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*/
1 /* Copyright (C) 1995-1997, 2000, 2006-2007, 2009-2021 Free Software
2 Foundation, Inc.
3 Contributed by Bernd Schmidt <crux@Pool.Informatik.RWTH-Aachen.DE>, 1997.
4
5 NOTE: The canonical source of this file is maintained with the GNU C
6 Library. Bugs can be reported to bug-glibc@gnu.org.
7
8 This file is free software: you can redistribute it and/or modify
9 it under the terms of the GNU Lesser General Public License as
10 published by the Free Software Foundation; either version 2.1 of the
11 License, or (at your option) any later version.
12
13 This file is distributed in the hope that it will be useful,
14 but WITHOUT ANY WARRANTY; without even the implied warranty of
15 MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
16 GNU Lesser General Public License for more details.
17
18 You should have received a copy of the GNU Lesser General Public License
19 along with this program. If not, see <https://www.gnu.org/licenses/>. */
20
21 /* Tree search for red/black trees.
22 The algorithm for adding nodes is taken from one of the many "Algorithms"
23 books by Robert Sedgewick, although the implementation differs.
24 The algorithm for deleting nodes can probably be found in a book named
25 "Introduction to Algorithms" by Cormen/Leiserson/Rivest. At least that's
26 the book that my professor took most algorithms from during the "Data
27 Structures" course...
28
29 Totally public domain. */
30
31 /* Red/black trees are binary trees in which the edges are colored either red
32 or black. They have the following properties:
33 1. The number of black edges on every path from the root to a leaf is
34 constant.
35 2. No two red edges are adjacent.
36 Therefore there is an upper bound on the length of every path, it's
37 O(log n) where n is the number of nodes in the tree. No path can be longer
38 than 1+2*P where P is the length of the shortest path in the tree.
39 Useful for the implementation:
40 3. If one of the children of a node is NULL, then the other one is red
41 (if it exists).
42
43 In the implementation, not the edges are colored, but the nodes. The color
44 interpreted as the color of the edge leading to this node. The color is
45 meaningless for the root node, but we color the root node black for
46 convenience. All added nodes are red initially.
47
48 Adding to a red/black tree is rather easy. The right place is searched
49 with a usual binary tree search. Additionally, whenever a node N is
50 reached that has two red successors, the successors are colored black and
51 the node itself colored red. This moves red edges up the tree where they
52 pose less of a problem once we get to really insert the new node. Changing
53 N's color to red may violate rule 2, however, so rotations may become
54 necessary to restore the invariants. Adding a new red leaf may violate
55 the same rule, so afterwards an additional check is run and the tree
56 possibly rotated.
57
58 Deleting is hairy. There are mainly two nodes involved: the node to be
59 deleted (n1), and another node that is to be unchained from the tree (n2).
60 If n1 has a successor (the node with a smallest key that is larger than
61 n1), then the successor becomes n2 and its contents are copied into n1,
62 otherwise n1 becomes n2.
63 Unchaining a node may violate rule 1: if n2 is black, one subtree is
64 missing one black edge afterwards. The algorithm must try to move this
65 error upwards towards the root, so that the subtree that does not have
66 enough black edges becomes the whole tree. Once that happens, the error
67 has disappeared. It may not be necessary to go all the way up, since it
68 is possible that rotations and recoloring can fix the error before that.
69
70 Although the deletion algorithm must walk upwards through the tree, we
71 do not store parent pointers in the nodes. Instead, delete allocates a
72 small array of parent pointers and fills it while descending the tree.
73 Since we know that the length of a path is O(log n), where n is the number
74 of nodes, this is likely to use less memory. */
75
76 /* Tree rotations look like this:
77 A C
78 / \ / \
79 B C A G
80 / \ / \ --> / \
81 D E F G B F
82 / \
83 D E
84
85 In this case, A has been rotated left. This preserves the ordering of the
86 binary tree. */
87
88 /* Don't use __attribute__ __nonnull__ in this compilation unit. Otherwise gcc
89 optimizes away the rootp == NULL tests below. */
90 #define _GL_ARG_NONNULL(params)
91
92 #include <config.h>
93
94 /* Specification. */
95 #include <search.h>
96
97 #include <stdlib.h>
98
99 typedef int (*__compar_fn_t) (const void *, const void *);
100 typedef void (*__action_fn_t) (const void *, VISIT, int);
101
102 #ifndef weak_alias
103 # define __tsearch tsearch
104 # define __tfind tfind
105 # define __tdelete tdelete
106 # define __twalk twalk
107 #endif
108
109 #ifndef internal_function
110 /* Inside GNU libc we mark some function in a special way. In other
111 environments simply ignore the marking. */
112 # define internal_function
113 #endif
114
115 typedef struct node_t
116 {
117 /* Callers expect this to be the first element in the structure - do not
118 move! */
119 const void *key;
120 struct node_t *left;
121 struct node_t *right;
122 unsigned int red:1;
123 } *node;
124 typedef const struct node_t *const_node;
125
126 #undef DEBUGGING
127
128 #ifdef DEBUGGING
129
130 /* Routines to check tree invariants. */
131
132 #include <assert.h>
133
134 #define CHECK_TREE(a) check_tree(a)
135
136 static void
137 check_tree_recurse (node p, int d_sofar, int d_total)
/* ![[next]](../icons/right.png)
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*/
138 {
139 if (p == NULL)
140 {
141 assert (d_sofar == d_total);
142 return;
143 }
144
145 check_tree_recurse (p->left, d_sofar + (p->left && !p->left->red), d_total);
146 check_tree_recurse (p->right, d_sofar + (p->right && !p->right->red), d_total);
147 if (p->left)
148 assert (!(p->left->red && p->red));
149 if (p->right)
150 assert (!(p->right->red && p->red));
151 }
152
153 static void
154 check_tree (node root)
/* ![[previous]](../icons/left.png)
*/
155 {
156 int cnt = 0;
157 node p;
158 if (root == NULL)
159 return;
160 root->red = 0;
161 for (p = root->left; p; p = p->left)
162 cnt += !p->red;
163 check_tree_recurse (root, 0, cnt);
164 }
165
166
167 #else
168
169 #define CHECK_TREE(a)
170
171 #endif
172
173 #if GNULIB_defined_tsearch
174
175 /* Possibly "split" a node with two red successors, and/or fix up two red
176 edges in a row. ROOTP is a pointer to the lowest node we visited, PARENTP
177 and GPARENTP pointers to its parent/grandparent. P_R and GP_R contain the
178 comparison values that determined which way was taken in the tree to reach
179 ROOTP. MODE is 1 if we need not do the split, but must check for two red
180 edges between GPARENTP and ROOTP. */
181 static void
182 maybe_split_for_insert (node *rootp, node *parentp, node *gparentp,
/* */
183 int p_r, int gp_r, int mode)
184 {
185 node root = *rootp;
186 node *rp, *lp;
187 rp = &(*rootp)->right;
188 lp = &(*rootp)->left;
189
190 /* See if we have to split this node (both successors red). */
191 if (mode == 1
192 || ((*rp) != NULL && (*lp) != NULL && (*rp)->red && (*lp)->red))
193 {
194 /* This node becomes red, its successors black. */
195 root->red = 1;
196 if (*rp)
197 (*rp)->red = 0;
198 if (*lp)
199 (*lp)->red = 0;
200
201 /* If the parent of this node is also red, we have to do
202 rotations. */
203 if (parentp != NULL && (*parentp)->red)
204 {
205 node gp = *gparentp;
206 node p = *parentp;
207 /* There are two main cases:
208 1. The edge types (left or right) of the two red edges differ.
209 2. Both red edges are of the same type.
210 There exist two symmetries of each case, so there is a total of
211 4 cases. */
212 if ((p_r > 0) != (gp_r > 0))
213 {
214 /* Put the child at the top of the tree, with its parent
215 and grandparent as successors. */
216 p->red = 1;
217 gp->red = 1;
218 root->red = 0;
219 if (p_r < 0)
220 {
221 /* Child is left of parent. */
222 p->left = *rp;
223 *rp = p;
224 gp->right = *lp;
225 *lp = gp;
226 }
227 else
228 {
229 /* Child is right of parent. */
230 p->right = *lp;
231 *lp = p;
232 gp->left = *rp;
233 *rp = gp;
234 }
235 *gparentp = root;
236 }
237 else
238 {
239 *gparentp = *parentp;
240 /* Parent becomes the top of the tree, grandparent and
241 child are its successors. */
242 p->red = 0;
243 gp->red = 1;
244 if (p_r < 0)
245 {
246 /* Left edges. */
247 gp->left = p->right;
248 p->right = gp;
249 }
250 else
251 {
252 /* Right edges. */
253 gp->right = p->left;
254 p->left = gp;
255 }
256 }
257 }
258 }
259 }
260
261 /* Find or insert datum into search tree.
262 KEY is the key to be located, ROOTP is the address of tree root,
263 COMPAR the ordering function. */
264 void *
265 __tsearch (const void *key, void **vrootp, __compar_fn_t compar)
/* */
266 {
267 node q;
268 node *parentp = NULL, *gparentp = NULL;
269 node *rootp = (node *) vrootp;
270 node *nextp;
271 int r = 0, p_r = 0, gp_r = 0; /* No they might not, Mr Compiler. */
272
273 if (rootp == NULL)
274 return NULL;
275
276 /* This saves some additional tests below. */
277 if (*rootp != NULL)
278 (*rootp)->red = 0;
279
280 CHECK_TREE (*rootp);
281
282 nextp = rootp;
283 while (*nextp != NULL)
284 {
285 node root = *rootp;
286 r = (*compar) (key, root->key);
287 if (r == 0)
288 return root;
289
290 maybe_split_for_insert (rootp, parentp, gparentp, p_r, gp_r, 0);
291 /* If that did any rotations, parentp and gparentp are now garbage.
292 That doesn't matter, because the values they contain are never
293 used again in that case. */
294
295 nextp = r < 0 ? &root->left : &root->right;
296 if (*nextp == NULL)
297 break;
298
299 gparentp = parentp;
300 parentp = rootp;
301 rootp = nextp;
302
303 gp_r = p_r;
304 p_r = r;
305 }
306
307 q = (struct node_t *) malloc (sizeof (struct node_t));
308 if (q != NULL)
309 {
310 *nextp = q; /* link new node to old */
311 q->key = key; /* initialize new node */
312 q->red = 1;
313 q->left = q->right = NULL;
314
315 if (nextp != rootp)
316 /* There may be two red edges in a row now, which we must avoid by
317 rotating the tree. */
318 maybe_split_for_insert (nextp, rootp, parentp, r, p_r, 1);
319 }
320
321 return q;
322 }
323 #ifdef weak_alias
324 weak_alias (__tsearch, tsearch)
/* */
325 #endif
326
327
328 /* Find datum in search tree.
329 KEY is the key to be located, ROOTP is the address of tree root,
330 COMPAR the ordering function. */
331 void *
332 __tfind (const void *key, void *const *vrootp, __compar_fn_t compar)
333 {
334 node *rootp = (node *) vrootp;
335
336 if (rootp == NULL)
337 return NULL;
338
339 CHECK_TREE (*rootp);
340
341 while (*rootp != NULL)
342 {
343 node root = *rootp;
344 int r;
345
346 r = (*compar) (key, root->key);
347 if (r == 0)
348 return root;
349
350 rootp = r < 0 ? &root->left : &root->right;
351 }
352 return NULL;
353 }
354 #ifdef weak_alias
355 weak_alias (__tfind, tfind)
/* */
356 #endif
357
358
359 /* Delete node with given key.
360 KEY is the key to be deleted, ROOTP is the address of the root of tree,
361 COMPAR the comparison function. */
362 void *
363 __tdelete (const void *key, void **vrootp, __compar_fn_t compar)
364 {
365 node p, q, r, retval;
366 int cmp;
367 node *rootp = (node *) vrootp;
368 node root, unchained;
369 /* Stack of nodes so we remember the parents without recursion. It's
370 _very_ unlikely that there are paths longer than 40 nodes. The tree
371 would need to have around 250.000 nodes. */
372 int stacksize = 100;
373 int sp = 0;
374 node *nodestack[100];
375
376 if (rootp == NULL)
377 return NULL;
378 p = *rootp;
379 if (p == NULL)
380 return NULL;
381
382 CHECK_TREE (p);
383
384 while ((cmp = (*compar) (key, (*rootp)->key)) != 0)
385 {
386 if (sp == stacksize)
387 abort ();
388
389 nodestack[sp++] = rootp;
390 p = *rootp;
391 rootp = ((cmp < 0)
392 ? &(*rootp)->left
393 : &(*rootp)->right);
394 if (*rootp == NULL)
395 return NULL;
396 }
397
398 /* This is bogus if the node to be deleted is the root... this routine
399 really should return an integer with 0 for success, -1 for failure
400 and errno = ESRCH or something. */
401 retval = p;
402
403 /* We don't unchain the node we want to delete. Instead, we overwrite
404 it with its successor and unchain the successor. If there is no
405 successor, we really unchain the node to be deleted. */
406
407 root = *rootp;
408
409 r = root->right;
410 q = root->left;
411
412 if (q == NULL || r == NULL)
413 unchained = root;
414 else
415 {
416 node *parent = rootp, *up = &root->right;
417 for (;;)
418 {
419 if (sp == stacksize)
420 abort ();
421 nodestack[sp++] = parent;
422 parent = up;
423 if ((*up)->left == NULL)
424 break;
425 up = &(*up)->left;
426 }
427 unchained = *up;
428 }
429
430 /* We know that either the left or right successor of UNCHAINED is NULL.
431 R becomes the other one, it is chained into the parent of UNCHAINED. */
432 r = unchained->left;
433 if (r == NULL)
434 r = unchained->right;
435 if (sp == 0)
436 *rootp = r;
437 else
438 {
439 q = *nodestack[sp-1];
440 if (unchained == q->right)
441 q->right = r;
442 else
443 q->left = r;
444 }
445
446 if (unchained != root)
447 root->key = unchained->key;
448 if (!unchained->red)
449 {
450 /* Now we lost a black edge, which means that the number of black
451 edges on every path is no longer constant. We must balance the
452 tree. */
453 /* NODESTACK now contains all parents of R. R is likely to be NULL
454 in the first iteration. */
455 /* NULL nodes are considered black throughout - this is necessary for
456 correctness. */
457 while (sp > 0 && (r == NULL || !r->red))
458 {
459 node *pp = nodestack[sp - 1];
460 p = *pp;
461 /* Two symmetric cases. */
462 if (r == p->left)
463 {
464 /* Q is R's brother, P is R's parent. The subtree with root
465 R has one black edge less than the subtree with root Q. */
466 q = p->right;
467 if (q->red)
468 {
469 /* If Q is red, we know that P is black. We rotate P left
470 so that Q becomes the top node in the tree, with P below
471 it. P is colored red, Q is colored black.
472 This action does not change the black edge count for any
473 leaf in the tree, but we will be able to recognize one
474 of the following situations, which all require that Q
475 is black. */
476 q->red = 0;
477 p->red = 1;
478 /* Left rotate p. */
479 p->right = q->left;
480 q->left = p;
481 *pp = q;
482 /* Make sure pp is right if the case below tries to use
483 it. */
484 nodestack[sp++] = pp = &q->left;
485 q = p->right;
486 }
487 /* We know that Q can't be NULL here. We also know that Q is
488 black. */
489 if ((q->left == NULL || !q->left->red)
490 && (q->right == NULL || !q->right->red))
491 {
492 /* Q has two black successors. We can simply color Q red.
493 The whole subtree with root P is now missing one black
494 edge. Note that this action can temporarily make the
495 tree invalid (if P is red). But we will exit the loop
496 in that case and set P black, which both makes the tree
497 valid and also makes the black edge count come out
498 right. If P is black, we are at least one step closer
499 to the root and we'll try again the next iteration. */
500 q->red = 1;
501 r = p;
502 }
503 else
504 {
505 /* Q is black, one of Q's successors is red. We can
506 repair the tree with one operation and will exit the
507 loop afterwards. */
508 if (q->right == NULL || !q->right->red)
509 {
510 /* The left one is red. We perform the same action as
511 in maybe_split_for_insert where two red edges are
512 adjacent but point in different directions:
513 Q's left successor (let's call it Q2) becomes the
514 top of the subtree we are looking at, its parent (Q)
515 and grandparent (P) become its successors. The former
516 successors of Q2 are placed below P and Q.
517 P becomes black, and Q2 gets the color that P had.
518 This changes the black edge count only for node R and
519 its successors. */
520 node q2 = q->left;
521 q2->red = p->red;
522 p->right = q2->left;
523 q->left = q2->right;
524 q2->right = q;
525 q2->left = p;
526 *pp = q2;
527 p->red = 0;
528 }
529 else
530 {
531 /* It's the right one. Rotate P left. P becomes black,
532 and Q gets the color that P had. Q's right successor
533 also becomes black. This changes the black edge
534 count only for node R and its successors. */
535 q->red = p->red;
536 p->red = 0;
537
538 q->right->red = 0;
539
540 /* left rotate p */
541 p->right = q->left;
542 q->left = p;
543 *pp = q;
544 }
545
546 /* We're done. */
547 sp = 1;
548 r = NULL;
549 }
550 }
551 else
552 {
553 /* Comments: see above. */
554 q = p->left;
555 if (q->red)
556 {
557 q->red = 0;
558 p->red = 1;
559 p->left = q->right;
560 q->right = p;
561 *pp = q;
562 nodestack[sp++] = pp = &q->right;
563 q = p->left;
564 }
565 if ((q->right == NULL || !q->right->red)
566 && (q->left == NULL || !q->left->red))
567 {
568 q->red = 1;
569 r = p;
570 }
571 else
572 {
573 if (q->left == NULL || !q->left->red)
574 {
575 node q2 = q->right;
576 q2->red = p->red;
577 p->left = q2->right;
578 q->right = q2->left;
579 q2->left = q;
580 q2->right = p;
581 *pp = q2;
582 p->red = 0;
583 }
584 else
585 {
586 q->red = p->red;
587 p->red = 0;
588 q->left->red = 0;
589 p->left = q->right;
590 q->right = p;
591 *pp = q;
592 }
593 sp = 1;
594 r = NULL;
595 }
596 }
597 --sp;
598 }
599 if (r != NULL)
600 r->red = 0;
601 }
602
603 free (unchained);
604 return retval;
605 }
606 #ifdef weak_alias
607 weak_alias (__tdelete, tdelete)
/* */
608 #endif
609
610 #endif /* GNULIB_defined_tsearch */
611
612
613 #if GNULIB_defined_twalk
614
615 /* Walk the nodes of a tree.
616 ROOT is the root of the tree to be walked, ACTION the function to be
617 called at each node. LEVEL is the level of ROOT in the whole tree. */
618 static void
619 internal_function
620 trecurse (const void *vroot, __action_fn_t action, int level)
621 {
622 const_node root = (const_node) vroot;
623
624 if (root->left == NULL && root->right == NULL)
625 (*action) (root, leaf, level);
626 else
627 {
628 (*action) (root, preorder, level);
629 if (root->left != NULL)
630 trecurse (root->left, action, level + 1);
631 (*action) (root, postorder, level);
632 if (root->right != NULL)
633 trecurse (root->right, action, level + 1);
634 (*action) (root, endorder, level);
635 }
636 }
637
638
639 /* Walk the nodes of a tree.
640 ROOT is the root of the tree to be walked, ACTION the function to be
641 called at each node. */
642 void
643 __twalk (const void *vroot, __action_fn_t action)
/* */
644 {
645 const_node root = (const_node) vroot;
646
647 CHECK_TREE (root);
648
649 if (root != NULL && action != NULL)
650 trecurse (root, action, 0);
651 }
652 #ifdef weak_alias
653 weak_alias (__twalk, twalk)
/* */
654 #endif
655
656 #endif /* GNULIB_defined_twalk */
657
658
659 #ifdef _LIBC
660
661 /* The standardized functions miss an important functionality: the
662 tree cannot be removed easily. We provide a function to do this. */
663 static void
664 internal_function
665 tdestroy_recurse (node root, __free_fn_t freefct)
666 {
667 if (root->left != NULL)
668 tdestroy_recurse (root->left, freefct);
669 if (root->right != NULL)
670 tdestroy_recurse (root->right, freefct);
671 (*freefct) ((void *) root->key);
672 /* Free the node itself. */
673 free (root);
674 }
675
676 void
677 __tdestroy (void *vroot, __free_fn_t freefct)
/* ![[last]](../icons/n_last.png)
*/
678 {
679 node root = (node) vroot;
680
681 CHECK_TREE (root);
682
683 if (root != NULL)
684 tdestroy_recurse (root, freefct);
685 }
686 weak_alias (__tdestroy, tdestroy)
687
688 #endif /* _LIBC */
![[bottom]](../icons/n_bottom.png){kind=icon}
*/