1 .\" $OpenBSD: tree.3,v 1.7 2002/06/12 01:09:20 provos Exp $
3 .\" Copyright 2002 Niels Provos <provos@citi.umich.edu>
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41 .Nm SPLAY_INITIALIZER ,
55 .Nm RB_PROTOTYPE_STATIC ,
56 .Nm RB_PROTOTYPE_INSERT ,
57 .Nm RB_PROTOTYPE_INSERT_COLOR ,
58 .Nm RB_PROTOTYPE_REMOVE ,
59 .Nm RB_PROTOTYPE_REMOVE_COLOR ,
60 .Nm RB_PROTOTYPE_FIND ,
61 .Nm RB_PROTOTYPE_NFIND ,
62 .Nm RB_PROTOTYPE_NEXT ,
63 .Nm RB_PROTOTYPE_PREV ,
64 .Nm RB_PROTOTYPE_MINMAX ,
65 .Nm RB_PROTOTYPE_REINSERT ,
67 .Nm RB_GENERATE_STATIC ,
68 .Nm RB_GENERATE_INSERT ,
69 .Nm RB_GENERATE_INSERT_COLOR ,
70 .Nm RB_GENERATE_REMOVE ,
71 .Nm RB_GENERATE_REMOVE_COLOR ,
72 .Nm RB_GENERATE_FIND ,
73 .Nm RB_GENERATE_NFIND ,
74 .Nm RB_GENERATE_NEXT ,
75 .Nm RB_GENERATE_PREV ,
76 .Nm RB_GENERATE_MINMAX ,
77 .Nm RB_GENERATE_REINSERT ,
95 .Nm RB_FOREACH_REVERSE ,
96 .Nm RB_FOREACH_REVERSE_FROM ,
97 .Nm RB_FOREACH_REVERSE_SAFE ,
102 .Nd "implementations of splay and red-black trees"
105 .Fn SPLAY_PROTOTYPE NAME TYPE FIELD CMP
106 .Fn SPLAY_GENERATE NAME TYPE FIELD CMP
108 .Fn SPLAY_HEAD HEADNAME TYPE
110 .Fn SPLAY_INITIALIZER "SPLAY_HEAD *head"
111 .Fn SPLAY_ROOT "SPLAY_HEAD *head"
113 .Fn SPLAY_EMPTY "SPLAY_HEAD *head"
115 .Fn SPLAY_NEXT NAME "SPLAY_HEAD *head" "struct TYPE *elm"
117 .Fn SPLAY_MIN NAME "SPLAY_HEAD *head"
119 .Fn SPLAY_MAX NAME "SPLAY_HEAD *head"
121 .Fn SPLAY_FIND NAME "SPLAY_HEAD *head" "struct TYPE *elm"
123 .Fn SPLAY_LEFT "struct TYPE *elm" "SPLAY_ENTRY NAME"
125 .Fn SPLAY_RIGHT "struct TYPE *elm" "SPLAY_ENTRY NAME"
126 .Fn SPLAY_FOREACH VARNAME NAME "SPLAY_HEAD *head"
128 .Fn SPLAY_INIT "SPLAY_HEAD *head"
130 .Fn SPLAY_INSERT NAME "SPLAY_HEAD *head" "struct TYPE *elm"
132 .Fn SPLAY_REMOVE NAME "SPLAY_HEAD *head" "struct TYPE *elm"
133 .Fn RB_PROTOTYPE NAME TYPE FIELD CMP
134 .Fn RB_PROTOTYPE_STATIC NAME TYPE FIELD CMP
135 .Fn RB_PROTOTYPE_INSERT NAME TYPE ATTR
136 .Fn RB_PROTOTYPE_INSERT_COLOR NAME TYPE ATTR
137 .Fn RB_PROTOTYPE_REMOVE NAME TYPE ATTR
138 .Fn RB_PROTOTYPE_REMOVE_COLOR NAME TYPE ATTR
139 .Fn RB_PROTOTYPE_FIND NAME TYPE ATTR
140 .Fn RB_PROTOTYPE_NFIND NAME TYPE ATTR
141 .Fn RB_PROTOTYPE_NEXT NAME TYPE ATTR
142 .Fn RB_PROTOTYPE_PREV NAME TYPE ATTR
143 .Fn RB_PROTOTYPE_MINMAX NAME TYPE ATTR
144 .Fn RB_PROTOTYPE_REINSERT NAME TYPE ATTR
145 .Fn RB_GENERATE NAME TYPE FIELD CMP
146 .Fn RB_GENERATE_STATIC NAME TYPE FIELD CMP
147 .Fn RB_GENERATE_INSERT NAME TYPE FIELD CMP ATTR
148 .Fn RB_GENERATE_INSERT_COLOR NAME TYPE FIELD ATTR
149 .Fn RB_GENERATE_REMOVE NAME TYPE FIELD ATTR
150 .Fn RB_GENERATE_REMOVE_COLOR NAME TYPE FIELD ATTR
151 .Fn RB_GENERATE_FIND NAME TYPE FIELD CMP ATTR
152 .Fn RB_GENERATE_NFIND NAME TYPE FIELD CMP ATTR
153 .Fn RB_GENERATE_NEXT NAME TYPE FIELD ATTR
154 .Fn RB_GENERATE_PREV NAME TYPE FIELD ATTR
155 .Fn RB_GENERATE_MINMAX NAME TYPE FIELD ATTR
156 .Fn RB_GENERATE_REINSERT NAME TYPE FIELD CMP ATTR
158 .Fn RB_HEAD HEADNAME TYPE
159 .Fn RB_INITIALIZER "RB_HEAD *head"
161 .Fn RB_ROOT "RB_HEAD *head"
163 .Fn RB_EMPTY "RB_HEAD *head"
165 .Fn RB_NEXT NAME "RB_HEAD *head" "struct TYPE *elm"
167 .Fn RB_PREV NAME "RB_HEAD *head" "struct TYPE *elm"
169 .Fn RB_MIN NAME "RB_HEAD *head"
171 .Fn RB_MAX NAME "RB_HEAD *head"
173 .Fn RB_FIND NAME "RB_HEAD *head" "struct TYPE *elm"
175 .Fn RB_NFIND NAME "RB_HEAD *head" "struct TYPE *elm"
177 .Fn RB_LEFT "struct TYPE *elm" "RB_ENTRY NAME"
179 .Fn RB_RIGHT "struct TYPE *elm" "RB_ENTRY NAME"
181 .Fn RB_PARENT "struct TYPE *elm" "RB_ENTRY NAME"
182 .Fn RB_FOREACH VARNAME NAME "RB_HEAD *head"
183 .Fn RB_FOREACH_FROM "VARNAME" "NAME" "POS_VARNAME"
184 .Fn RB_FOREACH_SAFE "VARNAME" "NAME" "RB_HEAD *head" "TEMP_VARNAME"
185 .Fn RB_FOREACH_REVERSE VARNAME NAME "RB_HEAD *head"
186 .Fn RB_FOREACH_REVERSE_FROM "VARNAME" "NAME" "POS_VARNAME"
187 .Fn RB_FOREACH_REVERSE_SAFE "VARNAME" "NAME" "RB_HEAD *head" "TEMP_VARNAME"
189 .Fn RB_INIT "RB_HEAD *head"
191 .Fn RB_INSERT NAME "RB_HEAD *head" "struct TYPE *elm"
193 .Fn RB_REMOVE NAME "RB_HEAD *head" "struct TYPE *elm"
195 .Fn RB_REINSERT NAME "RB_HEAD *head" "struct TYPE *elm"
197 These macros define data structures for different types of trees:
198 splay trees and red-black trees.
200 In the macro definitions,
202 is the name tag of a user defined structure that must contain a field of type
210 is the name tag of a user defined structure that must be declared
217 has to be a unique name prefix for every tree that is defined.
219 The function prototypes are declared with
220 .Fn SPLAY_PROTOTYPE ,
223 .Fn RB_PROTOTYPE_STATIC .
224 The function bodies are generated with
228 .Fn RB_GENERATE_STATIC .
229 See the examples below for further explanation of how these macros are used.
231 A splay tree is a self-organizing data structure.
232 Every operation on the tree causes a splay to happen.
233 The splay moves the requested
234 node to the root of the tree and partly rebalances it.
236 This has the benefit that request locality causes faster lookups as
237 the requested nodes move to the top of the tree.
238 On the other hand, every lookup causes memory writes.
240 The Balance Theorem bounds the total access time for
244 inserts on an initially empty tree as
245 .Fn O "\*[lp]m + n\*[rp]lg n" .
247 amortized cost for a sequence of
249 accesses to a splay tree is
252 A splay tree is headed by a structure defined by the
256 structure is declared as follows:
257 .Bd -ragged -offset indent
258 .Fn SPLAY_HEAD HEADNAME TYPE
264 is the name of the structure to be defined, and struct
266 is the type of the elements to be inserted into the tree.
270 macro declares a structure that allows elements to be connected in the tree.
272 In order to use the functions that manipulate the tree structure,
273 their prototypes need to be declared with the
278 is a unique identifier for this particular tree.
281 argument is the type of the structure that is being managed
285 argument is the name of the element defined by
288 The function bodies are generated with the
291 It takes the same arguments as the
293 macro, but should be used only once.
298 argument is the name of a function used to compare tree nodes
300 The function takes two arguments of type
301 .Vt "struct TYPE *" .
302 If the first argument is smaller than the second, the function returns a
303 value smaller than zero.
304 If they are equal, the function returns zero.
305 Otherwise, it should return a value greater than zero.
307 function defines the order of the tree elements.
311 macro initializes the tree referenced by
314 The splay tree can also be initialized statically by using the
315 .Fn SPLAY_INITIALIZER
317 .Bd -ragged -offset indent
318 .Fn SPLAY_HEAD HEADNAME TYPE
321 .Fn SPLAY_INITIALIZER &head ;
326 macro inserts the new element
332 macro removes the element
334 from the tree pointed by
339 macro can be used to find a particular element in the tree.
340 .Bd -literal -offset indent
341 struct TYPE find, *res;
343 res = SPLAY_FIND(NAME, head, &find);
352 macros can be used to traverse the tree:
353 .Bd -literal -offset indent
354 for (np = SPLAY_MIN(NAME, &head); np != NULL; np = SPLAY_NEXT(NAME, &head, np))
357 Or, for simplicity, one can use the
360 .Bd -ragged -offset indent
361 .Fn SPLAY_FOREACH np NAME head
366 macro should be used to check whether a splay tree is empty.
368 A red-black tree is a binary search tree with the node color as an
370 It fulfills a set of conditions:
371 .Bl -enum -offset indent
373 Every search path from the root to a leaf consists of the same number of
376 Each red node (except for the root) has a black parent.
378 Each leaf node is black.
381 Every operation on a red-black tree is bounded as
383 The maximum height of a red-black tree is
386 A red-black tree is headed by a structure defined by the
390 structure is declared as follows:
391 .Bd -ragged -offset indent
392 .Fn RB_HEAD HEADNAME TYPE
398 is the name of the structure to be defined, and struct
400 is the type of the elements to be inserted into the tree.
404 macro declares a structure that allows elements to be connected in the tree.
406 In order to use the functions that manipulate the tree structure,
407 their prototypes need to be declared with the
410 .Fn RB_PROTOTYPE_STATIC
414 is a unique identifier for this particular tree.
417 argument is the type of the structure that is being managed
421 argument is the name of the element defined by
423 Individual prototypes can be declared with
424 .Fn RB_PROTOTYPE_INSERT ,
425 .Fn RB_PROTOTYPE_INSERT_COLOR ,
426 .Fn RB_PROTOTYPE_REMOVE ,
427 .Fn RB_PROTOTYPE_REMOVE_COLOR ,
428 .Fn RB_PROTOTYPE_FIND ,
429 .Fn RB_PROTOTYPE_NFIND ,
430 .Fn RB_PROTOTYPE_NEXT ,
431 .Fn RB_PROTOTYPE_PREV ,
432 .Fn RB_PROTOTYPE_MINMAX ,
434 .Fn RB_PROTOTYPE_REINSERT
435 in case not all functions are required.
436 The individual prototype macros expect
444 argument must be empty for global functions or
446 for static functions.
448 The function bodies are generated with the
451 .Fn RB_GENERATE_STATIC
453 These macros take the same arguments as the
456 .Fn RB_PROTOTYPE_STATIC
457 macros, but should be used only once.
458 As an alternative individual function bodies are generated with the
459 .Fn RB_GENERATE_INSERT ,
460 .Fn RB_GENERATE_INSERT_COLOR ,
461 .Fn RB_GENERATE_REMOVE ,
462 .Fn RB_GENERATE_REMOVE_COLOR ,
463 .Fn RB_GENERATE_FIND ,
464 .Fn RB_GENERATE_NFIND ,
465 .Fn RB_GENERATE_NEXT ,
466 .Fn RB_GENERATE_PREV ,
467 .Fn RB_GENERATE_MINMAX ,
469 .Fn RB_GENERATE_REINSERT
475 argument is the name of a function used to compare tree nodes
477 The function takes two arguments of type
478 .Vt "struct TYPE *" .
479 If the first argument is smaller than the second, the function returns a
480 value smaller than zero.
481 If they are equal, the function returns zero.
482 Otherwise, it should return a value greater than zero.
484 function defines the order of the tree elements.
488 macro initializes the tree referenced by
491 The red-black tree can also be initialized statically by using the
494 .Bd -ragged -offset indent
495 .Fn RB_HEAD HEADNAME TYPE
498 .Fn RB_INITIALIZER &head ;
503 macro inserts the new element
509 macro removes the element
511 from the tree pointed by
518 macros can be used to find a particular element in the tree.
519 .Bd -literal -offset indent
520 struct TYPE find, *res;
522 res = RB_FIND(NAME, head, &find);
532 macros can be used to traverse the tree:
534 .Dl "for (np = RB_MIN(NAME, &head); np != NULL; np = RB_NEXT(NAME, &head, np))"
536 Or, for simplicity, one can use the
539 .Fn RB_FOREACH_REVERSE
541 .Bd -ragged -offset indent
542 .Fn RB_FOREACH np NAME head
548 .Fn RB_FOREACH_REVERSE_SAFE
549 traverse the tree referenced by head
550 in a forward or reverse direction respectively,
551 assigning each element in turn to np.
552 However, unlike their unsafe counterparts,
553 they permit both the removal of np
554 as well as freeing it from within the loop safely
555 without interfering with the traversal.
560 .Fn RB_FOREACH_REVERSE_FROM
561 may be used to continue an interrupted traversal
562 in a forward or reverse direction respectively.
563 The head pointer is not required.
564 The pointer to the node from where to resume the traversal
565 should be passed as their last argument,
566 and will be overwritten to provide safe traversal.
570 macro should be used to check whether a red-black tree is empty.
574 macro updates the position of the element
577 This must be called if a member of a
579 is modified in a way that affects comparison, such as by modifying
581 This is a lower overhead alternative to removing the element
582 and reinserting it again.
584 The following example demonstrates how to declare a red-black tree
586 Values are inserted into it and the contents of the tree are printed
588 Lastly, the internal structure of the tree is printed.
589 .Bd -literal -offset 3n
590 #include <sys/tree.h>
596 RB_ENTRY(node) entry;
601 intcmp(struct node *e1, struct node *e2)
603 return (e1->i < e2->i ? -1 : e1->i > e2->i);
606 RB_HEAD(inttree, node) head = RB_INITIALIZER(&head);
607 RB_GENERATE(inttree, node, entry, intcmp)
610 20, 16, 17, 13, 3, 6, 1, 8, 2, 4, 10, 19, 5, 9, 12, 15, 18,
615 print_tree(struct node *n)
617 struct node *left, *right;
623 left = RB_LEFT(n, entry);
624 right = RB_RIGHT(n, entry);
625 if (left == NULL && right == NULL)
642 for (i = 0; i < sizeof(testdata) / sizeof(testdata[0]); i++) {
643 if ((n = malloc(sizeof(struct node))) == NULL)
646 RB_INSERT(inttree, &head, n);
649 RB_FOREACH(n, inttree, &head) {
650 printf("%d\en", n->i);
652 print_tree(RB_ROOT(&head));
658 Trying to free a tree in the following way is a common error:
659 .Bd -literal -offset indent
660 SPLAY_FOREACH(var, NAME, head) {
661 SPLAY_REMOVE(NAME, head, var);
671 macro refers to a pointer that may have been reallocated already.
672 Proper code needs a second variable.
673 .Bd -literal -offset indent
674 for (var = SPLAY_MIN(NAME, head); var != NULL; var = nxt) {
675 nxt = SPLAY_NEXT(NAME, head, var);
676 SPLAY_REMOVE(NAME, head, var);
687 if the element was inserted in the tree successfully, otherwise they
688 return a pointer to the element with the colliding key.
694 return the pointer to the removed element otherwise they return
696 to indicate an error.
701 The tree macros first appeared in
704 The author of the tree macros is