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+.\"	$OpenBSD: tree.3,v 1.7 2002/06/12 01:09:20 provos 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.
+.\" * 3. All advertising materials mentioning features or use of this software
+.\" *    must display the following acknowledgement:
+.\" *      This product includes software developed by Niels Provos.
+.\" * 4. The name of the author may not be used to endorse or promote products
+.\" *    derived from this software without specific prior written permission.
+.\" *
+.\" * 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 February 24, 2002
+.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_GENERATE,
+.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_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
+.Fd #include <sys/tree.h>
+.Pp
+.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"
+.Pp
+.Fn RB_PROTOTYPE "NAME" "TYPE" "FIELD" "CMP"
+.Fn RB_GENERATE "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_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 defines 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
+.Li SPLAY_ENTRY ,
+or
+.Li 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 either
+.Li SPLAY_PROTOTYPE,
+or
+.Li RB_PROTOTYPE .
+The function bodies are generated with either
+.Li SPLAY_GENERATE,
+or
+.Li RB_GENERATE .
+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 noded
+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.
+.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 -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.
+.Pp
+.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 -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
+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
+macro. It takes the same arguments as the
+.Fn RB_PROTOTYPE
+macro, 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
+.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 redblack 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.
+.Pp
+The
+.Fn RB_REMOVE
+macro removes the element
+.Fa elm
+from the tree pointed by
+.Fa head .
+.Pp
+The
+.Fn RB_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 = 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:
+.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
+macro:
+.Bd -literal -offset indent
+RB_FOREACH(np, NAME, head)
+.Ed
+.Pp
+The
+.Fn RB_EMPTY
+macro should be used to check whether a splay tree is empty.
+.Pp
+.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 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
+.Va 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
+.Va NULL
+to indicate an error.
+.Sh AUTHORS
+The author of the tree macros is Niels Provos.