TREE(3) BSD Library Functions Manual TREE(3)NAME
SPLAY_PROTOTYPE, SPLAY_GENERATE, SPLAY_ENTRY, SPLAY_HEAD,
SPLAY_INITIALIZER, SPLAY_ROOT, SPLAY_EMPTY, SPLAY_NEXT, SPLAY_MIN,
SPLAY_MAX, SPLAY_FIND, SPLAY_LEFT, SPLAY_RIGHT, SPLAY_FOREACH,
SPLAY_INIT, SPLAY_INSERT, SPLAY_REMOVE, RB_PROTOTYPE,
RB_PROTOTYPE_STATIC, RB_GENERATE, RB_GENERATE_STATIC, RB_ENTRY, RB_HEAD,
RB_INITIALIZER, RB_ROOT, RB_EMPTY, RB_NEXT, RB_PREV, RB_MIN, RB_MAX,
RB_FIND, RB_NFIND, RB_LEFT, RB_RIGHT, RB_PARENT, RB_FOREACH,
RB_FOREACH_REVERSE, RB_INIT, RB_INSERT, RB_REMOVE — implementations of
splay and red-black trees
SYNOPSIS
#include <sys/tree.h>
SPLAY_PROTOTYPE(NAME, TYPE, FIELD, CMP);
SPLAY_GENERATE(NAME, TYPE, FIELD, CMP);
SPLAY_ENTRY(TYPE);
SPLAY_HEAD(HEADNAME, TYPE);
struct TYPE *
SPLAY_INITIALIZER(SPLAY_HEAD *head);
SPLAY_ROOT(SPLAY_HEAD *head);
bool
SPLAY_EMPTY(SPLAY_HEAD *head);
struct TYPE *
SPLAY_NEXT(NAME, SPLAY_HEAD *head, struct TYPE *elm);
struct TYPE *
SPLAY_MIN(NAME, SPLAY_HEAD *head);
struct TYPE *
SPLAY_MAX(NAME, SPLAY_HEAD *head);
struct TYPE *
SPLAY_FIND(NAME, SPLAY_HEAD *head, struct TYPE *elm);
struct TYPE *
SPLAY_LEFT(struct TYPE *elm, SPLAY_ENTRY NAME);
struct TYPE *
SPLAY_RIGHT(struct TYPE *elm, SPLAY_ENTRY NAME);
SPLAY_FOREACH(VARNAME, NAME, SPLAY_HEAD *head);
void
SPLAY_INIT(SPLAY_HEAD *head);
struct TYPE *
SPLAY_INSERT(NAME, SPLAY_HEAD *head, struct TYPE *elm);
struct TYPE *
SPLAY_REMOVE(NAME, SPLAY_HEAD *head, struct TYPE *elm);
RB_PROTOTYPE(NAME, TYPE, FIELD, CMP);
RB_PROTOTYPE_STATIC(NAME, TYPE, FIELD, CMP);
RB_GENERATE(NAME, TYPE, FIELD, CMP);
RB_GENERATE_STATIC(NAME, TYPE, FIELD, CMP);
RB_ENTRY(TYPE);
RB_HEAD(HEADNAME, TYPE);
RB_INITIALIZER(RB_HEAD *head);
struct TYPE *
RB_ROOT(RB_HEAD *head);
bool
RB_EMPTY(RB_HEAD *head);
struct TYPE *
RB_NEXT(NAME, RB_HEAD *head, struct TYPE *elm);
struct TYPE *
RB_PREV(NAME, RB_HEAD *head, struct TYPE *elm);
struct TYPE *
RB_MIN(NAME, RB_HEAD *head);
struct TYPE *
RB_MAX(NAME, RB_HEAD *head);
struct TYPE *
RB_FIND(NAME, RB_HEAD *head, struct TYPE *elm);
struct TYPE *
RB_NFIND(NAME, RB_HEAD *head, struct TYPE *elm);
struct TYPE *
RB_LEFT(struct TYPE *elm, RB_ENTRY NAME);
struct TYPE *
RB_RIGHT(struct TYPE *elm, RB_ENTRY NAME);
struct TYPE *
RB_PARENT(struct TYPE *elm, RB_ENTRY NAME);
RB_FOREACH(VARNAME, NAME, RB_HEAD *head);
RB_FOREACH_REVERSE(VARNAME, NAME, RB_HEAD *head);
void
RB_INIT(RB_HEAD *head);
struct TYPE *
RB_INSERT(NAME, RB_HEAD *head, struct TYPE *elm);
struct TYPE *
RB_REMOVE(NAME, RB_HEAD *head, struct TYPE *elm);
DESCRIPTION
These macros define data structures for different types of trees: splay
trees and red-black trees.
In the macro definitions, TYPE is the name tag of a user defined struc‐
ture that must contain a field of type SPLAY_ENTRY, or RB_ENTRY, named
ENTRYNAME. The argument HEADNAME is the name tag of a user defined
structure that must be declared using the macros SPLAY_HEAD(), or
RB_HEAD(). The argument NAME has to be a unique name prefix for every
tree that is defined.
The function prototypes are declared with SPLAY_PROTOTYPE(),
RB_PROTOTYPE(), or RB_PROTOTYPE_STATIC(). The function bodies are gener‐
ated with SPLAY_GENERATE(), RB_GENERATE(), or RB_GENERATE_STATIC(). See
the examples below for further explanation of how these macros are used.
SPLAY TREES
A splay tree is a self-organizing data structure. Every operation on the
tree causes a splay to happen. The splay moves the requested node to the
root of the tree and partly rebalances it.
This has the benefit that request locality causes faster lookups as the
requested nodes move to the top of the tree. On the other hand, every
lookup causes memory writes.
The Balance Theorem bounds the total access time for m operations and n
inserts on an initially empty tree as O((m + n)lg n). The amortized cost
for a sequence of m accesses to a splay tree is O(lg n).
A splay tree is headed by a structure defined by the SPLAY_HEAD() macro.
A structure is declared as follows:
SPLAY_HEAD(HEADNAME, TYPE) head;
where HEADNAME is the name of the structure to be defined, and struct
TYPE is the type of the elements to be inserted into the tree.
The SPLAY_ENTRY() macro declares a structure that allows elements to be
connected in the tree.
In order to use the functions that manipulate the tree structure, their
prototypes need to be declared with the SPLAY_PROTOTYPE() macro, where
NAME is a unique identifier for this particular tree. The TYPE argument
is the type of the structure that is being managed by the tree. The
FIELD argument is the name of the element defined by SPLAY_ENTRY().
The function bodies are generated with the SPLAY_GENERATE() macro. It
takes the same arguments as the SPLAY_PROTOTYPE() macro, but should be
used only once.
Finally, the CMP argument is the name of a function used to compare tree
nodes with each other. The function takes two arguments of type struct
TYPE *. If the first argument is smaller than the second, the function
returns a value smaller than zero. If they are equal, the function
returns zero. Otherwise, it should return a value greater than zero.
The compare function defines the order of the tree elements.
The SPLAY_INIT() macro initializes the tree referenced by head.
The splay tree can also be initialized statically by using the
SPLAY_INITIALIZER() macro like this:
SPLAY_HEAD(HEADNAME, TYPE) head = SPLAY_INITIALIZER(&head);
The SPLAY_INSERT() macro inserts the new element elm into the tree.
The SPLAY_REMOVE() macro removes the element elm from the tree pointed by
head.
The SPLAY_FIND() macro can be used to find a particular element in the
tree.
struct TYPE find, *res;
find.key = 30;
res = SPLAY_FIND(NAME, head, &find);
The SPLAY_ROOT(), SPLAY_MIN(), SPLAY_MAX(), and SPLAY_NEXT() macros can
be used to traverse the tree:
for (np = SPLAY_MIN(NAME, &head); np != NULL; np = SPLAY_NEXT(NAME, &head, np))
Or, for simplicity, one can use the SPLAY_FOREACH() macro:
SPLAY_FOREACH(np, NAME, head)
The SPLAY_EMPTY() macro should be used to check whether a splay tree is
empty.
RED-BLACK TREES
A red-black tree is a binary search tree with the node color as an extra
attribute. It fulfills a set of conditions:
1. Every search path from the root to a leaf consists of the same
number of black nodes.
2. Each red node (except for the root) has a black parent.
3. Each leaf node is black.
Every operation on a red-black tree is bounded as O(lg n). The maximum
height of a red-black tree is 2lg(n + 1).
A red-black tree is headed by a structure defined by the RB_HEAD() macro.
A structure is declared as follows:
RB_HEAD(HEADNAME, TYPE) head;
where HEADNAME is the name of the structure to be defined, and struct
TYPE is the type of the elements to be inserted into the tree.
The RB_ENTRY() macro declares a structure that allows elements to be con‐
nected in the tree.
In order to use the functions that manipulate the tree structure, their
prototypes need to be declared with the RB_PROTOTYPE() or
RB_PROTOTYPE_STATIC() macro, where NAME is a unique identifier for this
particular tree. The TYPE argument is the type of the structure that is
being managed by the tree. The FIELD argument is the name of the element
defined by RB_ENTRY().
The function bodies are generated with the RB_GENERATE() or
RB_GENERATE_STATIC() macro. These macros take the same arguments as the
RB_PROTOTYPE() and RB_PROTOTYPE_STATIC() macros, but should be used only
once.
Finally, the CMP argument is the name of a function used to compare tree
noded with each other. The function takes two arguments of type struct
TYPE *. If the first argument is smaller than the second, the function
returns a value smaller than zero. If they are equal, the function
returns zero. Otherwise, it should return a value greater than zero.
The compare function defines the order of the tree elements.
The RB_INIT() macro initializes the tree referenced by head.
The red-black tree can also be initialized statically by using the
RB_INITIALIZER() macro like this:
RB_HEAD(HEADNAME, TYPE) head = RB_INITIALIZER(&head);
The RB_INSERT() macro inserts the new element elm into the tree.
The RB_REMOVE() macro removes the element elm from the tree pointed by
head.
The RB_FIND() and RB_NFIND() macros can be used to find a particular ele‐
ment in the tree.
struct TYPE find, *res;
find.key = 30;
res = RB_FIND(NAME, head, &find);
The RB_ROOT(), RB_MIN(), RB_MAX(), RB_NEXT(), and RB_PREV() macros can be
used to traverse the tree:
for (np = RB_MIN(NAME, &head); np != NULL; np = RB_NEXT(NAME,
&head, np))
Or, for simplicity, one can use the RB_FOREACH() or RB_FOREACH_REVERSE()
macro:
RB_FOREACH(np, NAME, head)
The RB_EMPTY() macro should be used to check whether a red-black tree is
empty.
NOTES
Trying to free a tree in the following way is a common error:
SPLAY_FOREACH(var, NAME, head) {
SPLAY_REMOVE(NAME, head, var);
free(var);
}
free(head);
Since var is freed, the FOREACH() macro refers to a pointer that may have
been reallocated already. Proper code needs a second variable.
for (var = SPLAY_MIN(NAME, head); var != NULL; var = nxt) {
nxt = SPLAY_NEXT(NAME, head, var);
SPLAY_REMOVE(NAME, head, var);
free(var);
}
Both RB_INSERT() and SPLAY_INSERT() return NULL if the element was
inserted in the tree successfully, otherwise they return a pointer to the
element with the colliding key.
Accordingly, RB_REMOVE() and SPLAY_REMOVE() return the pointer to the
removed element otherwise they return NULL to indicate an error.
AUTHORS
The author of the tree macros is Niels Provos.
BSD December 27, 2007 BSD
