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 each node. 197This data can be used to augment some new functionality to rbtree. 198Augmented rbtree is an optional feature built on top of basic rbtree 199infrastructure. rbtree user who wants this feature will have an augment 200callback function in rb_root initialized. 201 202This callback function will be called from rbtree core routines whenever 203a node has a change in one or both of its children. It is the responsibility 204of the callback function to recalculate the additional data that is in the 205rb node using new children information. Note that if this new additional 206data affects the parent node's additional data, then callback function has 207to handle it and do the recursive updates. 208 209 210Interval tree is an example of augmented rb tree. Reference - 211"Introduction to Algorithms" by Cormen, Leiserson, Rivest and Stein. 212More details about interval trees: 213 214Classical rbtree has a single key and it cannot be directly used to store 215interval ranges like [lo:hi] and do a quick lookup for any overlap with a new 216lo:hi or to find whether there is an exact match for a new lo:hi. 217 218However, rbtree can be augmented to store such interval ranges in a structured 219way making it possible to do efficient lookup and exact match. 220 221This "extra information" stored in each node is the maximum hi 222(max_hi) value among all the nodes that are its descendents. This 223information can be maintained at each node just be looking at the node 224and its immediate children. And this will be used in O(log n) lookup 225for lowest match (lowest start address among all possible matches) 226with something like: 227 228find_lowest_match(lo, hi, node) 229{ 230 lowest_match = NULL; 231 while (node) { 232 if (max_hi(node->left) > lo) { 233 // Lowest overlap if any must be on left side 234 node = node->left; 235 } else if (overlap(lo, hi, node)) { 236 lowest_match = node; 237 break; 238 } else if (lo > node->lo) { 239 // Lowest overlap if any must be on right side 240 node = node->right; 241 } else { 242 break; 243 } 244 } 245 return lowest_match; 246} 247 248Finding exact match will be to first find lowest match and then to follow 249successor nodes looking for exact match, until the start of a node is beyond 250the hi value we are looking for. 251