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_GENERATE

RB_ENTRY, RB_HEAD, RB_INITIALIZER, RB_ROOT, RB_EMPTY,

RB_NEXT

RB_MAX, RB_FIND, RB_LEFT, RB_RIGHT, RB_PARENT, RB_FOREACH,

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_GENERATE(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_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_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);
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 de

fined structure 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

RB_HEAD(). The argument NAME has to be a unique name prefix

for every
tree that is defined.

The function prototypes are declared with either

SPLAY_PROTOTYPE

RB_PROTOTYPE(). The function bodies are generated with ei

ther
SPLAY_GENERATE(), or RB_GENERATE(). 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 op

eration on the
tree causes a splay to happen. The splay moves the request

ed 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 oper

ations 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

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 el

ements 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

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 ele

ment 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 con

sists 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

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 ele

ments 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 RB_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 RB_ENTRY().

The function bodies are generated with the RB_GENERATE()

macro. It takes
the same arguments as the RB_PROTOTYPE() macro, 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 us

ing 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() macro can be used to find a particular element

in the tree.

struct TYPE find, *res;
find.key = 30;
res = RB_FIND(NAME, head, &find);

The RB_ROOT(), RB_MIN(), RB_MAX(), and RB_NEXT() 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() 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 er

ror:

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 vari

able.

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 ele

ment 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 February 24, 2002

BSD