503 lines
12 KiB
Groff
503 lines
12 KiB
Groff
.\" $OpenBSD: tree.3,v 1.7 2002/06/12 01:09:20 provos Exp $
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.\"
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.\" Copyright 2002 Niels Provos <provos@citi.umich.edu>
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.\" All rights reserved.
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.\"
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.\" Redistribution and use in source and binary forms, with or without
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.\" modification, are permitted provided that the following conditions
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.\" are met:
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.\" 1. Redistributions of source code must retain the above copyright
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.\" notice, this list of conditions and the following disclaimer.
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.\" 2. Redistributions in binary form must reproduce the above copyright
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.\" notice, this list of conditions and the following disclaimer in the
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.\" documentation and/or other materials provided with the distribution.
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.\" 3. All advertising materials mentioning features or use of this software
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.\" must display the following acknowledgement:
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.\" This product includes software developed by Niels Provos.
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.\" 4. The name of the author may not be used to endorse or promote products
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.\" derived from this software without specific prior written permission.
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.\"
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.\" THIS SOFTWARE IS PROVIDED BY THE AUTHOR ``AS IS'' AND ANY EXPRESS OR
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.\" IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES
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.\" OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED.
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.\" IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR ANY DIRECT, INDIRECT,
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.\" INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT
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.\" NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE,
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.\" DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY
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.\" THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT
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.\" (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF
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.\" THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
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.\"
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.\" $FreeBSD$
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.\"
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.Dd December 27, 2007
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.Dt TREE 3
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.Os
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.Sh NAME
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.Nm SPLAY_PROTOTYPE ,
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.Nm SPLAY_GENERATE ,
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.Nm SPLAY_ENTRY ,
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.Nm SPLAY_HEAD ,
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.Nm SPLAY_INITIALIZER ,
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.Nm SPLAY_ROOT ,
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.Nm SPLAY_EMPTY ,
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.Nm SPLAY_NEXT ,
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.Nm SPLAY_MIN ,
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.Nm SPLAY_MAX ,
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.Nm SPLAY_FIND ,
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.Nm SPLAY_LEFT ,
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.Nm SPLAY_RIGHT ,
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.Nm SPLAY_FOREACH ,
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.Nm SPLAY_INIT ,
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.Nm SPLAY_INSERT ,
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.Nm SPLAY_REMOVE ,
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.Nm RB_PROTOTYPE ,
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.Nm RB_PROTOTYPE_STATIC ,
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.Nm RB_GENERATE ,
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.Nm RB_GENERATE_STATIC ,
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.Nm RB_ENTRY ,
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.Nm RB_HEAD ,
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.Nm RB_INITIALIZER ,
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.Nm RB_ROOT ,
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.Nm RB_EMPTY ,
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.Nm RB_NEXT ,
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.Nm RB_PREV ,
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.Nm RB_MIN ,
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.Nm RB_MAX ,
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.Nm RB_FIND ,
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.Nm RB_NFIND ,
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.Nm RB_LEFT ,
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.Nm RB_RIGHT ,
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.Nm RB_PARENT ,
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.Nm RB_FOREACH ,
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.Nm RB_FOREACH_REVERSE ,
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.Nm RB_INIT ,
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.Nm RB_INSERT ,
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.Nm RB_REMOVE
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.Nd "implementations of splay and red-black trees"
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.Sh SYNOPSIS
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.In sys/tree.h
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.Fn SPLAY_PROTOTYPE NAME TYPE FIELD CMP
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.Fn SPLAY_GENERATE NAME TYPE FIELD CMP
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.Fn SPLAY_ENTRY TYPE
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.Fn SPLAY_HEAD HEADNAME TYPE
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.Ft "struct TYPE *"
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.Fn SPLAY_INITIALIZER "SPLAY_HEAD *head"
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.Fn SPLAY_ROOT "SPLAY_HEAD *head"
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.Ft bool
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.Fn SPLAY_EMPTY "SPLAY_HEAD *head"
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.Ft "struct TYPE *"
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.Fn SPLAY_NEXT NAME "SPLAY_HEAD *head" "struct TYPE *elm"
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.Ft "struct TYPE *"
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.Fn SPLAY_MIN NAME "SPLAY_HEAD *head"
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.Ft "struct TYPE *"
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.Fn SPLAY_MAX NAME "SPLAY_HEAD *head"
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.Ft "struct TYPE *"
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.Fn SPLAY_FIND NAME "SPLAY_HEAD *head" "struct TYPE *elm"
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.Ft "struct TYPE *"
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.Fn SPLAY_LEFT "struct TYPE *elm" "SPLAY_ENTRY NAME"
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.Ft "struct TYPE *"
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.Fn SPLAY_RIGHT "struct TYPE *elm" "SPLAY_ENTRY NAME"
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.Fn SPLAY_FOREACH VARNAME NAME "SPLAY_HEAD *head"
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.Ft void
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.Fn SPLAY_INIT "SPLAY_HEAD *head"
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.Ft "struct TYPE *"
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.Fn SPLAY_INSERT NAME "SPLAY_HEAD *head" "struct TYPE *elm"
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.Ft "struct TYPE *"
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.Fn SPLAY_REMOVE NAME "SPLAY_HEAD *head" "struct TYPE *elm"
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.Fn RB_PROTOTYPE NAME TYPE FIELD CMP
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.Fn RB_PROTOTYPE_STATIC NAME TYPE FIELD CMP
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.Fn RB_GENERATE NAME TYPE FIELD CMP
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.Fn RB_GENERATE_STATIC NAME TYPE FIELD CMP
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.Fn RB_ENTRY TYPE
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.Fn RB_HEAD HEADNAME TYPE
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.Fn RB_INITIALIZER "RB_HEAD *head"
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.Ft "struct TYPE *"
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.Fn RB_ROOT "RB_HEAD *head"
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.Ft "bool"
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.Fn RB_EMPTY "RB_HEAD *head"
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.Ft "struct TYPE *"
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.Fn RB_NEXT NAME "RB_HEAD *head" "struct TYPE *elm"
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.Ft "struct TYPE *"
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.Fn RB_PREV NAME "RB_HEAD *head" "struct TYPE *elm"
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.Ft "struct TYPE *"
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.Fn RB_MIN NAME "RB_HEAD *head"
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.Ft "struct TYPE *"
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.Fn RB_MAX NAME "RB_HEAD *head"
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.Ft "struct TYPE *"
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.Fn RB_FIND NAME "RB_HEAD *head" "struct TYPE *elm"
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.Ft "struct TYPE *"
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.Fn RB_NFIND NAME "RB_HEAD *head" "struct TYPE *elm"
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.Ft "struct TYPE *"
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.Fn RB_LEFT "struct TYPE *elm" "RB_ENTRY NAME"
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.Ft "struct TYPE *"
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.Fn RB_RIGHT "struct TYPE *elm" "RB_ENTRY NAME"
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.Ft "struct TYPE *"
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.Fn RB_PARENT "struct TYPE *elm" "RB_ENTRY NAME"
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.Fn RB_FOREACH VARNAME NAME "RB_HEAD *head"
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.Fn RB_FOREACH_REVERSE VARNAME NAME "RB_HEAD *head"
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.Ft void
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.Fn RB_INIT "RB_HEAD *head"
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.Ft "struct TYPE *"
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.Fn RB_INSERT NAME "RB_HEAD *head" "struct TYPE *elm"
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.Ft "struct TYPE *"
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.Fn RB_REMOVE NAME "RB_HEAD *head" "struct TYPE *elm"
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.Sh DESCRIPTION
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These macros define data structures for different types of trees:
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splay trees and red-black trees.
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.Pp
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In the macro definitions,
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.Fa TYPE
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is the name tag of a user defined structure that must contain a field of type
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.Vt SPLAY_ENTRY ,
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or
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.Vt RB_ENTRY ,
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named
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.Fa ENTRYNAME .
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The argument
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.Fa HEADNAME
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is the name tag of a user defined structure that must be declared
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using the macros
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.Fn SPLAY_HEAD ,
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or
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.Fn RB_HEAD .
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The argument
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.Fa NAME
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has to be a unique name prefix for every tree that is defined.
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.Pp
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The function prototypes are declared with
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.Fn SPLAY_PROTOTYPE ,
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.Fn RB_PROTOTYPE ,
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or
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.Fn RB_PROTOTYPE_STATIC .
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The function bodies are generated with
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.Fn SPLAY_GENERATE ,
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.Fn RB_GENERATE ,
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or
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.Fn RB_GENERATE_STATIC .
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See the examples below for further explanation of how these macros are used.
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.Sh SPLAY TREES
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A splay tree is a self-organizing data structure.
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Every operation on the tree causes a splay to happen.
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The splay moves the requested
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node to the root of the tree and partly rebalances it.
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.Pp
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This has the benefit that request locality causes faster lookups as
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the requested nodes move to the top of the tree.
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On the other hand, every lookup causes memory writes.
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.Pp
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The Balance Theorem bounds the total access time for
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.Ar m
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operations and
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.Ar n
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inserts on an initially empty tree as
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.Fn O "\*[lp]m + n\*[rp]lg n" .
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The
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amortized cost for a sequence of
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.Ar m
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accesses to a splay tree is
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.Fn O "lg n" .
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.Pp
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A splay tree is headed by a structure defined by the
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.Fn SPLAY_HEAD
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macro.
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A
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structure is declared as follows:
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.Bd -ragged -offset indent
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.Fn SPLAY_HEAD HEADNAME TYPE
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.Va head ;
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.Ed
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.Pp
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where
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.Fa HEADNAME
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is the name of the structure to be defined, and struct
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.Fa TYPE
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is the type of the elements to be inserted into the tree.
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.Pp
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The
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.Fn SPLAY_ENTRY
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macro declares a structure that allows elements to be connected in the tree.
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.Pp
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In order to use the functions that manipulate the tree structure,
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their prototypes need to be declared with the
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.Fn SPLAY_PROTOTYPE
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macro,
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where
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.Fa NAME
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is a unique identifier for this particular tree.
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The
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.Fa TYPE
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argument is the type of the structure that is being managed
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by the tree.
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The
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.Fa FIELD
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argument is the name of the element defined by
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.Fn SPLAY_ENTRY .
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.Pp
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The function bodies are generated with the
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.Fn SPLAY_GENERATE
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macro.
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It takes the same arguments as the
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.Fn SPLAY_PROTOTYPE
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macro, but should be used only once.
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.Pp
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Finally,
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the
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.Fa CMP
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argument is the name of a function used to compare tree nodes
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with each other.
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The function takes two arguments of type
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.Vt "struct TYPE *" .
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If the first argument is smaller than the second, the function returns a
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value smaller than zero.
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If they are equal, the function returns zero.
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Otherwise, it should return a value greater than zero.
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The compare
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function defines the order of the tree elements.
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.Pp
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The
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.Fn SPLAY_INIT
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macro initializes the tree referenced by
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.Fa head .
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.Pp
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The splay tree can also be initialized statically by using the
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.Fn SPLAY_INITIALIZER
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macro like this:
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.Bd -ragged -offset indent
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.Fn SPLAY_HEAD HEADNAME TYPE
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.Va head
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=
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.Fn SPLAY_INITIALIZER &head ;
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.Ed
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.Pp
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The
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.Fn SPLAY_INSERT
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macro inserts the new element
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.Fa elm
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into the tree.
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.Pp
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The
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.Fn SPLAY_REMOVE
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macro removes the element
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.Fa elm
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from the tree pointed by
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.Fa head .
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.Pp
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The
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.Fn SPLAY_FIND
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macro can be used to find a particular element in the tree.
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.Bd -literal -offset indent
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struct TYPE find, *res;
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find.key = 30;
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res = SPLAY_FIND(NAME, head, &find);
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.Ed
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.Pp
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The
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.Fn SPLAY_ROOT ,
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.Fn SPLAY_MIN ,
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.Fn SPLAY_MAX ,
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and
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.Fn SPLAY_NEXT
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macros can be used to traverse the tree:
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.Bd -literal -offset indent
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for (np = SPLAY_MIN(NAME, &head); np != NULL; np = SPLAY_NEXT(NAME, &head, np))
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.Ed
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.Pp
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Or, for simplicity, one can use the
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.Fn SPLAY_FOREACH
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macro:
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.Bd -ragged -offset indent
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.Fn SPLAY_FOREACH np NAME head
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.Ed
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.Pp
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The
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.Fn SPLAY_EMPTY
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macro should be used to check whether a splay tree is empty.
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.Sh RED-BLACK TREES
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A red-black tree is a binary search tree with the node color as an
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extra attribute.
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It fulfills a set of conditions:
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.Bl -enum -offset indent
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.It
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Every search path from the root to a leaf consists of the same number of
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black nodes.
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.It
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Each red node (except for the root) has a black parent.
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.It
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Each leaf node is black.
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.El
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.Pp
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Every operation on a red-black tree is bounded as
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.Fn O "lg n" .
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The maximum height of a red-black tree is
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.Fn 2lg "n + 1" .
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.Pp
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A red-black tree is headed by a structure defined by the
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.Fn RB_HEAD
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macro.
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A
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structure is declared as follows:
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.Bd -ragged -offset indent
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.Fn RB_HEAD HEADNAME TYPE
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.Va head ;
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.Ed
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.Pp
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where
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.Fa HEADNAME
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is the name of the structure to be defined, and struct
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.Fa TYPE
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is the type of the elements to be inserted into the tree.
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.Pp
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The
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.Fn RB_ENTRY
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macro declares a structure that allows elements to be connected in the tree.
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.Pp
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In order to use the functions that manipulate the tree structure,
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their prototypes need to be declared with the
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.Fn RB_PROTOTYPE
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or
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.Fn RB_PROTOTYPE_STATIC
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macro,
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where
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.Fa NAME
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is a unique identifier for this particular tree.
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The
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.Fa TYPE
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argument is the type of the structure that is being managed
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by the tree.
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The
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.Fa FIELD
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argument is the name of the element defined by
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.Fn RB_ENTRY .
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.Pp
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The function bodies are generated with the
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.Fn RB_GENERATE
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or
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.Fn RB_GENERATE_STATIC
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macro.
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These macros take the same arguments as the
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.Fn RB_PROTOTYPE
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and
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.Fn RB_PROTOTYPE_STATIC
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macros, but should be used only once.
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.Pp
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Finally,
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the
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.Fa CMP
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argument is the name of a function used to compare tree noded
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with each other.
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The function takes two arguments of type
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.Vt "struct TYPE *" .
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If the first argument is smaller than the second, the function returns a
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value smaller than zero.
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If they are equal, the function returns zero.
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Otherwise, it should return a value greater than zero.
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The compare
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function defines the order of the tree elements.
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.Pp
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The
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.Fn RB_INIT
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macro initializes the tree referenced by
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.Fa head .
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.Pp
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The red-black tree can also be initialized statically by using the
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.Fn RB_INITIALIZER
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macro like this:
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.Bd -ragged -offset indent
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.Fn RB_HEAD HEADNAME TYPE
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.Va head
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=
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.Fn RB_INITIALIZER &head ;
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.Ed
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.Pp
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The
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.Fn RB_INSERT
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macro inserts the new element
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.Fa elm
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into the tree.
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.Pp
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The
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.Fn RB_REMOVE
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macro removes the element
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.Fa elm
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from the tree pointed by
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.Fa head .
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.Pp
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The
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.Fn RB_FIND
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and
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.Fn RB_NFIND
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macros can be used to find a particular element in the tree.
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.Bd -literal -offset indent
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struct TYPE find, *res;
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find.key = 30;
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res = RB_FIND(NAME, head, &find);
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.Ed
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.Pp
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The
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.Fn RB_ROOT ,
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.Fn RB_MIN ,
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.Fn RB_MAX ,
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.Fn RB_NEXT ,
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and
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.Fn RB_PREV
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macros can be used to traverse the tree:
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.Pp
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.Dl "for (np = RB_MIN(NAME, &head); np != NULL; np = RB_NEXT(NAME, &head, np))"
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.Pp
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Or, for simplicity, one can use the
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.Fn RB_FOREACH
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or
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.Fn RB_FOREACH_REVERSE
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macro:
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.Bd -ragged -offset indent
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.Fn RB_FOREACH np NAME head
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.Ed
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.Pp
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The
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.Fn RB_EMPTY
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macro should be used to check whether a red-black tree is empty.
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.Sh NOTES
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Trying to free a tree in the following way is a common error:
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.Bd -literal -offset indent
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SPLAY_FOREACH(var, NAME, head) {
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SPLAY_REMOVE(NAME, head, var);
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free(var);
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}
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free(head);
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.Ed
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.Pp
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Since
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.Va var
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is freed, the
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.Fn FOREACH
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macro refers to a pointer that may have been reallocated already.
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Proper code needs a second variable.
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.Bd -literal -offset indent
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for (var = SPLAY_MIN(NAME, head); var != NULL; var = nxt) {
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nxt = SPLAY_NEXT(NAME, head, var);
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SPLAY_REMOVE(NAME, head, var);
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free(var);
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}
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.Ed
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.Pp
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Both
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.Fn RB_INSERT
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and
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.Fn SPLAY_INSERT
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return
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.Dv NULL
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if the element was inserted in the tree successfully, otherwise they
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return a pointer to the element with the colliding key.
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.Pp
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Accordingly,
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.Fn RB_REMOVE
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and
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.Fn SPLAY_REMOVE
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return the pointer to the removed element otherwise they return
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.Dv NULL
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to indicate an error.
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.Sh AUTHORS
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The author of the tree macros is
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.An Niels Provos .
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