.\"	$OpenBSD: tree.3,v 1.7 2002/06/12 01:09:20 provos Exp $
.\"
.\" Copyright 2002 Niels Provos <provos@citi.umich.edu>
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.\" modification, are permitted provided that the following conditions
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.\" 3. All advertising materials mentioning features or use of this software
.\"    must display the following acknowledgement:
.\"      This product includes software developed by Niels Provos.
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.\" $FreeBSD$
.\"
.Dd September 28, 2019
.Dt ARB 3
.Os
.Sh NAME
.Nm ARB_PROTOTYPE ,
.Nm ARB_PROTOTYPE_STATIC ,
.Nm ARB_PROTOTYPE_INSERT ,
.Nm ARB_PROTOTYPE_INSERT_COLOR ,
.Nm ARB_PROTOTYPE_REMOVE ,
.Nm ARB_PROTOTYPE_REMOVE_COLOR ,
.Nm ARB_PROTOTYPE_FIND ,
.Nm ARB_PROTOTYPE_NFIND ,
.Nm ARB_PROTOTYPE_NEXT ,
.Nm ARB_PROTOTYPE_PREV ,
.Nm ARB_PROTOTYPE_MINMAX ,
.Nm ARB_PROTOTYPE_REINSERT ,
.Nm ARB_GENERATE ,
.Nm ARB_GENERATE_STATIC ,
.Nm ARB_GENERATE_INSERT ,
.Nm ARB_GENERATE_INSERT_COLOR ,
.Nm ARB_GENERATE_REMOVE ,
.Nm ARB_GENERATE_REMOVE_COLOR ,
.Nm ARB_GENERATE_FIND ,
.Nm ARB_GENERATE_NFIND ,
.Nm ARB_GENERATE_NEXT ,
.Nm ARB_GENERATE_PREV ,
.Nm ARB_GENERATE_MINMAX ,
.Nm ARB_GENERATE_REINSERT ,
.Nm ARB8_ENTRY ,
.Nm ARB16_ENTRY ,
.Nm ARB32_ENTRY ,
.Nm ARB8_HEAD ,
.Nm ARB16_HEAD ,
.Nm ARB32_HEAD ,
.Nm ARB_ALLOCSIZE ,
.Nm ARB_INITIALIZER ,
.Nm ARB_ROOT ,
.Nm ARB_EMPTY ,
.Nm ARB_FULL ,
.Nm ARB_CURNODES ,
.Nm ARB_MAXNODES ,
.Nm ARB_NEXT ,
.Nm ARB_PREV ,
.Nm ARB_MIN ,
.Nm ARB_MAX ,
.Nm ARB_FIND ,
.Nm ARB_NFIND ,
.Nm ARB_LEFT ,
.Nm ARB_LEFTIDX ,
.Nm ARB_RIGHT ,
.Nm ARB_RIGHTIDX ,
.Nm ARB_PARENT ,
.Nm ARB_PARENTIDX ,
.Nm ARB_GETFREE ,
.Nm ARB_FREEIDX ,
.Nm ARB_FOREACH ,
.Nm ARB_FOREACH_FROM ,
.Nm ARB_FOREACH_SAFE ,
.Nm ARB_FOREACH_REVERSE ,
.Nm ARB_FOREACH_REVERSE_FROM ,
.Nm ARB_FOREACH_REVERSE_SAFE ,
.Nm ARB_INIT ,
.Nm ARB_INSERT ,
.Nm ARB_REMOVE ,
.Nm ARB_REINSERT
.Nd "array-based red-black trees"
.Sh SYNOPSIS
.In sys/arb.h
.Fn ARB_PROTOTYPE NAME TYPE FIELD CMP
.Fn ARB_PROTOTYPE_STATIC NAME TYPE FIELD CMP
.Fn ARB_PROTOTYPE_INSERT NAME TYPE ATTR
.Fn ARB_PROTOTYPE_INSERT_COLOR NAME TYPE ATTR
.Fn ARB_PROTOTYPE_REMOVE NAME TYPE ATTR
.Fn ARB_PROTOTYPE_REMOVE_COLOR NAME TYPE ATTR
.Fn ARB_PROTOTYPE_FIND NAME TYPE ATTR
.Fn ARB_PROTOTYPE_NFIND NAME TYPE ATTR
.Fn ARB_PROTOTYPE_NEXT NAME TYPE ATTR
.Fn ARB_PROTOTYPE_PREV NAME TYPE ATTR
.Fn ARB_PROTOTYPE_MINMAX NAME TYPE ATTR
.Fn ARB_PROTOTYPE_REINSERT NAME TYPE ATTR
.Fn ARB_GENERATE NAME TYPE FIELD CMP
.Fn ARB_GENERATE_STATIC NAME TYPE FIELD CMP
.Fn ARB_GENERATE_INSERT NAME TYPE FIELD CMP ATTR
.Fn ARB_GENERATE_INSERT_COLOR NAME TYPE FIELD ATTR
.Fn ARB_GENERATE_REMOVE NAME TYPE FIELD ATTR
.Fn ARB_GENERATE_REMOVE_COLOR NAME TYPE FIELD ATTR
.Fn ARB_GENERATE_FIND NAME TYPE FIELD CMP ATTR
.Fn ARB_GENERATE_NFIND NAME TYPE FIELD CMP ATTR
.Fn ARB_GENERATE_NEXT NAME TYPE FIELD ATTR
.Fn ARB_GENERATE_PREV NAME TYPE FIELD ATTR
.Fn ARB_GENERATE_MINMAX NAME TYPE FIELD ATTR
.Fn ARB_GENERATE_REINSERT NAME TYPE FIELD CMP ATTR
.Fn ARB<8|16|32>_ENTRY
.Fn ARB<8|16|32>_HEAD HEADNAME TYPE
.Ft "size_t"
.Fn ARB_ALLOCSIZE "ARB_HEAD *head" "int<8|16|32>_t maxnodes" "struct TYPE *elm"
.Fn ARB_INITIALIZER "ARB_HEAD *head" "int<8|16|32>_t maxnodes"
.Ft "struct TYPE *"
.Fn ARB_ROOT "ARB_HEAD *head"
.Ft "bool"
.Fn ARB_EMPTY "ARB_HEAD *head"
.Ft "bool"
.Fn ARB_FULL "ARB_HEAD *head"
.Ft "int<8|16|32>_t"
.Fn ARB_CURNODES "ARB_HEAD *head"
.Ft "int<8|16|32>_t"
.Fn ARB_MAXNODES "ARB_HEAD *head"
.Ft "struct TYPE *"
.Fn ARB_NEXT NAME "ARB_HEAD *head" "struct TYPE *elm"
.Ft "struct TYPE *"
.Fn ARB_PREV NAME "ARB_HEAD *head" "struct TYPE *elm"
.Ft "struct TYPE *"
.Fn ARB_MIN NAME "ARB_HEAD *head"
.Ft "struct TYPE *"
.Fn ARB_MAX NAME "ARB_HEAD *head"
.Ft "struct TYPE *"
.Fn ARB_FIND NAME "ARB_HEAD *head" "struct TYPE *elm"
.Ft "struct TYPE *"
.Fn ARB_NFIND NAME "ARB_HEAD *head" "struct TYPE *elm"
.Ft "struct TYPE *"
.Fn ARB_LEFT "struct TYPE *elm" "ARB_ENTRY NAME"
.Ft "int<8|16|32>_t"
.Fn ARB_LEFTIDX "struct TYPE *elm" "ARB_ENTRY NAME"
.Ft "struct TYPE *"
.Fn ARB_RIGHT "struct TYPE *elm" "ARB_ENTRY NAME"
.Ft "int<8|16|32>_t"
.Fn ARB_RIGHTIDX "struct TYPE *elm" "ARB_ENTRY NAME"
.Ft "struct TYPE *"
.Fn ARB_PARENT "struct TYPE *elm" "ARB_ENTRY NAME"
.Ft "int<8|16|32>_t"
.Fn ARB_PARENTIDX "struct TYPE *elm" "ARB_ENTRY NAME"
.Ft "struct TYPE *"
.Fn ARB_GETFREE "ARB_HEAD *head" "FIELD"
.Ft "int<8|16|32>_t"
.Fn ARB_FREEIDX "ARB_HEAD *head"
.Fn ARB_FOREACH VARNAME NAME "ARB_HEAD *head"
.Fn ARB_FOREACH_FROM "VARNAME" "NAME" "POS_VARNAME"
.Fn ARB_FOREACH_SAFE "VARNAME" "NAME" "ARB_HEAD *head" "TEMP_VARNAME"
.Fn ARB_FOREACH_REVERSE VARNAME NAME "ARB_HEAD *head"
.Fn ARB_FOREACH_REVERSE_FROM "VARNAME" "NAME" "POS_VARNAME"
.Fn ARB_FOREACH_REVERSE_SAFE "VARNAME" "NAME" "ARB_HEAD *head" "TEMP_VARNAME"
.Ft void
.Fn ARB_INIT "struct TYPE *elm" "FIELD" "ARB_HEAD *head" "int<8|16|32>_t maxnodes"
.Ft "struct TYPE *"
.Fn ARB_INSERT NAME "ARB_HEAD *head" "struct TYPE *elm"
.Ft "struct TYPE *"
.Fn ARB_REMOVE NAME "ARB_HEAD *head" "struct TYPE *elm"
.Ft "struct TYPE *"
.Fn ARB_REINSERT NAME "ARB_HEAD *head" "struct TYPE *elm"
.Sh DESCRIPTION
These macros define data structures for and array-based red-black trees.
They use a single, continuous chunk of memory, and are useful
e.g., when the tree needs to be transferred between userspace and kernel.
.Pp
In the macro definitions,
.Fa TYPE
is the name tag of a user defined structure that must contain a field of type
.Vt ARB_ENTRY ,
named
.Fa ENTRYNAME .
The argument
.Fa HEADNAME
is the name tag of a user defined structure that must be declared
using the
.Fn ARB_HEAD
macro.
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 ARB_PROTOTYPE ,
or
.Fn ARB_PROTOTYPE_STATIC .
The function bodies are generated with
.Fn ARB_GENERATE ,
or
.Fn ARB_GENERATE_STATIC .
See the examples below for further explanation of how these macros are used.
.Pp
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
.Fn ARB_*
trees require entries to be allocated as an array, and uses array
indices to link entries together.
The maximum number of
.Fn ARB_*
tree entries is therefore constrained by the minimum of array size and choice of
signed integer data type used to store array indices.
Use
.Fn ARB_ALLOCSIZE
to compute the size of memory chunk to allocate.
.Pp
A red-black tree is headed by a structure defined by the
.Fn ARB_HEAD
macro.
A
structure is declared with either of the following:
.Bd -ragged -offset indent
.Fn ARB<8|16|32>_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 ARB_HEAD
variant includes a suffix denoting the signed integer data type size
.Pq in bits
used to store array indices.
For example,
.Fn ARB_HEAD8
creates a red-black tree head strucutre with 8-bit signed array indices capable
of indexing up to 128 entries.
.Pp
The
.Fn ARB_ENTRY
macro declares a structure that allows elements to be connected in the tree.
Similarly to the
.Fn ARB<8|16|32>_HEAD
macro, the
.Fn ARB_ENTRY
variant includes a suffix denoting the signed integer data type size
.Pq in bits
used to store array indices.
Entries should use the same number of bits as the tree head structure they will
be linked into.
.Pp
In order to use the functions that manipulate the tree structure,
their prototypes need to be declared with the
.Fn ARB_PROTOTYPE
or
.Fn ARB_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 ARB_ENTRY .
Individual prototypes can be declared with
.Fn ARB_PROTOTYPE_INSERT ,
.Fn ARB_PROTOTYPE_INSERT_COLOR ,
.Fn ARB_PROTOTYPE_REMOVE ,
.Fn ARB_PROTOTYPE_REMOVE_COLOR ,
.Fn ARB_PROTOTYPE_FIND ,
.Fn ARB_PROTOTYPE_NFIND ,
.Fn ARB_PROTOTYPE_NEXT ,
.Fn ARB_PROTOTYPE_PREV ,
.Fn ARB_PROTOTYPE_MINMAX ,
and
.Fn ARB_PROTOTYPE_REINSERT
in case not all functions are required.
The individual prototype macros expect
.Fa NAME ,
.Fa TYPE ,
and
.Fa ATTR
arguments.
The
.Fa ATTR
argument must be empty for global functions or
.Fa static
for static functions.
.Pp
The function bodies are generated with the
.Fn ARB_GENERATE
or
.Fn ARB_GENERATE_STATIC
macro.
These macros take the same arguments as the
.Fn ARB_PROTOTYPE
and
.Fn ARB_PROTOTYPE_STATIC
macros, but should be used only once.
As an alternative individual function bodies are generated with the
.Fn ARB_GENERATE_INSERT ,
.Fn ARB_GENERATE_INSERT_COLOR ,
.Fn ARB_GENERATE_REMOVE ,
.Fn ARB_GENERATE_REMOVE_COLOR ,
.Fn ARB_GENERATE_FIND ,
.Fn ARB_GENERATE_NFIND ,
.Fn ARB_GENERATE_NEXT ,
.Fn ARB_GENERATE_PREV ,
.Fn ARB_GENERATE_MINMAX ,
and
.Fn ARB_GENERATE_REINSERT
macros.
.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 ARB_INIT
macro initializes the tree referenced by
.Fa head ,
with the array length of
.Fa maxnodes .
.Pp
The red-black tree can also be initialized statically by using the
.Fn ARB_INITIALIZER
macro:
.Bd -ragged -offset indent
.Fn ARB<8|16|32>_HEAD HEADNAME TYPE
.Va head
=
.Fn ARB_INITIALIZER &head maxnodes ;
.Ed
.Pp
The
.Fn ARB_INSERT
macro inserts the new element
.Fa elm
into the tree.
.Pp
The
.Fn ARB_REMOVE
macro removes the element
.Fa elm
from the tree pointed by
.Fa head .
.Pp
The
.Fn ARB_FIND
and
.Fn ARB_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 ARB_ROOT ,
.Fn ARB_MIN ,
.Fn ARB_MAX ,
.Fn ARB_NEXT ,
and
.Fn ARB_PREV
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 ARB_FOREACH
or
.Fn ARB_FOREACH_REVERSE
macro:
.Bd -ragged -offset indent
.Fn RB_FOREACH np NAME head
.Ed
.Pp
The macros
.Fn ARB_FOREACH_SAFE
and
.Fn ARB_FOREACH_REVERSE_SAFE
traverse the tree referenced by head
in a forward or reverse direction respectively,
assigning each element in turn to np.
However, unlike their unsafe counterparts,
they permit both the removal of np
as well as freeing it from within the loop safely
without interfering with the traversal.
.Pp
Both
.Fn ARB_FOREACH_FROM
and
.Fn ARB_FOREACH_REVERSE_FROM
may be used to continue an interrupted traversal
in a forward or reverse direction respectively.
The head pointer is not required.
The pointer to the node from where to resume the traversal
should be passed as their last argument,
and will be overwritten to provide safe traversal.
.Pp
The
.Fn ARB_EMPTY
macro should be used to check whether a red-black tree is empty.
.Pp
Given that ARB trees have an intrinsic upper bound on the number of entries,
some ARB-specific additional macros are defined.
The
.Fn ARB_FULL
macro returns a boolean indicating whether the current number of tree entries
equals the tree's maximum.
The
.Fn ARB_CURNODES
and
.Fn ARB_MAXNODES
macros return the current and maximum number of entries respectively.
The
.Fn ARB_GETFREE
macro returns a pointer to the next free entry in the array of entries, ready to
be linked into the tree.
The
.Fn ARB_INSERT
returns
.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 ARB_REMOVE
returns the pointer to the removed element otherwise they return
.Dv NULL
to indicate an error.
.Pp
The
.Fn RB_REINSERT
macro updates the position of the element
.Fa elm
in the tree.
This must be called if a member of a
.Nm tree
is modified in a way that affects comparison, such as by modifying
a node's key.
This is a lower overhead alternative to removing the element
and reinserting it again.
.Sh SEE ALSO
.Xr queue 3 ,
.Xr tree 3
.Sh HISTORY
The
.Nm ARB
macros first appeared in
.Fx 13.0 .
.Sh AUTHORS
The
.Nm ARB
macros were implemented by
.An Lawrence Stewart Aq Mt lstewart@FreeBSD.org ,
based on
.Xr tree 3
macros written by
.An Niels Provos .