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Based on kernel version 4.15. Page generated on 2018-01-29 10:00 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	Cached rbtrees
197	--------------
198	
199	Computing the leftmost (smallest) node is quite a common task for binary
200	search trees, such as for traversals or users relying on a the particular
201	order for their own logic. To this end, users can use 'struct rb_root_cached'
202	to optimize O(logN) rb_first() calls to a simple pointer fetch avoiding
203	potentially expensive tree iterations. This is done at negligible runtime
204	overhead for maintanence; albeit larger memory footprint.
205	
206	Similar to the rb_root structure, cached rbtrees are initialized to be
207	empty via:
208	
209	  struct rb_root_cached mytree = RB_ROOT_CACHED;
210	
211	Cached rbtree is simply a regular rb_root with an extra pointer to cache the
212	leftmost node. This allows rb_root_cached to exist wherever rb_root does,
213	which permits augmented trees to be supported as well as only a few extra
214	interfaces:
215	
216	  struct rb_node *rb_first_cached(struct rb_root_cached *tree);
217	  void rb_insert_color_cached(struct rb_node *, struct rb_root_cached *, bool);
218	  void rb_erase_cached(struct rb_node *node, struct rb_root_cached *);
219	
220	Both insert and erase calls have their respective counterpart of augmented
221	trees:
222	
223	  void rb_insert_augmented_cached(struct rb_node *node, struct rb_root_cached *,
224					  bool, struct rb_augment_callbacks *);
225	  void rb_erase_augmented_cached(struct rb_node *, struct rb_root_cached *,
226					 struct rb_augment_callbacks *);
227	
228	
229	Support for Augmented rbtrees
230	-----------------------------
231	
232	Augmented rbtree is an rbtree with "some" additional data stored in
233	each node, where the additional data for node N must be a function of
234	the contents of all nodes in the subtree rooted at N. This data can
235	be used to augment some new functionality to rbtree. Augmented rbtree
236	is an optional feature built on top of basic rbtree infrastructure.
237	An rbtree user who wants this feature will have to call the augmentation
238	functions with the user provided augmentation callback when inserting
239	and erasing nodes.
240	
241	C files implementing augmented rbtree manipulation must include
242	<linux/rbtree_augmented.h> instead of <linux/rbtree.h>. Note that
243	linux/rbtree_augmented.h exposes some rbtree implementations details
244	you are not expected to rely on; please stick to the documented APIs
245	there and do not include <linux/rbtree_augmented.h> from header files
246	either so as to minimize chances of your users accidentally relying on
247	such implementation details.
248	
249	On insertion, the user must update the augmented information on the path
250	leading to the inserted node, then call rb_link_node() as usual and
251	rb_augment_inserted() instead of the usual rb_insert_color() call.
252	If rb_augment_inserted() rebalances the rbtree, it will callback into
253	a user provided function to update the augmented information on the
254	affected subtrees.
255	
256	When erasing a node, the user must call rb_erase_augmented() instead of
257	rb_erase(). rb_erase_augmented() calls back into user provided functions
258	to updated the augmented information on affected subtrees.
259	
260	In both cases, the callbacks are provided through struct rb_augment_callbacks.
261	3 callbacks must be defined:
262	
263	- A propagation callback, which updates the augmented value for a given
264	  node and its ancestors, up to a given stop point (or NULL to update
265	  all the way to the root).
266	
267	- A copy callback, which copies the augmented value for a given subtree
268	  to a newly assigned subtree root.
269	
270	- A tree rotation callback, which copies the augmented value for a given
271	  subtree to a newly assigned subtree root AND recomputes the augmented
272	  information for the former subtree root.
273	
274	The compiled code for rb_erase_augmented() may inline the propagation and
275	copy callbacks, which results in a large function, so each augmented rbtree
276	user should have a single rb_erase_augmented() call site in order to limit
277	compiled code size.
278	
279	
280	Sample usage
281	^^^^^^^^^^^^
282	
283	Interval tree is an example of augmented rb tree. Reference -
284	"Introduction to Algorithms" by Cormen, Leiserson, Rivest and Stein.
285	More details about interval trees:
286	
287	Classical rbtree has a single key and it cannot be directly used to store
288	interval ranges like [lo:hi] and do a quick lookup for any overlap with a new
289	lo:hi or to find whether there is an exact match for a new lo:hi.
290	
291	However, rbtree can be augmented to store such interval ranges in a structured
292	way making it possible to do efficient lookup and exact match.
293	
294	This "extra information" stored in each node is the maximum hi
295	(max_hi) value among all the nodes that are its descendants. This
296	information can be maintained at each node just be looking at the node
297	and its immediate children. And this will be used in O(log n) lookup
298	for lowest match (lowest start address among all possible matches)
299	with something like::
300	
301	  struct interval_tree_node *
302	  interval_tree_first_match(struct rb_root *root,
303				    unsigned long start, unsigned long last)
304	  {
305		struct interval_tree_node *node;
306	
307		if (!root->rb_node)
308			return NULL;
309		node = rb_entry(root->rb_node, struct interval_tree_node, rb);
310	
311		while (true) {
312			if (node->rb.rb_left) {
313				struct interval_tree_node *left =
314					rb_entry(node->rb.rb_left,
315						 struct interval_tree_node, rb);
316				if (left->__subtree_last >= start) {
317					/*
318					 * Some nodes in left subtree satisfy Cond2.
319					 * Iterate to find the leftmost such node N.
320					 * If it also satisfies Cond1, that's the match
321					 * we are looking for. Otherwise, there is no
322					 * matching interval as nodes to the right of N
323					 * can't satisfy Cond1 either.
324					 */
325					node = left;
326					continue;
327				}
328			}
329			if (node->start <= last) {		/* Cond1 */
330				if (node->last >= start)	/* Cond2 */
331					return node;	/* node is leftmost match */
332				if (node->rb.rb_right) {
333					node = rb_entry(node->rb.rb_right,
334						struct interval_tree_node, rb);
335					if (node->__subtree_last >= start)
336						continue;
337				}
338			}
339			return NULL;	/* No match */
340		}
341	  }
342	
343	Insertion/removal are defined using the following augmented callbacks::
344	
345	  static inline unsigned long
346	  compute_subtree_last(struct interval_tree_node *node)
347	  {
348		unsigned long max = node->last, subtree_last;
349		if (node->rb.rb_left) {
350			subtree_last = rb_entry(node->rb.rb_left,
351				struct interval_tree_node, rb)->__subtree_last;
352			if (max < subtree_last)
353				max = subtree_last;
354		}
355		if (node->rb.rb_right) {
356			subtree_last = rb_entry(node->rb.rb_right,
357				struct interval_tree_node, rb)->__subtree_last;
358			if (max < subtree_last)
359				max = subtree_last;
360		}
361		return max;
362	  }
363	
364	  static void augment_propagate(struct rb_node *rb, struct rb_node *stop)
365	  {
366		while (rb != stop) {
367			struct interval_tree_node *node =
368				rb_entry(rb, struct interval_tree_node, rb);
369			unsigned long subtree_last = compute_subtree_last(node);
370			if (node->__subtree_last == subtree_last)
371				break;
372			node->__subtree_last = subtree_last;
373			rb = rb_parent(&node->rb);
374		}
375	  }
376	
377	  static void augment_copy(struct rb_node *rb_old, struct rb_node *rb_new)
378	  {
379		struct interval_tree_node *old =
380			rb_entry(rb_old, struct interval_tree_node, rb);
381		struct interval_tree_node *new =
382			rb_entry(rb_new, struct interval_tree_node, rb);
383	
384		new->__subtree_last = old->__subtree_last;
385	  }
386	
387	  static void augment_rotate(struct rb_node *rb_old, struct rb_node *rb_new)
388	  {
389		struct interval_tree_node *old =
390			rb_entry(rb_old, struct interval_tree_node, rb);
391		struct interval_tree_node *new =
392			rb_entry(rb_new, struct interval_tree_node, rb);
393	
394		new->__subtree_last = old->__subtree_last;
395		old->__subtree_last = compute_subtree_last(old);
396	  }
397	
398	  static const struct rb_augment_callbacks augment_callbacks = {
399		augment_propagate, augment_copy, augment_rotate
400	  };
401	
402	  void interval_tree_insert(struct interval_tree_node *node,
403				    struct rb_root *root)
404	  {
405		struct rb_node **link = &root->rb_node, *rb_parent = NULL;
406		unsigned long start = node->start, last = node->last;
407		struct interval_tree_node *parent;
408	
409		while (*link) {
410			rb_parent = *link;
411			parent = rb_entry(rb_parent, struct interval_tree_node, rb);
412			if (parent->__subtree_last < last)
413				parent->__subtree_last = last;
414			if (start < parent->start)
415				link = &parent->rb.rb_left;
416			else
417				link = &parent->rb.rb_right;
418		}
419	
420		node->__subtree_last = last;
421		rb_link_node(&node->rb, rb_parent, link);
422		rb_insert_augmented(&node->rb, root, &augment_callbacks);
423	  }
424	
425	  void interval_tree_remove(struct interval_tree_node *node,
426				    struct rb_root *root)
427	  {
428		rb_erase_augmented(&node->rb, root, &augment_callbacks);
429	  }
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