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.\" Copyright 2002 Niels Provos <provos@citi.umich.edu>
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.\"      This product includes software developed by Niels Provos.
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.Dd January 17, 2006
.Dt TREE 3
.Os
.Sh NAME
.Nm SPLAY_PROTOTYPE ,
.Nm SPLAY_GENERATE ,
.Nm SPLAY_ENTRY ,
.Nm SPLAY_HEAD ,
.Nm SPLAY_INITIALIZER ,
.Nm SPLAY_ROOT ,
.Nm SPLAY_EMPTY ,
.Nm SPLAY_NEXT ,
.Nm SPLAY_MIN ,
.Nm SPLAY_MAX ,
.Nm SPLAY_FIND ,
.Nm SPLAY_LEFT ,
.Nm SPLAY_RIGHT ,
.Nm SPLAY_FOREACH ,
.Nm SPLAY_INIT ,
.Nm SPLAY_INSERT ,
.Nm SPLAY_REMOVE ,
.Nm RB_PROTOTYPE ,
.Nm RB_PROTOTYPE_STATIC ,
.Nm RB_GENERATE ,
.Nm RB_GENERATE_STATIC ,
.Nm RB_ENTRY ,
.Nm RB_HEAD ,
.Nm RB_INITIALIZER ,
.Nm RB_ROOT ,
.Nm RB_EMPTY ,
.Nm RB_NEXT ,
.Nm RB_MIN ,
.Nm RB_MAX ,
.Nm RB_FIND ,
.Nm RB_NFIND ,
.Nm RB_LEFT ,
.Nm RB_RIGHT ,
.Nm RB_PARENT ,
.Nm RB_FOREACH ,
.Nm RB_INIT ,
.Nm RB_INSERT ,
.Nm RB_REMOVE
.Nd "implementations of splay and red-black trees"
.Sh SYNOPSIS
.In sys/tree.h
.Fn SPLAY_PROTOTYPE NAME TYPE FIELD CMP
.Fn SPLAY_GENERATE NAME TYPE FIELD CMP
.Fn SPLAY_ENTRY TYPE
.Fn SPLAY_HEAD HEADNAME TYPE
.Ft "struct TYPE *"
.Fn SPLAY_INITIALIZER "SPLAY_HEAD *head"
.Fn SPLAY_ROOT "SPLAY_HEAD *head"
.Ft bool
.Fn SPLAY_EMPTY "SPLAY_HEAD *head"
.Ft "struct TYPE *"
.Fn SPLAY_NEXT NAME "SPLAY_HEAD *head" "struct TYPE *elm"
.Ft "struct TYPE *"
.Fn SPLAY_MIN NAME "SPLAY_HEAD *head"
.Ft "struct TYPE *"
.Fn SPLAY_MAX NAME "SPLAY_HEAD *head"
.Ft "struct TYPE *"
.Fn SPLAY_FIND NAME "SPLAY_HEAD *head" "struct TYPE *elm"
.Ft "struct TYPE *"
.Fn SPLAY_LEFT "struct TYPE *elm" "SPLAY_ENTRY NAME"
.Ft "struct TYPE *"
.Fn SPLAY_RIGHT "struct TYPE *elm" "SPLAY_ENTRY NAME"
.Fn SPLAY_FOREACH VARNAME NAME "SPLAY_HEAD *head"
.Ft void
.Fn SPLAY_INIT "SPLAY_HEAD *head"
.Ft "struct TYPE *"
.Fn SPLAY_INSERT NAME "SPLAY_HEAD *head" "struct TYPE *elm"
.Ft "struct TYPE *"
.Fn SPLAY_REMOVE NAME "SPLAY_HEAD *head" "struct TYPE *elm"
.Fn RB_PROTOTYPE NAME TYPE FIELD CMP
.Fn RB_PROTOTYPE_STATIC NAME TYPE FIELD CMP
.Fn RB_GENERATE NAME TYPE FIELD CMP
.Fn RB_GENERATE_STATIC NAME TYPE FIELD CMP
.Fn RB_ENTRY TYPE
.Fn RB_HEAD HEADNAME TYPE
.Fn RB_INITIALIZER "RB_HEAD *head"
.Ft "struct TYPE *"
.Fn RB_ROOT "RB_HEAD *head"
.Ft "bool"
.Fn RB_EMPTY "RB_HEAD *head"
.Ft "struct TYPE *"
.Fn RB_NEXT NAME "RB_HEAD *head" "struct TYPE *elm"
.Ft "struct TYPE *"
.Fn RB_MIN NAME "RB_HEAD *head"
.Ft "struct TYPE *"
.Fn RB_MAX NAME "RB_HEAD *head"
.Ft "struct TYPE *"
.Fn RB_FIND NAME "RB_HEAD *head" "struct TYPE *elm"
.Ft "struct TYPE *"
.Fn RB_NFIND NAME "RB_HEAD *head" "struct TYPE *elm"
.Ft "struct TYPE *"
.Fn RB_LEFT "struct TYPE *elm" "RB_ENTRY NAME"
.Ft "struct TYPE *"
.Fn RB_RIGHT "struct TYPE *elm" "RB_ENTRY NAME"
.Ft "struct TYPE *"
.Fn RB_PARENT "struct TYPE *elm" "RB_ENTRY NAME"
.Fn RB_FOREACH VARNAME NAME "RB_HEAD *head"
.Ft void
.Fn RB_INIT "RB_HEAD *head"
.Ft "struct TYPE *"
.Fn RB_INSERT NAME "RB_HEAD *head" "struct TYPE *elm"
.Ft "struct TYPE *"
.Fn RB_REMOVE NAME "RB_HEAD *head" "struct TYPE *elm"
.Sh DESCRIPTION
These macros define data structures for different types of trees:
splay trees and red-black trees.
.Pp
In the macro definitions,
.Fa TYPE
is the name tag of a user defined structure that must contain a field of type
.Vt SPLAY_ENTRY ,
or
.Vt RB_ENTRY ,
named
.Fa ENTRYNAME .
The argument
.Fa HEADNAME
is the name tag of a user defined structure that must be declared
using the macros
.Fn SPLAY_HEAD ,
or
.Fn RB_HEAD .
The argument
.Fa NAME
has to be a unique name prefix for every tree that is defined.
.Pp
The function prototypes are declared with
.Fn SPLAY_PROTOTYPE ,
.Fn RB_PROTOTYPE ,
or
.Fn RB_PROTOTYPE_STATIC .
The function bodies are generated with
.Fn SPLAY_GENERATE ,
.Fn RB_GENERATE ,
or
.Fn RB_GENERATE_STATIC .
See the examples below for further explanation of how these macros are used.
.Sh 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.
.Pp
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.
.Pp
The Balance Theorem bounds the total access time for
.Ar m
operations and
.Ar n
inserts on an initially empty tree as
.Fn O "\*[lp]m + n\*[rp]lg n" .
The
amortized cost for a sequence of
.Ar m
accesses to a splay tree is
.Fn O "lg n" .
.Pp
A splay tree is headed by a structure defined by the
.Fn SPLAY_HEAD
macro.
A
structure is declared as follows:
.Bd -ragged -offset indent
.Fn SPLAY_HEAD HEADNAME TYPE
.Va head ;
.Ed
.Pp
where
.Fa HEADNAME
is the name of the structure to be defined, and struct
.Fa TYPE
is the type of the elements to be inserted into the tree.
.Pp
The
.Fn SPLAY_ENTRY
macro declares a structure that allows elements to be connected in the tree.
.Pp
In order to use the functions that manipulate the tree structure,
their prototypes need to be declared with the
.Fn SPLAY_PROTOTYPE
macro,
where
.Fa NAME
is a unique identifier for this particular tree.
The
.Fa TYPE
argument is the type of the structure that is being managed
by the tree.
The
.Fa FIELD
argument is the name of the element defined by
.Fn SPLAY_ENTRY .
.Pp
The function bodies are generated with the
.Fn SPLAY_GENERATE
macro.
It takes the same arguments as the
.Fn SPLAY_PROTOTYPE
macro, but should be used only once.
.Pp
Finally,
the
.Fa CMP
argument is the name of a function used to compare tree nodes
with each other.
The function takes two arguments of type
.Vt "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.
.Pp
The
.Fn SPLAY_INIT
macro initializes the tree referenced by
.Fa head .
.Pp
The splay tree can also be initialized statically by using the
.Fn SPLAY_INITIALIZER
macro like this:
.Bd -ragged -offset indent
.Fn SPLAY_HEAD HEADNAME TYPE
.Va head
=
.Fn SPLAY_INITIALIZER &head ;
.Ed
.Pp
The
.Fn SPLAY_INSERT
macro inserts the new element
.Fa elm
into the tree.
.Pp
The
.Fn SPLAY_REMOVE
macro removes the element
.Fa elm
from the tree pointed by
.Fa head .
.Pp
The
.Fn SPLAY_FIND
macro can be used to find a particular element in the tree.
.Bd -literal -offset indent
struct TYPE find, *res;
find.key = 30;
res = SPLAY_FIND(NAME, head, &find);
.Ed
.Pp
The
.Fn SPLAY_ROOT ,
.Fn SPLAY_MIN ,
.Fn SPLAY_MAX ,
and
.Fn SPLAY_NEXT
macros can be used to traverse the tree:
.Bd -literal -offset indent
for (np = SPLAY_MIN(NAME, &head); np != NULL; np = SPLAY_NEXT(NAME, &head, np))
.Ed
.Pp
Or, for simplicity, one can use the
.Fn SPLAY_FOREACH
macro:
.Bd -ragged -offset indent
.Fn SPLAY_FOREACH np NAME head
.Ed
.Pp
The
.Fn SPLAY_EMPTY
macro should be used to check whether a splay tree is empty.
.Sh 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:
.Bl -enum -offset indent
.It
Every search path from the root to a leaf consists of the same number of
black nodes.
.It
Each red node (except for the root) has a black parent.
.It
Each leaf node is black.
.El
.Pp
Every operation on a red-black tree is bounded as
.Fn O "lg n" .
The maximum height of a red-black tree is
.Fn 2lg "n + 1" .
.Pp
A red-black tree is headed by a structure defined by the
.Fn RB_HEAD
macro.
A
structure is declared as follows:
.Bd -ragged -offset indent
.Fn RB_HEAD HEADNAME TYPE
.Va head ;
.Ed
.Pp
where
.Fa HEADNAME
is the name of the structure to be defined, and struct
.Fa TYPE
is the type of the elements to be inserted into the tree.
.Pp
The
.Fn RB_ENTRY
macro declares a structure that allows elements to be connected in the tree.
.Pp
In order to use the functions that manipulate the tree structure,
their prototypes need to be declared with the
.Fn RB_PROTOTYPE
or
.Fn RB_PROTOTYPE_STATIC
macro,
where
.Fa NAME
is a unique identifier for this particular tree.
The
.Fa TYPE
argument is the type of the structure that is being managed
by the tree.
The
.Fa FIELD
argument is the name of the element defined by
.Fn RB_ENTRY .
.Pp
The function bodies are generated with the
.Fn RB_GENERATE
or
.Fn RB_GENERATE_STATIC
macro.
These macros take the same arguments as the
.Fn RB_PROTOTYPE
and
.Fn RB_PROTOTYPE_STATIC
macros, but should be used only once.
.Pp
Finally,
the
.Fa CMP
argument is the name of a function used to compare tree noded
with each other.
The function takes two arguments of type
.Vt "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.
.Pp
The
.Fn RB_INIT
macro initializes the tree referenced by
.Fa head .
.Pp
The red-black tree can also be initialized statically by using the
.Fn RB_INITIALIZER
macro like this:
.Bd -ragged -offset indent
.Fn RB_HEAD HEADNAME TYPE
.Va head
=
.Fn RB_INITIALIZER &head ;
.Ed
.Pp
The
.Fn RB_INSERT
macro inserts the new element
.Fa elm
into the tree.
.Pp
The
.Fn RB_REMOVE
macro removes the element
.Fa elm
from the tree pointed by
.Fa head .
.Pp
The
.Fn RB_FIND
and
.Fn RB_NFIND
macros can be used to find a particular element in the tree.
.Bd -literal -offset indent
struct TYPE find, *res;
find.key = 30;
res = RB_FIND(NAME, head, &find);
.Ed
.Pp
The
.Fn RB_ROOT ,
.Fn RB_MIN ,
.Fn RB_MAX ,
and
.Fn RB_NEXT
macros can be used to traverse the tree:
.Pp
.Dl "for (np = RB_MIN(NAME, &head); np != NULL; np = RB_NEXT(NAME, &head, np))"
.Pp
Or, for simplicity, one can use the
.Fn RB_FOREACH
macro:
.Bd -ragged -offset indent
.Fn RB_FOREACH np NAME head
.Ed
.Pp
The
.Fn RB_EMPTY
macro should be used to check whether a red-black tree is empty.
.Sh NOTES
Trying to free a tree in the following way is a common error:
.Bd -literal -offset indent
SPLAY_FOREACH(var, NAME, head) {
	SPLAY_REMOVE(NAME, head, var);
	free(var);
}
free(head);
.Ed
.Pp
Since
.Va var
is freed, the
.Fn FOREACH
macro refers to a pointer that may have been reallocated already.
Proper code needs a second variable.
.Bd -literal -offset indent
for (var = SPLAY_MIN(NAME, head); var != NULL; var = nxt) {
	nxt = SPLAY_NEXT(NAME, head, var);
	SPLAY_REMOVE(NAME, head, var);
	free(var);
}
.Ed
.Pp
Both
.Fn RB_INSERT
and
.Fn SPLAY_INSERT
return
.Dv NULL
if the element was inserted in the tree successfully, otherwise they
return a pointer to the element with the colliding key.
.Pp
Accordingly,
.Fn RB_REMOVE
and
.Fn SPLAY_REMOVE
return the pointer to the removed element otherwise they return
.Dv NULL
to indicate an error.
.Sh AUTHORS
The author of the tree macros is
.An Niels Provos .