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1 // SPDX-License-Identifier: GPL-2.0-or-later
2 /*
3 Red Black Trees
4 (C) 1999 Andrea Arcangeli <andrea@suse.de>
5 (C) 2002 David Woodhouse <dwmw2@infradead.org>
6 (C) 2012 Michel Lespinasse <walken@google.com>
7
8
9 linux/lib/rbtree.c
10 */
11
12 #include <linux/rbtree_augmented.h>
13 #include <linux/export.h>
14
15 /*
16 * red-black trees properties: http://en.wikipedia.org/wiki/Rbtree
17 *
18 * 1) A node is either red or black
19 * 2) The root is black
20 * 3) All leaves (NULL) are black
21 * 4) Both children of every red node are black
22 * 5) Every simple path from root to leaves contains the same number
23 * of black nodes.
24 *
25 * 4 and 5 give the O(log n) guarantee, since 4 implies you cannot have two
26 * consecutive red nodes in a path and every red node is therefore followed by
27 * a black. So if B is the number of black nodes on every simple path (as per
28 * 5), then the longest possible path due to 4 is 2B.
29 *
30 * We shall indicate color with case, where black nodes are uppercase and red
31 * nodes will be lowercase. Unknown color nodes shall be drawn as red within
32 * parentheses and have some accompanying text comment.
33 */
34
35 /*
36 * Notes on lockless lookups:
37 *
38 * All stores to the tree structure (rb_left and rb_right) must be done using
39 * WRITE_ONCE(). And we must not inadvertently cause (temporary) loops in the
40 * tree structure as seen in program order.
41 *
42 * These two requirements will allow lockless iteration of the tree -- not
43 * correct iteration mind you, tree rotations are not atomic so a lookup might
44 * miss entire subtrees.
45 *
46 * But they do guarantee that any such traversal will only see valid elements
47 * and that it will indeed complete -- does not get stuck in a loop.
48 *
49 * It also guarantees that if the lookup returns an element it is the 'correct'
50 * one. But not returning an element does _NOT_ mean it's not present.
51 *
52 * NOTE:
53 *
54 * Stores to __rb_parent_color are not important for simple lookups so those
55 * are left undone as of now. Nor did I check for loops involving parent
56 * pointers.
57 */
58
59 static inline void rb_set_black(struct rb_node *rb)
60 {
61 rb->__rb_parent_color |= RB_BLACK;
62 }
63
64 static inline struct rb_node *rb_red_parent(struct rb_node *red)
65 {
66 return (struct rb_node *)red->__rb_parent_color;
67 }
68
69 /*
70 * Helper function for rotations:
71 * - old's parent and color get assigned to new
72 * - old gets assigned new as a parent and 'color' as a color.
73 */
74 static inline void
75 __rb_rotate_set_parents(struct rb_node *old, struct rb_node *new,
76 struct rb_root *root, int color)
77 {
78 struct rb_node *parent = rb_parent(old);
79 new->__rb_parent_color = old->__rb_parent_color;
80 rb_set_parent_color(old, new, color);
81 __rb_change_child(old, new, parent, root);
82 }
83
84 static __always_inline void
85 __rb_insert(struct rb_node *node, struct rb_root *root,
86 bool newleft, struct rb_node **leftmost,
87 void (*augment_rotate)(struct rb_node *old, struct rb_node *new))
88 {
89 struct rb_node *parent = rb_red_parent(node), *gparent, *tmp;
90
91 if (newleft)
92 *leftmost = node;
93
94 while (true) {
95 /*
96 * Loop invariant: node is red.
97 */
98 if (unlikely(!parent)) {
99 /*
100 * The inserted node is root. Either this is the
101 * first node, or we recursed at Case 1 below and
102 * are no longer violating 4).
103 */
104 rb_set_parent_color(node, NULL, RB_BLACK);
105 break;
106 }
107
108 /*
109 * If there is a black parent, we are done.
110 * Otherwise, take some corrective action as,
111 * per 4), we don't want a red root or two
112 * consecutive red nodes.
113 */
114 if(rb_is_black(parent))
115 break;
116
117 gparent = rb_red_parent(parent);
118
119 tmp = gparent->rb_right;
120 if (parent != tmp) { /* parent == gparent->rb_left */
121 if (tmp && rb_is_red(tmp)) {
122 /*
123 * Case 1 - node's uncle is red (color flips).
124 *
125 * G g
126 * / \ / \
127 * p u --> P U
128 * / /
129 * n n
130 *
131 * However, since g's parent might be red, and
132 * 4) does not allow this, we need to recurse
133 * at g.
134 */
135 rb_set_parent_color(tmp, gparent, RB_BLACK);
136 rb_set_parent_color(parent, gparent, RB_BLACK);
137 node = gparent;
138 parent = rb_parent(node);
139 rb_set_parent_color(node, parent, RB_RED);
140 continue;
141 }
142
143 tmp = parent->rb_right;
144 if (node == tmp) {
145 /*
146 * Case 2 - node's uncle is black and node is
147 * the parent's right child (left rotate at parent).
148 *
149 * G G
150 * / \ / \
151 * p U --> n U
152 * \ /
153 * n p
154 *
155 * This still leaves us in violation of 4), the
156 * continuation into Case 3 will fix that.
157 */
158 tmp = node->rb_left;
159 WRITE_ONCE(parent->rb_right, tmp);
160 WRITE_ONCE(node->rb_left, parent);
161 if (tmp)
162 rb_set_parent_color(tmp, parent,
163 RB_BLACK);
164 rb_set_parent_color(parent, node, RB_RED);
165 augment_rotate(parent, node);
166 parent = node;
167 tmp = node->rb_right;
168 }
169
170 /*
171 * Case 3 - node's uncle is black and node is
172 * the parent's left child (right rotate at gparent).
173 *
174 * G P
175 * / \ / \
176 * p U --> n g
177 * / \
178 * n U
179 */
180 WRITE_ONCE(gparent->rb_left, tmp); /* == parent->rb_right */
181 WRITE_ONCE(parent->rb_right, gparent);
182 if (tmp)
183 rb_set_parent_color(tmp, gparent, RB_BLACK);
184 __rb_rotate_set_parents(gparent, parent, root, RB_RED);
185 augment_rotate(gparent, parent);
186 break;
187 } else {
188 tmp = gparent->rb_left;
189 if (tmp && rb_is_red(tmp)) {
190 /* Case 1 - color flips */
191 rb_set_parent_color(tmp, gparent, RB_BLACK);
192 rb_set_parent_color(parent, gparent, RB_BLACK);
193 node = gparent;
194 parent = rb_parent(node);
195 rb_set_parent_color(node, parent, RB_RED);
196 continue;
197 }
198
199 tmp = parent->rb_left;
200 if (node == tmp) {
201 /* Case 2 - right rotate at parent */
202 tmp = node->rb_right;
203 WRITE_ONCE(parent->rb_left, tmp);
204 WRITE_ONCE(node->rb_right, parent);
205 if (tmp)
206 rb_set_parent_color(tmp, parent,
207 RB_BLACK);
208 rb_set_parent_color(parent, node, RB_RED);
209 augment_rotate(parent, node);
210 parent = node;
211 tmp = node->rb_left;
212 }
213
214 /* Case 3 - left rotate at gparent */
215 WRITE_ONCE(gparent->rb_right, tmp); /* == parent->rb_left */
216 WRITE_ONCE(parent->rb_left, gparent);
217 if (tmp)
218 rb_set_parent_color(tmp, gparent, RB_BLACK);
219 __rb_rotate_set_parents(gparent, parent, root, RB_RED);
220 augment_rotate(gparent, parent);
221 break;
222 }
223 }
224 }
225
226 /*
227 * Inline version for rb_erase() use - we want to be able to inline
228 * and eliminate the dummy_rotate callback there
229 */
230 static __always_inline void
231 ____rb_erase_color(struct rb_node *parent, struct rb_root *root,
232 void (*augment_rotate)(struct rb_node *old, struct rb_node *new))
233 {
234 struct rb_node *node = NULL, *sibling, *tmp1, *tmp2;
235
236 while (true) {
237 /*
238 * Loop invariants:
239 * - node is black (or NULL on first iteration)
240 * - node is not the root (parent is not NULL)
241 * - All leaf paths going through parent and node have a
242 * black node count that is 1 lower than other leaf paths.
243 */
244 sibling = parent->rb_right;
245 if (node != sibling) { /* node == parent->rb_left */
246 if (rb_is_red(sibling)) {
247 /*
248 * Case 1 - left rotate at parent
249 *
250 * P S
251 * / \ / \
252 * N s --> p Sr
253 * / \ / \
254 * Sl Sr N Sl
255 */
256 tmp1 = sibling->rb_left;
257 WRITE_ONCE(parent->rb_right, tmp1);
258 WRITE_ONCE(sibling->rb_left, parent);
259 rb_set_parent_color(tmp1, parent, RB_BLACK);
260 __rb_rotate_set_parents(parent, sibling, root,
261 RB_RED);
262 augment_rotate(parent, sibling);
263 sibling = tmp1;
264 }
265 tmp1 = sibling->rb_right;
266 if (!tmp1 || rb_is_black(tmp1)) {
267 tmp2 = sibling->rb_left;
268 if (!tmp2 || rb_is_black(tmp2)) {
269 /*
270 * Case 2 - sibling color flip
271 * (p could be either color here)
272 *
273 * (p) (p)
274 * / \ / \
275 * N S --> N s
276 * / \ / \
277 * Sl Sr Sl Sr
278 *
279 * This leaves us violating 5) which
280 * can be fixed by flipping p to black
281 * if it was red, or by recursing at p.
282 * p is red when coming from Case 1.
283 */
284 rb_set_parent_color(sibling, parent,
285 RB_RED);
286 if (rb_is_red(parent))
287 rb_set_black(parent);
288 else {
289 node = parent;
290 parent = rb_parent(node);
291 if (parent)
292 continue;
293 }
294 break;
295 }
296 /*
297 * Case 3 - right rotate at sibling
298 * (p could be either color here)
299 *
300 * (p) (p)
301 * / \ / \
302 * N S --> N sl
303 * / \ \
304 * sl Sr S
305 * \
306 * Sr
307 *
308 * Note: p might be red, and then both
309 * p and sl are red after rotation(which
310 * breaks property 4). This is fixed in
311 * Case 4 (in __rb_rotate_set_parents()
312 * which set sl the color of p
313 * and set p RB_BLACK)
314 *
315 * (p) (sl)
316 * / \ / \
317 * N sl --> P S
318 * \ / \
319 * S N Sr
320 * \
321 * Sr
322 */
323 tmp1 = tmp2->rb_right;
324 WRITE_ONCE(sibling->rb_left, tmp1);
325 WRITE_ONCE(tmp2->rb_right, sibling);
326 WRITE_ONCE(parent->rb_right, tmp2);
327 if (tmp1)
328 rb_set_parent_color(tmp1, sibling,
329 RB_BLACK);
330 augment_rotate(sibling, tmp2);
331 tmp1 = sibling;
332 sibling = tmp2;
333 }
334 /*
335 * Case 4 - left rotate at parent + color flips
336 * (p and sl could be either color here.
337 * After rotation, p becomes black, s acquires
338 * p's color, and sl keeps its color)
339 *
340 * (p) (s)
341 * / \ / \
342 * N S --> P Sr
343 * / \ / \
344 * (sl) sr N (sl)
345 */
346 tmp2 = sibling->rb_left;
347 WRITE_ONCE(parent->rb_right, tmp2);
348 WRITE_ONCE(sibling->rb_left, parent);
349 rb_set_parent_color(tmp1, sibling, RB_BLACK);
350 if (tmp2)
351 rb_set_parent(tmp2, parent);
352 __rb_rotate_set_parents(parent, sibling, root,
353 RB_BLACK);
354 augment_rotate(parent, sibling);
355 break;
356 } else {
357 sibling = parent->rb_left;
358 if (rb_is_red(sibling)) {
359 /* Case 1 - right rotate at parent */
360 tmp1 = sibling->rb_right;
361 WRITE_ONCE(parent->rb_left, tmp1);
362 WRITE_ONCE(sibling->rb_right, parent);
363 rb_set_parent_color(tmp1, parent, RB_BLACK);
364 __rb_rotate_set_parents(parent, sibling, root,
365 RB_RED);
366 augment_rotate(parent, sibling);
367 sibling = tmp1;
368 }
369 tmp1 = sibling->rb_left;
370 if (!tmp1 || rb_is_black(tmp1)) {
371 tmp2 = sibling->rb_right;
372 if (!tmp2 || rb_is_black(tmp2)) {
373 /* Case 2 - sibling color flip */
374 rb_set_parent_color(sibling, parent,
375 RB_RED);
376 if (rb_is_red(parent))
377 rb_set_black(parent);
378 else {
379 node = parent;
380 parent = rb_parent(node);
381 if (parent)
382 continue;
383 }
384 break;
385 }
386 /* Case 3 - left rotate at sibling */
387 tmp1 = tmp2->rb_left;
388 WRITE_ONCE(sibling->rb_right, tmp1);
389 WRITE_ONCE(tmp2->rb_left, sibling);
390 WRITE_ONCE(parent->rb_left, tmp2);
391 if (tmp1)
392 rb_set_parent_color(tmp1, sibling,
393 RB_BLACK);
394 augment_rotate(sibling, tmp2);
395 tmp1 = sibling;
396 sibling = tmp2;
397 }
398 /* Case 4 - right rotate at parent + color flips */
399 tmp2 = sibling->rb_right;
400 WRITE_ONCE(parent->rb_left, tmp2);
401 WRITE_ONCE(sibling->rb_right, parent);
402 rb_set_parent_color(tmp1, sibling, RB_BLACK);
403 if (tmp2)
404 rb_set_parent(tmp2, parent);
405 __rb_rotate_set_parents(parent, sibling, root,
406 RB_BLACK);
407 augment_rotate(parent, sibling);
408 break;
409 }
410 }
411 }
412
413 /* Non-inline version for rb_erase_augmented() use */
414 void __rb_erase_color(struct rb_node *parent, struct rb_root *root,
415 void (*augment_rotate)(struct rb_node *old, struct rb_node *new))
416 {
417 ____rb_erase_color(parent, root, augment_rotate);
418 }
419
420 /*
421 * Non-augmented rbtree manipulation functions.
422 *
423 * We use dummy augmented callbacks here, and have the compiler optimize them
424 * out of the rb_insert_color() and rb_erase() function definitions.
425 */
426
427 static inline void dummy_propagate(struct rb_node *node, struct rb_node *stop) {}
428 static inline void dummy_copy(struct rb_node *old, struct rb_node *new) {}
429 static inline void dummy_rotate(struct rb_node *old, struct rb_node *new) {}
430
431 static const struct rb_augment_callbacks dummy_callbacks = {
432 .propagate = dummy_propagate,
433 .copy = dummy_copy,
434 .rotate = dummy_rotate
435 };
436
437 void rb_insert_color(struct rb_node *node, struct rb_root *root)
438 {
439 __rb_insert(node, root, false, NULL, dummy_rotate);
440 }
441
442 void rb_erase(struct rb_node *node, struct rb_root *root)
443 {
444 struct rb_node *rebalance;
445 rebalance = __rb_erase_augmented(node, root,
446 NULL, &dummy_callbacks);
447 if (rebalance)
448 ____rb_erase_color(rebalance, root, dummy_rotate);
449 }
450
451 void rb_insert_color_cached(struct rb_node *node,
452 struct rb_root_cached *root, bool leftmost)
453 {
454 __rb_insert(node, &root->rb_root, leftmost,
455 &root->rb_leftmost, dummy_rotate);
456 }
457
458 void rb_erase_cached(struct rb_node *node, struct rb_root_cached *root)
459 {
460 struct rb_node *rebalance;
461 rebalance = __rb_erase_augmented(node, &root->rb_root,
462 &root->rb_leftmost, &dummy_callbacks);
463 if (rebalance)
464 ____rb_erase_color(rebalance, &root->rb_root, dummy_rotate);
465 }
466
467 /*
468 * Augmented rbtree manipulation functions.
469 *
470 * This instantiates the same __always_inline functions as in the non-augmented
471 * case, but this time with user-defined callbacks.
472 */
473
474 void __rb_insert_augmented(struct rb_node *node, struct rb_root *root,
475 bool newleft, struct rb_node **leftmost,
476 void (*augment_rotate)(struct rb_node *old, struct rb_node *new))
477 {
478 __rb_insert(node, root, newleft, leftmost, augment_rotate);
479 }
480
481 /*
482 * This function returns the first node (in sort order) of the tree.
483 */
484 struct rb_node *rb_first(const struct rb_root *root)
485 {
486 struct rb_node *n;
487
488 n = root->rb_node;
489 if (!n)
490 return NULL;
491 while (n->rb_left)
492 n = n->rb_left;
493 return n;
494 }
495
496 struct rb_node *rb_last(const struct rb_root *root)
497 {
498 struct rb_node *n;
499
500 n = root->rb_node;
501 if (!n)
502 return NULL;
503 while (n->rb_right)
504 n = n->rb_right;
505 return n;
506 }
507
508 struct rb_node *rb_next(const struct rb_node *node)
509 {
510 struct rb_node *parent;
511
512 if (RB_EMPTY_NODE(node))
513 return NULL;
514
515 /*
516 * If we have a right-hand child, go down and then left as far
517 * as we can.
518 */
519 if (node->rb_right) {
520 node = node->rb_right;
521 while (node->rb_left)
522 node=node->rb_left;
523 return (struct rb_node *)node;
524 }
525
526 /*
527 * No right-hand children. Everything down and left is smaller than us,
528 * so any 'next' node must be in the general direction of our parent.
529 * Go up the tree; any time the ancestor is a right-hand child of its
530 * parent, keep going up. First time it's a left-hand child of its
531 * parent, said parent is our 'next' node.
532 */
533 while ((parent = rb_parent(node)) && node == parent->rb_right)
534 node = parent;
535
536 return parent;
537 }
538
539 struct rb_node *rb_prev(const struct rb_node *node)
540 {
541 struct rb_node *parent;
542
543 if (RB_EMPTY_NODE(node))
544 return NULL;
545
546 /*
547 * If we have a left-hand child, go down and then right as far
548 * as we can.
549 */
550 if (node->rb_left) {
551 node = node->rb_left;
552 while (node->rb_right)
553 node=node->rb_right;
554 return (struct rb_node *)node;
555 }
556
557 /*
558 * No left-hand children. Go up till we find an ancestor which
559 * is a right-hand child of its parent.
560 */
561 while ((parent = rb_parent(node)) && node == parent->rb_left)
562 node = parent;
563
564 return parent;
565 }
566
567 void rb_replace_node(struct rb_node *victim, struct rb_node *new,
568 struct rb_root *root)
569 {
570 struct rb_node *parent = rb_parent(victim);
571
572 /* Copy the pointers/colour from the victim to the replacement */
573 *new = *victim;
574
575 /* Set the surrounding nodes to point to the replacement */
576 if (victim->rb_left)
577 rb_set_parent(victim->rb_left, new);
578 if (victim->rb_right)
579 rb_set_parent(victim->rb_right, new);
580 __rb_change_child(victim, new, parent, root);
581 }
582
583 void rb_replace_node_cached(struct rb_node *victim, struct rb_node *new,
584 struct rb_root_cached *root)
585 {
586 rb_replace_node(victim, new, &root->rb_root);
587
588 if (root->rb_leftmost == victim)
589 root->rb_leftmost = new;
590 }
591
592 static struct rb_node *rb_left_deepest_node(const struct rb_node *node)
593 {
594 for (;;) {
595 if (node->rb_left)
596 node = node->rb_left;
597 else if (node->rb_right)
598 node = node->rb_right;
599 else
600 return (struct rb_node *)node;
601 }
602 }
603
604 struct rb_node *rb_next_postorder(const struct rb_node *node)
605 {
606 const struct rb_node *parent;
607 if (!node)
608 return NULL;
609 parent = rb_parent(node);
610
611 /* If we're sitting on node, we've already seen our children */
612 if (parent && node == parent->rb_left && parent->rb_right) {
613 /* If we are the parent's left node, go to the parent's right
614 * node then all the way down to the left */
615 return rb_left_deepest_node(parent->rb_right);
616 } else
617 /* Otherwise we are the parent's right node, and the parent
618 * should be next */
619 return (struct rb_node *)parent;
620 }
621
622 struct rb_node *rb_first_postorder(const struct rb_root *root)
623 {
624 if (!root->rb_node)
625 return NULL;
626
627 return rb_left_deepest_node(root->rb_node);
628 }