1Red-black Trees (rbtree) in Linux 2January 18, 2007 3Rob Landley <rob@landley.net> 4============================= 5 6What are red-black trees, and what are they for? 7------------------------------------------------ 8 9Red-black trees are a type of self-balancing binary search tree, used for 10storing sortable key/value data pairs. This differs from radix trees (which 11are used to efficiently store sparse arrays and thus use long integer indexes 12to insert/access/delete nodes) and hash tables (which are not kept sorted to 13be easily traversed in order, and must be tuned for a specific size and 14hash function where rbtrees scale gracefully storing arbitrary keys). 15 16Red-black trees are similar to AVL trees, but provide faster real-time bounded 17worst case performance for insertion and deletion (at most two rotations and 18three rotations, respectively, to balance the tree), with slightly slower 19(but still O(log n)) lookup time. 20 21To 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 32This document covers use of the Linux rbtree implementation. For more 33information 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 41Linux implementation of red-black trees 42--------------------------------------- 43 44Linux's rbtree implementation lives in the file "lib/rbtree.c". To use it, 45"#include <linux/rbtree.h>". 46 47The Linux rbtree implementation is optimized for speed, and thus has one 48less layer of indirection (and better cache locality) than more traditional 49tree implementations. Instead of using pointers to separate rb_node and data 50structures, each instance of struct rb_node is embedded in the data structure 51it organizes. And instead of using a comparison callback function pointer, 52users are expected to write their own tree search and insert functions 53which call the provided rbtree functions. Locking is also left up to the 54user of the rbtree code. 55 56Creating a new rbtree 57--------------------- 58 59Data 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 66When dealing with a pointer to the embedded struct rb_node, the containing data 67structure may be accessed with the standard container_of() macro. In addition, 68individual members may be accessed directly via rb_entry(node, type, member). 69 70At the root of each rbtree is an rb_root structure, which is initialized to be 71empty via: 72 73 struct rb_root mytree = RB_ROOT; 74 75Searching for a value in an rbtree 76---------------------------------- 77 78Writing a search function for your tree is fairly straightforward: start at the 79root, compare each value, and follow the left or right branch as necessary. 80 81Example: 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 103Inserting data into an rbtree 104----------------------------- 105 106Inserting data in the tree involves first searching for the place to insert the 107new node, then inserting the node and rebalancing ("recoloring") the tree. 108 109The search for insertion differs from the previous search by finding the 110location of the pointer on which to graft the new node. The new node also 111needs a link to its parent node for rebalancing purposes. 112 113Example: 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 140Removing or replacing existing data in an rbtree 141------------------------------------------------ 142 143To remove an existing node from a tree, call: 144 145 void rb_erase(struct rb_node *victim, struct rb_root *tree); 146 147Example: 148 149 struct mytype *data = mysearch(&mytree, "walrus"); 150 151 if (data) { 152 rb_erase(&data->node, &mytree); 153 myfree(data); 154 } 155 156To 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 161Replacing a node this way does not re-sort the tree: If the new node doesn't 162have the same key as the old node, the rbtree will probably become corrupted. 163 164Iterating through the elements stored in an rbtree (in sort order) 165------------------------------------------------------------------ 166 167Four functions are provided for iterating through an rbtree's contents in 168sorted order. These work on arbitrary trees, and should not need to be 169modified 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 176To start iterating, call rb_first() or rb_last() with a pointer to the root 177of the tree, which will return a pointer to the node structure contained in 178the first or last element in the tree. To continue, fetch the next or previous 179node by calling rb_next() or rb_prev() on the current node. This will return 180NULL when there are no more nodes left. 181 182The iterator functions return a pointer to the embedded struct rb_node, from 183which the containing data structure may be accessed with the container_of() 184macro, and individual members may be accessed directly via 185rb_entry(node, type, member). 186 187Example: 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 193Support for Augmented rbtrees 194----------------------------- 195 196Augmented rbtree is an rbtree with "some" additional data stored in 197each node, where the additional data for node N must be a function of 198the contents of all nodes in the subtree rooted at N. This data can 199be used to augment some new functionality to rbtree. Augmented rbtree 200is an optional feature built on top of basic rbtree infrastructure. 201An rbtree user who wants this feature will have to call the augmentation 202functions with the user provided augmentation callback when inserting 203and erasing nodes. 204 205C files implementing augmented rbtree manipulation must include 206<linux/rbtree_augmented.h> instead of <linus/rbtree.h>. Note that 207linux/rbtree_augmented.h exposes some rbtree implementations details 208you are not expected to rely on; please stick to the documented APIs 209there and do not include <linux/rbtree_augmented.h> from header files 210either so as to minimize chances of your users accidentally relying on 211such implementation details. 212 213On insertion, the user must update the augmented information on the path 214leading to the inserted node, then call rb_link_node() as usual and 215rb_augment_inserted() instead of the usual rb_insert_color() call. 216If rb_augment_inserted() rebalances the rbtree, it will callback into 217a user provided function to update the augmented information on the 218affected subtrees. 219 220When erasing a node, the user must call rb_erase_augmented() instead of 221rb_erase(). rb_erase_augmented() calls back into user provided functions 222to updated the augmented information on affected subtrees. 223 224In both cases, the callbacks are provided through struct rb_augment_callbacks. 2253 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 238The compiled code for rb_erase_augmented() may inline the propagation and 239copy callbacks, which results in a large function, so each augmented rbtree 240user should have a single rb_erase_augmented() call site in order to limit 241compiled code size. 242 243 244Sample usage: 245 246Interval tree is an example of augmented rb tree. Reference - 247"Introduction to Algorithms" by Cormen, Leiserson, Rivest and Stein. 248More details about interval trees: 249 250Classical rbtree has a single key and it cannot be directly used to store 251interval ranges like [lo:hi] and do a quick lookup for any overlap with a new 252lo:hi or to find whether there is an exact match for a new lo:hi. 253 254However, rbtree can be augmented to store such interval ranges in a structured 255way making it possible to do efficient lookup and exact match. 256 257This "extra information" stored in each node is the maximum hi 258(max_hi) value among all the nodes that are its descendants. This 259information can be maintained at each node just be looking at the node 260and its immediate children. And this will be used in O(log n) lookup 261for lowest match (lowest start address among all possible matches) 262with something like: 263 264struct interval_tree_node * 265interval_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 306Insertion/removal are defined using the following augmented callbacks: 307 308static inline unsigned long 309compute_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 327static 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 340static 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 350static 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 361static const struct rb_augment_callbacks augment_callbacks = { 362 augment_propagate, augment_copy, augment_rotate 363}; 364 365void 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 388void interval_tree_remove(struct interval_tree_node *node, 389 struct rb_root *root) 390{ 391 rb_erase_augmented(&node->rb, root, &augment_callbacks); 392} 393