What are red-black trees, and what are they for?¶

Red-black trees are a type of self-balancing binary search tree, used forstoring sortable key/value data pairs. This differs from radix trees (whichare used to efficiently store sparse arrays and thus use long integer indexesto insert/access/delete nodes) and hash tables (which are not kept sorted tobe easily traversed in order, and must be tuned for a specific size andhash function where rbtrees scale gracefully storing arbitrary keys).

Red-black trees are similar to AVL trees, but provide faster real-time boundedworst case performance for insertion and deletion (at most two rotations andthree rotations, respectively, to balance the tree), with slightly slower(but still O(log n)) lookup time.

To quote Linux Weekly News:

There are a number of red-black trees in use in the kernel.The deadline and CFQ I/O schedulers employ rbtrees totrack requests; the packet CD/DVD driver does the same.The high-resolution timer code uses an rbtree to organize outstandingtimer requests. The ext3 filesystem tracks directory entries in ared-black tree. Virtual memory areas (VMAs) are tracked with red-blacktrees, as are epoll file descriptors, cryptographic keys, and networkpackets in the “hierarchical token bucket” scheduler.

This document covers use of the Linux rbtree implementation. For moreinformation on the nature and implementation of Red Black Trees, see:

Linux Weekly News article on red-black trees

https://lwn.net/Articles/184495/

Wikipedia entry on red-black trees

https://en.wikipedia.org/wiki/Red-black_tree

Linux implementation of red-black trees¶

Linux’s rbtree implementation lives in the file “lib/rbtree.c”. To use it,“#include <linux/rbtree.h>”.

The Linux rbtree implementation is optimized for speed, and thus has oneless layer of indirection (and better cache locality) than more traditionaltree implementations. Instead of using pointers to separate rb_node and datastructures, each instance of struct rb_node is embedded in the data structureit organizes. And instead of using a comparison callback function pointer,users are expected to write their own tree search and insert functionswhich call the provided rbtree functions. Locking is also left up to theuser of the rbtree code.

Creating a new rbtree¶

Data nodes in an rbtree tree are structures containing a struct rb_node member:

struct mytype {      struct rb_node node;      char *keystring;};

When dealing with a pointer to the embedded struct rb_node, the containing datastructure may be accessed with the standardcontainer_of() macro. In addition,individual members may be accessed directly via rb_entry(node, type, member).

At the root of each rbtree is an rb_root structure, which is initialized to beempty via:

struct rb_root mytree = RB_ROOT;

Searching for a value in an rbtree¶

Writing a search function for your tree is fairly straightforward: start at theroot, compare each value, and follow the left or right branch as necessary.

Example:

struct mytype *my_search(struct rb_root *root, char *string){      struct rb_node *node = root->rb_node;      while (node) {              struct mytype *data = container_of(node, struct mytype, node);              int result;              result = strcmp(string, data->keystring);              if (result < 0)                      node = node->rb_left;              else if (result > 0)                      node = node->rb_right;              else                      return data;      }      return NULL;}

Inserting data into an rbtree¶

Inserting data in the tree involves first searching for the place to insert thenew node, then inserting the node and rebalancing (“recoloring”) the tree.

The search for insertion differs from the previous search by finding thelocation of the pointer on which to graft the new node. The new node alsoneeds a link to its parent node for rebalancing purposes.

Example:

int my_insert(struct rb_root *root, struct mytype *data){      struct rb_node **new = &(root->rb_node), *parent = NULL;      /* Figure out where to put new node */      while (*new) {              struct mytype *this = container_of(*new, struct mytype, node);              int result = strcmp(data->keystring, this->keystring);              parent = *new;              if (result < 0)                      new = &((*new)->rb_left);              else if (result > 0)                      new = &((*new)->rb_right);              else                      return FALSE;      }      /* Add new node and rebalance tree. */      rb_link_node(&data->node, parent, new);      rb_insert_color(&data->node, root);      return TRUE;}

Removing or replacing existing data in an rbtree¶

To remove an existing node from a tree, call:

void rb_erase(struct rb_node *victim, struct rb_root *tree);

Example:

struct mytype *data = mysearch(&mytree, "walrus");if (data) {      rb_erase(&data->node, &mytree);      myfree(data);}

To replace an existing node in a tree with a new one with the same key, call:

void rb_replace_node(struct rb_node *old, struct rb_node *new,                      struct rb_root *tree);

Replacing a node this way does not re-sort the tree: If the new node doesn’thave the same key as the old node, the rbtree will probably become corrupted.

Iterating through the elements stored in an rbtree (in sort order)¶

Four functions are provided for iterating through an rbtree’s contents insorted order. These work on arbitrary trees, and should not need to bemodified or wrapped (except for locking purposes):

struct rb_node *rb_first(struct rb_root *tree);struct rb_node *rb_last(struct rb_root *tree);struct rb_node *rb_next(struct rb_node *node);struct rb_node *rb_prev(struct rb_node *node);

To start iterating, call rb_first() or rb_last() with a pointer to the rootof the tree, which will return a pointer to the node structure contained inthe first or last element in the tree. To continue, fetch the next or previousnode by calling rb_next() or rb_prev() on the current node. This will returnNULL when there are no more nodes left.

The iterator functions return a pointer to the embedded struct rb_node, fromwhich the containing data structure may be accessed with thecontainer_of()macro, and individual members may be accessed directly viarb_entry(node, type, member).

Example:

struct rb_node *node;for (node = rb_first(&mytree); node; node = rb_next(node))      printk("key=%s\n", rb_entry(node, struct mytype, node)->keystring);

Cached rbtrees¶

Computing the leftmost (smallest) node is quite a common task for binarysearch trees, such as for traversals or users relying on a the particularorder for their own logic. To this end, users can use ‘struct rb_root_cached’to optimize O(logN) rb_first() calls to a simple pointer fetch avoidingpotentially expensive tree iterations. This is done at negligible runtimeoverhead for maintanence; albeit larger memory footprint.

Similar to the rb_root structure, cached rbtrees are initialized to beempty via:

struct rb_root_cached mytree = RB_ROOT_CACHED;

Cached rbtree is simply a regular rb_root with an extra pointer to cache theleftmost node. This allows rb_root_cached to exist wherever rb_root does,which permits augmented trees to be supported as well as only a few extrainterfaces:

struct rb_node *rb_first_cached(struct rb_root_cached *tree);void rb_insert_color_cached(struct rb_node *, struct rb_root_cached *, bool);void rb_erase_cached(struct rb_node *node, struct rb_root_cached *);

Both insert and erase calls have their respective counterpart of augmentedtrees:

void rb_insert_augmented_cached(struct rb_node *node, struct rb_root_cached *,                                bool, struct rb_augment_callbacks *);void rb_erase_augmented_cached(struct rb_node *, struct rb_root_cached *,                               struct rb_augment_callbacks *);

Support for Augmented rbtrees¶

Augmented rbtree is an rbtree with “some” additional data stored ineach node, where the additional data for node N must be a function ofthe contents of all nodes in the subtree rooted at N. This data canbe used to augment some new functionality to rbtree. Augmented rbtreeis an optional feature built on top of basic rbtree infrastructure.An rbtree user who wants this feature will have to call the augmentationfunctions with the user provided augmentation callback when insertingand erasing nodes.

C files implementing augmented rbtree manipulation must include<linux/rbtree_augmented.h> instead of <linux/rbtree.h>. Note thatlinux/rbtree_augmented.h exposes some rbtree implementations detailsyou are not expected to rely on; please stick to the documented APIsthere and do not include <linux/rbtree_augmented.h> from header fileseither so as to minimize chances of your users accidentally relying onsuch implementation details.

On insertion, the user must update the augmented information on the pathleading to the inserted node, then call rb_link_node() as usual andrb_augment_inserted() instead of the usual rb_insert_color() call.If rb_augment_inserted() rebalances the rbtree, it will callback intoa user provided function to update the augmented information on theaffected subtrees.

When erasing a node, the user must call rb_erase_augmented() instead ofrb_erase(). rb_erase_augmented() calls back into user provided functionsto updated the augmented information on affected subtrees.

In both cases, the callbacks are provided through struct rb_augment_callbacks.3 callbacks must be defined:

• A propagation callback, which updates the augmented value for a givennode and its ancestors, up to a given stop point (or NULL to updateall the way to the root).
• A copy callback, which copies the augmented value for a given subtreeto a newly assigned subtree root.
• A tree rotation callback, which copies the augmented value for a givensubtree to a newly assigned subtree root AND recomputes the augmentedinformation for the former subtree root.

The compiled code for rb_erase_augmented() may inline the propagation andcopy callbacks, which results in a large function, so each augmented rbtreeuser should have a single rb_erase_augmented() call site in order to limitcompiled code size.

struct interval_tree_node *interval_tree_first_match(struct rb_root *root,                          unsigned long start, unsigned long last){      struct interval_tree_node *node;      if (!root->rb_node)              return NULL;      node = rb_entry(root->rb_node, struct interval_tree_node, rb);      while (true) {              if (node->rb.rb_left) {                      struct interval_tree_node *left =                              rb_entry(node->rb.rb_left,                                       struct interval_tree_node, rb);                      if (left->__subtree_last >= start) {                              /*                               * Some nodes in left subtree satisfy Cond2.                               * Iterate to find the leftmost such node N.                               * If it also satisfies Cond1, that's the match                               * we are looking for. Otherwise, there is no                               * matching interval as nodes to the right of N                               * can't satisfy Cond1 either.                               */                              node = left;                              continue;                      }              }              if (node->start <= last) {              /* Cond1 */                      if (node->last >= start)        /* Cond2 */                              return node;    /* node is leftmost match */                      if (node->rb.rb_right) {                              node = rb_entry(node->rb.rb_right,                                      struct interval_tree_node, rb);                              if (node->__subtree_last >= start)                                      continue;                      }              }              return NULL;    /* No match */      }}

static inline unsigned longcompute_subtree_last(struct interval_tree_node *node){      unsigned long max = node->last, subtree_last;      if (node->rb.rb_left) {              subtree_last = rb_entry(node->rb.rb_left,                      struct interval_tree_node, rb)->__subtree_last;              if (max < subtree_last)                      max = subtree_last;      }      if (node->rb.rb_right) {              subtree_last = rb_entry(node->rb.rb_right,                      struct interval_tree_node, rb)->__subtree_last;              if (max < subtree_last)                      max = subtree_last;      }      return max;}static void augment_propagate(struct rb_node *rb, struct rb_node *stop){      while (rb != stop) {              struct interval_tree_node *node =                      rb_entry(rb, struct interval_tree_node, rb);              unsigned long subtree_last = compute_subtree_last(node);              if (node->__subtree_last == subtree_last)                      break;              node->__subtree_last = subtree_last;              rb = rb_parent(&node->rb);      }}static void augment_copy(struct rb_node *rb_old, struct rb_node *rb_new){      struct interval_tree_node *old =              rb_entry(rb_old, struct interval_tree_node, rb);      struct interval_tree_node *new =              rb_entry(rb_new, struct interval_tree_node, rb);      new->__subtree_last = old->__subtree_last;}static void augment_rotate(struct rb_node *rb_old, struct rb_node *rb_new){      struct interval_tree_node *old =              rb_entry(rb_old, struct interval_tree_node, rb);      struct interval_tree_node *new =              rb_entry(rb_new, struct interval_tree_node, rb);      new->__subtree_last = old->__subtree_last;      old->__subtree_last = compute_subtree_last(old);}static const struct rb_augment_callbacks augment_callbacks = {      augment_propagate, augment_copy, augment_rotate};void interval_tree_insert(struct interval_tree_node *node,                          struct rb_root *root){      struct rb_node **link = &root->rb_node, *rb_parent = NULL;      unsigned long start = node->start, last = node->last;      struct interval_tree_node *parent;      while (*link) {              rb_parent = *link;              parent = rb_entry(rb_parent, struct interval_tree_node, rb);              if (parent->__subtree_last < last)                      parent->__subtree_last = last;              if (start < parent->start)                      link = &parent->rb.rb_left;              else                      link = &parent->rb.rb_right;      }      node->__subtree_last = last;      rb_link_node(&node->rb, rb_parent, link);      rb_insert_augmented(&node->rb, root, &augment_callbacks);}void interval_tree_remove(struct interval_tree_node *node,                          struct rb_root *root){      rb_erase_augmented(&node->rb, root, &augment_callbacks);}