.\" $OpenBSD: tree.3,v 1.14 2007/12/24 12:04:13 otto Exp $ .\"/* .\" * Copyright 2002 Niels Provos .\" * 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 $Mdocdate: December 24 2007 $ .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 .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 "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_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 "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_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 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 .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 trees 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. .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 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 trees 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 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. .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 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 #include #include #include struct node { RB_ENTRY(node) entry; int i; }; int intcmp(struct node *e1, struct node *e2) { return (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 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.