former subtree root.
240	
241	The compiled code for rb_erase_augmented() may inline the propagation and
242	copy callbacks, which results in a large function, so each augmented rbtree
243	user should have a single rb_erase_augmented() call site in order to limit
244	compiled code size.
245	
246	
247	Sample usage
248	^^^^^^^^^^^^
249	
250	Interval tree is an example of augmented rb tree. Reference -
251	"Introduction to Algorithms" by Cormen, Leiserson, Rivest and Stein.
252	More details about interval trees:
253	
254	Classical rbtree has a single key and it cannot be directly used to store
255	interval ranges like [lo:hi] and do a quick lookup for any overlap with a new
256	lo:hi or to find whether there is an exact match for a new lo:hi.
257	
258	However, rbtree can be augmented to store such interval ranges in a structured
259	way making it possible to do efficient lookup and exact match.
260	
261	This "extra information" stored in each node is the maximum hi
262	(max_hi) value among all the nodes that are its descendants. This
263	information can be maintained at each node just be looking at the node
264	and its immediate children. And this will be used in O(log n) lookup
265	for lowest match (lowest start address among all possible matches)
266	with something like::
267	
268	  struct interval_tree_node *
269	  interval_tree_first_match(struct rb_root *root,
270				    unsigned long start, unsigned long last)
271	  {
272		struct interval_tree_node *node;
273	
274		if (!root->rb_node)
275			return NULL;
276		node = rb_entry(root->rb_node, struct interval_tree_node, rb);
277	
278		while (true) {
279			if (node->rb.rb_left) {
280				struct interval_tree_node *left =
281					rb_entry(node->rb.rb_left,
282						 struct interval_tree_node, rb);
283				if (left->__subtree_last >= start) {
284					/*
285					 * Some nodes in left subtree satisfy Cond2.
286					 * Iterate to find the leftmost such node N.
287					 * If it also satisfies Cond1, that's the match
288					 * we are looking for. Otherwise, there is no
289					 * matching interval as nodes to the right of N
290					 * can't satisfy Cond1 either.
291					 */
292					node = left;
293					continue;
294				}
295			}
296			if (node->start <= last) {		/* Cond1 */
297				if (node->last >= start)	/* Cond2 */
298					return node;	/* node is leftmost match */
299				if (node->rb.rb_right) {
300					node = rb_entry(node->rb.rb_right,
301						struct interval_tree_node, rb);
302					if (node->__subtree_last >= start)
303						continue;
304				}
305			}
306			return NULL;	/* No match */
307		}
308	  }
309	
310	Insertion/removal are defined using the following augmented callbacks::
311	
312	  static inline unsigned long
313	  compute_subtree_last(struct interval_tree_node *node)
314	  {
315		unsigned long max = node->last, subtree_last;
316		if (node->rb.rb_left) {
317			subtree_last = rb_entry(node->rb.rb_left,
318				struct interval_tree_node, rb)->__subtree_last;
319			if (max < subtree_last)
320				max = subtree_last;
321		}
322		if (node->rb.rb_right) {
323			subtree_last = rb_entry(node->rb.rb_right,
324				struct interval_tree_node, rb)->__subtree_last;
325			if (max < subtree_last)
326				max = subtree_last;
327		}
328		return max;
329	  }
330	
331	  static void augment_propagate(struct rb_node *rb, struct rb_node *stop)
332	  {
333		while (rb != stop) {
334			struct interval_tree_node *node =
335				rb_entry(rb, struct interval_tree_node, rb);
336			unsigned long subtree_last = compute_subtree_last(node);
337			if (node->__subtree_last == subtree_last)
338				break;
339			node->__subtree_last = subtree_last;
340			rb = rb_parent(&node->rb);
341		}
342	  }
343	
344	  static void augment_copy(struct rb_node *rb_old, struct rb_node *rb_new)
345	  {
346		struct interval_tree_node *old =
347			rb_entry(rb_old, struct interval_tree_node, rb);
348		struct interval_tree_node *new =
349			rb_entry(rb_new, struct interval_tree_node, rb);
350	
351		new->__subtree_last = old->__subtree_last;
352	  }
353	
354	  static void augment_rotate(struct rb_node *rb_old, struct rb_node *rb_new)
355	  {
356		struct interval_tree_node *old =
357			rb_entry(rb_old, struct interval_tree_node, rb);
358		struct interval_tree_node *new =
359			rb_entry(rb_new, struct interval_tree_node, rb);
360	
361		new->__subtree_last = old->__subtree_last;
362		old->__subtree_last = compute_subtree_last(old);
363	  }
364	
365	  static const struct rb_augment_callbacks augment_callbacks = {
366		augment_propagate, augment_copy, augment_rotate
367	  };
368	
369	  void interval_tree_insert(struct interval_tree_node *node,
370				    struct rb_root *root)
371	  {
372		struct rb_node **link = &root->rb_node, *rb_parent = NULL;
373		unsigned long start = node->start, last = node->last;
374		struct interval_tree_node *parent;
375	
376		while (*link) {
377			rb_parent = *link;
378			parent = rb_entry(rb_parent, struct interval_tree_node, rb);
379			if (parent->__subtree_last < last)
380				parent->__subtree_last = last;
381			if (start < parent->start)
382				link = &parent->rb.rb_left;
383			else
384				link = &parent->rb.rb_right;
385		}
386	
387		node->__subtree_last = last;
388		rb_link_node(&node->rb, rb_parent, link);
389		rb_insert_augmented(&node->rb, root, &augment_callbacks);
390	  }
391	
392	  void interval_tree_remove(struct interval_tree_node *node,
393				    struct rb_root *root)
394	  {
395		rb_erase_augmented(&node->rb, root, &augment_callbacks);
396	  }
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