.\" $OpenBSD: tree.3,v 1.21 2010/05/05 18:17:41 nicm 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. .\" * .\" * 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: May 5 2010 $ .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_REVERSE , .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_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_REVERSE "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 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 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 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, .Va 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. .Va 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, .Va 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 by .Fa head . .Fn RB_REMOVE returns .Fa elm . .Pp The .Fn RB_FIND 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 .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 ? -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 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 .Sh AUTHORS The author of the tree macros is Niels Provos.