1================================= 2Red-black Trees (rbtree) in Linux 3================================= 4 5 6:Date: January 18, 2007 7:Author: Rob Landley <rob@landley.net> 8 9What are red-black trees, and what are they for? 10------------------------------------------------ 11 12Red-black trees are a type of self-balancing binary search tree, used for 13storing sortable key/value data pairs. This differs from radix trees (which 14are used to efficiently store sparse arrays and thus use long integer indexes 15to insert/access/delete nodes) and hash tables (which are not kept sorted to 16be easily traversed in order, and must be tuned for a specific size and 17hash function where rbtrees scale gracefully storing arbitrary keys). 18 19Red-black trees are similar to AVL trees, but provide faster real-time bounded 20worst case performance for insertion and deletion (at most two rotations and 21three rotations, respectively, to balance the tree), with slightly slower 22(but still O(log n)) lookup time. 23 24To 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 35This document covers use of the Linux rbtree implementation. For more 36information on the nature and implementation of Red Black Trees, see: 37 38 Linux Weekly News article on red-black trees 39 https://lwn.net/Articles/184495/ 40 41 Wikipedia entry on red-black trees 42 https://en.wikipedia.org/wiki/Red-black_tree 43 44Linux implementation of red-black trees 45--------------------------------------- 46 47Linux's rbtree implementation lives in the file "lib/rbtree.c". To use it, 48"#include <linux/rbtree.h>". 49 50The Linux rbtree implementation is optimized for speed, and thus has one 51less layer of indirection (and better cache locality) than more traditional 52tree implementations. Instead of using pointers to separate rb_node and data 53structures, each instance of struct rb_node is embedded in the data structure 54it organizes. And instead of using a comparison callback function pointer, 55users are expected to write their own tree search and insert functions 56which call the provided rbtree functions. Locking is also left up to the 57user of the rbtree code. 58 59Creating a new rbtree 60--------------------- 61 62Data 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 69When dealing with a pointer to the embedded struct rb_node, the containing data 70structure may be accessed with the standard container_of() macro. In addition, 71individual members may be accessed directly via rb_entry(node, type, member). 72 73At the root of each rbtree is an rb_root structure, which is initialized to be 74empty via: 75 76 struct rb_root mytree = RB_ROOT; 77 78Searching for a value in an rbtree 79---------------------------------- 80 81Writing a search function for your tree is fairly straightforward: start at the 82root, compare each value, and follow the left or right branch as necessary. 83 84Example:: 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 106Inserting data into an rbtree 107----------------------------- 108 109Inserting data in the tree involves first searching for the place to insert the 110new node, then inserting the node and rebalancing ("recoloring") the tree. 111 112The search for insertion differs from the previous search by finding the 113location of the pointer on which to graft the new node. The new node also 114needs a link to its parent node for rebalancing purposes. 115 116Example:: 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 143Removing or replacing existing data in an rbtree 144------------------------------------------------ 145 146To remove an existing node from a tree, call:: 147 148 void rb_erase(struct rb_node *victim, struct rb_root *tree); 149 150Example:: 151 152 struct mytype *data = mysearch(&mytree, "walrus"); 153 154 if (data) { 155 rb_erase(&data->node, &mytree); 156 myfree(data); 157 } 158 159To 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 164Replacing a node this way does not re-sort the tree: If the new node doesn't 165have the same key as the old node, the rbtree will probably become corrupted. 166 167Iterating through the elements stored in an rbtree (in sort order) 168------------------------------------------------------------------ 169 170Four functions are provided for iterating through an rbtree's contents in 171sorted order. These work on arbitrary trees, and should not need to be 172modified 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 179To start iterating, call rb_first() or rb_last() with a pointer to the root 180of the tree, which will return a pointer to the node structure contained in 181the first or last element in the tree. To continue, fetch the next or previous 182node by calling rb_next() or rb_prev() on the current node. This will return 183NULL when there are no more nodes left. 184 185The iterator functions return a pointer to the embedded struct rb_node, from 186which the containing data structure may be accessed with the container_of() 187macro, and individual members may be accessed directly via 188rb_entry(node, type, member). 189 190Example:: 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 196Cached rbtrees 197-------------- 198 199Computing the leftmost (smallest) node is quite a common task for binary 200search trees, such as for traversals or users relying on a the particular 201order for their own logic. To this end, users can use 'struct rb_root_cached' 202to optimize O(logN) rb_first() calls to a simple pointer fetch avoiding 203potentially expensive tree iterations. This is done at negligible runtime 204overhead for maintanence; albeit larger memory footprint. 205 206Similar to the rb_root structure, cached rbtrees are initialized to be 207empty via:: 208 209 struct rb_root_cached mytree = RB_ROOT_CACHED; 210 211Cached rbtree is simply a regular rb_root with an extra pointer to cache the 212leftmost node. This allows rb_root_cached to exist wherever rb_root does, 213which permits augmented trees to be supported as well as only a few extra 214interfaces:: 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 220Both insert and erase calls have their respective counterpart of augmented 221trees:: 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 229Support for Augmented rbtrees 230----------------------------- 231 232Augmented rbtree is an rbtree with "some" additional data stored in 233each node, where the additional data for node N must be a function of 234the contents of all nodes in the subtree rooted at N. This data can 235be used to augment some new functionality to rbtree. Augmented rbtree 236is an optional feature built on top of basic rbtree infrastructure. 237An rbtree user who wants this feature will have to call the augmentation 238functions with the user provided augmentation callback when inserting 239and erasing nodes. 240 241C files implementing augmented rbtree manipulation must include 242<linux/rbtree_augmented.h> instead of <linux/rbtree.h>. Note that 243linux/rbtree_augmented.h exposes some rbtree implementations details 244you are not expected to rely on; please stick to the documented APIs 245there and do not include <linux/rbtree_augmented.h> from header files 246either so as to minimize chances of your users accidentally relying on 247such implementation details. 248 249On insertion, the user must update the augmented information on the path 250leading to the inserted node, then call rb_link_node() as usual and 251rb_augment_inserted() instead of the usual rb_insert_color() call. 252If rb_augment_inserted() rebalances the rbtree, it will callback into 253a user provided function to update the augmented information on the 254affected subtrees. 255 256When erasing a node, the user must call rb_erase_augmented() instead of 257rb_erase(). rb_erase_augmented() calls back into user provided functions 258to updated the augmented information on affected subtrees. 259 260In both cases, the callbacks are provided through struct rb_augment_callbacks. 2613 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 274The compiled code for rb_erase_augmented() may inline the propagation and 275copy callbacks, which results in a large function, so each augmented rbtree 276user should have a single rb_erase_augmented() call site in order to limit 277compiled code size. 278 279 280Sample usage 281^^^^^^^^^^^^ 282 283Interval tree is an example of augmented rb tree. Reference - 284"Introduction to Algorithms" by Cormen, Leiserson, Rivest and Stein. 285More details about interval trees: 286 287Classical rbtree has a single key and it cannot be directly used to store 288interval ranges like [lo:hi] and do a quick lookup for any overlap with a new 289lo:hi or to find whether there is an exact match for a new lo:hi. 290 291However, rbtree can be augmented to store such interval ranges in a structured 292way making it possible to do efficient lookup and exact match. 293 294This "extra information" stored in each node is the maximum hi 295(max_hi) value among all the nodes that are its descendants. This 296information can be maintained at each node just be looking at the node 297and its immediate children. And this will be used in O(log n) lookup 298for lowest match (lowest start address among all possible matches) 299with 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 343Insertion/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 } 430