.\"	$NetBSD: tree.3,v 1.14 2019/05/24 21:32:05 wiz Exp $
.\"	$OpenBSD: tree.3,v 1.23 2011/07/09 08:43:01 jmc Exp $
.\"/*
.\" * Copyright 2002 Niels Provos <provos@citi.umich.edu>
.\" * All rights reserved.
.\" *
.\" * Redistribution and use in source and binary forms, with or without
.\" * modification, are permitted provided that the following conditions
.\" * are met:
.\" * 1. Redistributions of source code must retain the above copyright
.\" *    notice, this list of conditions and the following disclaimer.
.\" * 2. Redistributions in binary form must reproduce the above copyright
.\" *    notice, this list of conditions and the following disclaimer in the
.\" *    documentation and/or other materials provided with the distribution.
.\" *
.\" * THIS SOFTWARE IS PROVIDED BY THE AUTHOR ``AS IS'' AND ANY EXPRESS OR
.\" * IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES
.\" * OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED.
.\" * IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR ANY DIRECT, INDIRECT,
.\" * INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT
.\" * NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE,
.\" * DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY
.\" * THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT
.\" * (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF
.\" * THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
.\" */
.Dd May 25, 2019
.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_PREV ,
.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_FOREACH_SAFE ,
.Nm RB_FOREACH_REVERSE ,
.Nm RB_FOREACH_REVERSE_SAFE ,
.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 "int"
.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"
.Pp
.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 "int"
.Fn RB_EMPTY "RB_HEAD *head"
.Ft "struct TYPE *"
.Fn RB_NEXT "NAME" "RB_HEAD *head" "struct TYPE *elm"
.Ft "struct TYPE *"
.Fn RB_PREV "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"
.Fn RB_FOREACH_SAFE "VARNAME" "NAME" "RB_HEAD *head" "TEMP_VARNAME"
.Fn RB_FOREACH_REVERSE "VARNAME" "NAME" "RB_HEAD *head"
.Fn RB_FOREACH_REVERSE_SAFE "VARNAME" "NAME" "RB_HEAD *head" "TEMP_VARNAME"
.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
.Bf -symbolic
This is a legacy interface; for new code,
.Xr rbtree 3
is preferred.
.Ef
.Pp
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 named
.Fa FIELD ,
of type
.Li SPLAY_ENTRY
or
.Li RB_ENTRY .
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
.Li SPLAY_PROTOTYPE ,
.Li RB_PROTOTYPE ,
or
.Li RB_PROTOTYPE_STATIC .
The function bodies are generated with
.Li SPLAY_GENERATE ,
.Li RB_GENERATE ,
or
.Li 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 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).
.Pp
A splay tree is headed by a structure defined by the
.Fn SPLAY_HEAD
macro.
A
.Fa SPLAY_HEAD
structure is declared as follows:
.Bd -literal -offset indent
SPLAY_HEAD(HEADNAME, TYPE) 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
.Fa "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 -literal -offset indent
SPLAY_HEAD(HEADNAME, TYPE) head = SPLAY_INITIALIZER(&head);
.Ed
.Pp
The
.Fn SPLAY_INSERT
macro inserts the new element
.Fa elm
into the tree.
Upon success,
.Dv NULL
is returned.
If a matching element already exists in the tree, the insertion is
aborted, and a pointer to the existing element is returned.
.Pp
The
.Fn SPLAY_REMOVE
macro removes the element
.Fa elm
from the tree pointed by
.Fa head .
Upon success, a pointer to the removed element is returned.
.Dv NULL
is returned if
.Fa elm
is not present in the tree.
.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 -literal -offset indent
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:
.Pp
.Bl -enum -compact -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 O(lg n).
The maximum height of a red-black tree is 2lg (n+1).
.Pp
A red-black tree is headed by a structure defined by the
.Fn RB_HEAD
macro.
A
.Fa RB_HEAD
structure is declared as follows:
.Bd -literal -offset indent
RB_HEAD(HEADNAME, TYPE) 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
macros,
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
macros.
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 trees' nodes
with each other.
The function takes two arguments of type
.Fa "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 -literal -offset indent
RB_HEAD(HEADNAME, TYPE) head = RB_INITIALIZER(&head);
.Ed
.Pp
The
.Fn RB_INSERT
macro inserts the new element
.Fa elm
into the tree.
Upon success,
.Dv NULL
is returned.
If a matching element already exists in the tree, the insertion is
aborted, and a pointer to the existing element is returned.
.Pp
The
.Fn RB_REMOVE
macro removes the element
.Fa elm
from the tree pointed to by
.Fa head .
The element must be present in that tree.
.Fn RB_REMOVE
returns
.Fa elm .
.Pp
The
.Fn RB_FIND
macro can be used to find a particular element in the tree.
and
.Fn RB_NFIND
macros can be used to find a particular element in the tree.
.Fn RB_FIND
finds the node with the same key as
.Fa elm .
.Fn RB_NFIND
finds the first node greater than or equal to the search key.
.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 ,
.Fn RB_NEXT ,
and
.Fn RB_PREV
macros can be used to traverse the tree:
.Bd -literal -offset indent
for (np = RB_MIN(NAME, &head); np != NULL; np = RB_NEXT(NAME, &head, np))
.Ed
.Pp
Or, for simplicity, one can use the
.Fn RB_FOREACH
or
.Fn RB_FOREACH_REVERSE
macros:
.Bd -literal -offset indent
RB_FOREACH(np, NAME, &head)
.Ed
.Pp
The macros
.Fn RB_FOREACH_SAFE
and
.Fn RB_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
The
.Fn RB_EMPTY
macro should be used to check whether a red-black tree is empty.
.Sh EXAMPLES
The following example demonstrates how to declare a red-black tree
holding integers.
Values are inserted into it and the contents of the tree are printed
in order.
Lastly, the internal structure of the tree is printed.
.Bd -literal -offset 3n
#include <sys/tree.h>
#include <err.h>
#include <stdio.h>
#include <stdlib.h>

struct node {
	RB_ENTRY(node) entry;
	int i;
};

int
intcmp(struct node *e1, struct node *e2)
{
	return (e1->i < e2->i ? -1 : e1->i > e2->i);
}

RB_HEAD(inttree, node) head = RB_INITIALIZER(&head);
RB_GENERATE(inttree, node, entry, intcmp)

int testdata[] = {
	20, 16, 17, 13, 3, 6, 1, 8, 2, 4, 10, 19, 5, 9, 12, 15, 18,
	7, 11, 14
};

void
print_tree(struct node *n)
{
	struct node *left, *right;

	if (n == NULL) {
		printf("nil");
		return;
	}
	left = RB_LEFT(n, entry);
	right = RB_RIGHT(n, entry);
	if (left == NULL && right == NULL)
		printf("%d", n->i);
	else {
		printf("%d(", n->i);
		print_tree(left);
		printf(",");
		print_tree(right);
		printf(")");
	}
}

int
main()
{
	int i;
	struct node *n;

	for (i = 0; i < sizeof(testdata) / sizeof(testdata[0]); i++) {
		if ((n = malloc(sizeof(struct node))) == NULL)
			err(1, NULL);
		n->i = testdata[i];
		RB_INSERT(inttree, &head, n);
	}

	RB_FOREACH(n, inttree, &head) {
		printf("%d\en", n->i);
	}
	print_tree(RB_ROOT(&head));
	printf("\en");
	return (0);
}
.Ed
.Sh NOTES
Some of these macros or functions perform no error checking,
and invalid usage leads to undefined behaviour.
In the case of macros or functions that expect their arguments
to be elements that are present in the tree, passing an element
that is not present in the tree is invalid.
.Pp
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 free'd, 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 SEE ALSO
.Xr rbtree 3
.Sh AUTHORS
The author of the tree macros is
.An Niels Provos .