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 void (*augment_rotate)(struct rb_node *old, struct rb_node *new))
87 {
88 struct rb_node *parent = rb_red_parent(node), *gparent, *tmp;
89
90 while (true) {
91 /*
92 * Loop invariant: node is red.
93 */
94 if (unlikely(!parent)) {
95 /*
96 * The inserted node is root. Either this is the
97 * first node, or we recursed at Case 1 below and
98 * are no longer violating 4).
99 */
100 rb_set_parent_color(node, NULL, RB_BLACK);
101 break;
102 }
103
104 /*
105 * If there is a black parent, we are done.
106 * Otherwise, take some corrective action as,
107 * per 4), we don't want a red root or two
108 * consecutive red nodes.
109 */
110 if(rb_is_black(parent))
111 break;
112
113 gparent = rb_red_parent(parent);
114
115 tmp = gparent->rb_right;
116 if (parent != tmp) { /* parent == gparent->rb_left */
117 if (tmp && rb_is_red(tmp)) {
118 /*
119 * Case 1 - node's uncle is red (color flips).
120 *
121 * G g
122 * / \ / \
123 * p u --> P U
124 * / /
125 * n n
126 *
127 * However, since g's parent might be red, and
128 * 4) does not allow this, we need to recurse
129 * at g.
130 */
131 rb_set_parent_color(tmp, gparent, RB_BLACK);
132 rb_set_parent_color(parent, gparent, RB_BLACK);
133 node = gparent;
134 parent = rb_parent(node);
135 rb_set_parent_color(node, parent, RB_RED);
136 continue;
137 }
138
139 tmp = parent->rb_right;
140 if (node == tmp) {
141 /*
142 * Case 2 - node's uncle is black and node is
143 * the parent's right child (left rotate at parent).
144 *
145 * G G
146 * / \ / \
147 * p U --> n U
148 * \ /
149 * n p
150 *
151 * This still leaves us in violation of 4), the
152 * continuation into Case 3 will fix that.
153 */
154 tmp = node->rb_left;
155 WRITE_ONCE(parent->rb_right, tmp);
156 WRITE_ONCE(node->rb_left, parent);
157 if (tmp)
158 rb_set_parent_color(tmp, parent,
159 RB_BLACK);
160 rb_set_parent_color(parent, node, RB_RED);
161 augment_rotate(parent, node);
162 parent = node;
163 tmp = node->rb_right;
164 }
165
166 /*
167 * Case 3 - node's uncle is black and node is
168 * the parent's left child (right rotate at gparent).
169 *
170 * G P
171 * / \ / \
172 * p U --> n g
173 * / \
174 * n U
175 */
176 WRITE_ONCE(gparent->rb_left, tmp); /* == parent->rb_right */
177 WRITE_ONCE(parent->rb_right, gparent);
178 if (tmp)
179 rb_set_parent_color(tmp, gparent, RB_BLACK);
180 __rb_rotate_set_parents(gparent, parent, root, RB_RED);
181 augment_rotate(gparent, parent);
182 break;
183 } else {
184 tmp = gparent->rb_left;
185 if (tmp && rb_is_red(tmp)) {
186 /* Case 1 - color flips */
187 rb_set_parent_color(tmp, gparent, RB_BLACK);
188 rb_set_parent_color(parent, gparent, RB_BLACK);
189 node = gparent;
190 parent = rb_parent(node);
191 rb_set_parent_color(node, parent, RB_RED);
192 continue;
193 }
194
195 tmp = parent->rb_left;
196 if (node == tmp) {
197 /* Case 2 - right rotate at parent */
198 tmp = node->rb_right;
199 WRITE_ONCE(parent->rb_left, tmp);
200 WRITE_ONCE(node->rb_right, parent);
201 if (tmp)
202 rb_set_parent_color(tmp, parent,
203 RB_BLACK);
204 rb_set_parent_color(parent, node, RB_RED);
205 augment_rotate(parent, node);
206 parent = node;
207 tmp = node->rb_left;
208 }
209
210 /* Case 3 - left rotate at gparent */
211 WRITE_ONCE(gparent->rb_right, tmp); /* == parent->rb_left */
212 WRITE_ONCE(parent->rb_left, gparent);
213 if (tmp)
214 rb_set_parent_color(tmp, gparent, RB_BLACK);
215 __rb_rotate_set_parents(gparent, parent, root, RB_RED);
216 augment_rotate(gparent, parent);
217 break;
218 }
219 }
220 }
221
222 /*
223 * Inline version for rb_erase() use - we want to be able to inline
224 * and eliminate the dummy_rotate callback there
225 */
226 static __always_inline void
227 ____rb_erase_color(struct rb_node *parent, struct rb_root *root,
228 void (*augment_rotate)(struct rb_node *old, struct rb_node *new))
229 {
230 struct rb_node *node = NULL, *sibling, *tmp1, *tmp2;
231
232 while (true) {
233 /*
234 * Loop invariants:
235 * - node is black (or NULL on first iteration)
236 * - node is not the root (parent is not NULL)
237 * - All leaf paths going through parent and node have a
238 * black node count that is 1 lower than other leaf paths.
239 */
240 sibling = parent->rb_right;
241 if (node != sibling) { /* node == parent->rb_left */
242 if (rb_is_red(sibling)) {
243 /*
244 * Case 1 - left rotate at parent
245 *
246 * P S
247 * / \ / \
248 * N s --> p Sr
249 * / \ / \
250 * Sl Sr N Sl
251 */
252 tmp1 = sibling->rb_left;
253 WRITE_ONCE(parent->rb_right, tmp1);
254 WRITE_ONCE(sibling->rb_left, parent);
255 rb_set_parent_color(tmp1, parent, RB_BLACK);
256 __rb_rotate_set_parents(parent, sibling, root,
257 RB_RED);
258 augment_rotate(parent, sibling);
259 sibling = tmp1;
260 }
261 tmp1 = sibling->rb_right;
262 if (!tmp1 || rb_is_black(tmp1)) {
263 tmp2 = sibling->rb_left;
264 if (!tmp2 || rb_is_black(tmp2)) {
265 /*
266 * Case 2 - sibling color flip
267 * (p could be either color here)
268 *
269 * (p) (p)
270 * / \ / \
271 * N S --> N s
272 * / \ / \
273 * Sl Sr Sl Sr
274 *
275 * This leaves us violating 5) which
276 * can be fixed by flipping p to black
277 * if it was red, or by recursing at p.
278 * p is red when coming from Case 1.
279 */
280 rb_set_parent_color(sibling, parent,
281 RB_RED);
282 if (rb_is_red(parent))
283 rb_set_black(parent);
284 else {
285 node = parent;
286 parent = rb_parent(node);
287 if (parent)
288 continue;
289 }
290 break;
291 }
292 /*
293 * Case 3 - right rotate at sibling
294 * (p could be either color here)
295 *
296 * (p) (p)
297 * / \ / \
298 * N S --> N sl
299 * / \ \
300 * sl Sr S
301 * \
302 * Sr
303 *
304 * Note: p might be red, and then both
305 * p and sl are red after rotation(which
306 * breaks property 4). This is fixed in
307 * Case 4 (in __rb_rotate_set_parents()
308 * which set sl the color of p
309 * and set p RB_BLACK)
310 *
311 * (p) (sl)
312 * / \ / \
313 * N sl --> P S
314 * \ / \
315 * S N Sr
316 * \
317 * Sr
318 */
319 tmp1 = tmp2->rb_right;
320 WRITE_ONCE(sibling->rb_left, tmp1);
321 WRITE_ONCE(tmp2->rb_right, sibling);
322 WRITE_ONCE(parent->rb_right, tmp2);
323 if (tmp1)
324 rb_set_parent_color(tmp1, sibling,
325 RB_BLACK);
326 augment_rotate(sibling, tmp2);
327 tmp1 = sibling;
328 sibling = tmp2;
329 }
330 /*
331 * Case 4 - left rotate at parent + color flips
332 * (p and sl could be either color here.
333 * After rotation, p becomes black, s acquires
334 * p's color, and sl keeps its color)
335 *
336 * (p) (s)
337 * / \ / \
338 * N S --> P Sr
339 * / \ / \
340 * (sl) sr N (sl)
341 */
342 tmp2 = sibling->rb_left;
343 WRITE_ONCE(parent->rb_right, tmp2);
344 WRITE_ONCE(sibling->rb_left, parent);
345 rb_set_parent_color(tmp1, sibling, RB_BLACK);
346 if (tmp2)
347 rb_set_parent(tmp2, parent);
348 __rb_rotate_set_parents(parent, sibling, root,
349 RB_BLACK);
350 augment_rotate(parent, sibling);
351 break;
352 } else {
353 sibling = parent->rb_left;
354 if (rb_is_red(sibling)) {
355 /* Case 1 - right rotate at parent */
356 tmp1 = sibling->rb_right;
357 WRITE_ONCE(parent->rb_left, tmp1);
358 WRITE_ONCE(sibling->rb_right, parent);
359 rb_set_parent_color(tmp1, parent, RB_BLACK);
360 __rb_rotate_set_parents(parent, sibling, root,
361 RB_RED);
362 augment_rotate(parent, sibling);
363 sibling = tmp1;
364 }
365 tmp1 = sibling->rb_left;
366 if (!tmp1 || rb_is_black(tmp1)) {
367 tmp2 = sibling->rb_right;
368 if (!tmp2 || rb_is_black(tmp2)) {
369 /* Case 2 - sibling color flip */
370 rb_set_parent_color(sibling, parent,
371 RB_RED);
372 if (rb_is_red(parent))
373 rb_set_black(parent);
374 else {
375 node = parent;
376 parent = rb_parent(node);
377 if (parent)
378 continue;
379 }
380 break;
381 }
382 /* Case 3 - left rotate at sibling */
383 tmp1 = tmp2->rb_left;
384 WRITE_ONCE(sibling->rb_right, tmp1);
385 WRITE_ONCE(tmp2->rb_left, sibling);
386 WRITE_ONCE(parent->rb_left, tmp2);
387 if (tmp1)
388 rb_set_parent_color(tmp1, sibling,
389 RB_BLACK);
390 augment_rotate(sibling, tmp2);
391 tmp1 = sibling;
392 sibling = tmp2;
393 }
394 /* Case 4 - right rotate at parent + color flips */
395 tmp2 = sibling->rb_right;
396 WRITE_ONCE(parent->rb_left, tmp2);
397 WRITE_ONCE(sibling->rb_right, parent);
398 rb_set_parent_color(tmp1, sibling, RB_BLACK);
399 if (tmp2)
400 rb_set_parent(tmp2, parent);
401 __rb_rotate_set_parents(parent, sibling, root,
402 RB_BLACK);
403 augment_rotate(parent, sibling);
404 break;
405 }
406 }
407 }
408
409 /* Non-inline version for rb_erase_augmented() use */
410 void __rb_erase_color(struct rb_node *parent, struct rb_root *root,
411 void (*augment_rotate)(struct rb_node *old, struct rb_node *new))
412 {
413 ____rb_erase_color(parent, root, augment_rotate);
414 }
415
416 /*
417 * Non-augmented rbtree manipulation functions.
418 *
419 * We use dummy augmented callbacks here, and have the compiler optimize them
420 * out of the rb_insert_color() and rb_erase() function definitions.
421 */
422
423 static inline void dummy_propagate(struct rb_node *node, struct rb_node *stop) {}
424 static inline void dummy_copy(struct rb_node *old, struct rb_node *new) {}
425 static inline void dummy_rotate(struct rb_node *old, struct rb_node *new) {}
426
427 static const struct rb_augment_callbacks dummy_callbacks = {
428 .propagate = dummy_propagate,
429 .copy = dummy_copy,
430 .rotate = dummy_rotate
431 };
432
433 void rb_insert_color(struct rb_node *node, struct rb_root *root)
434 {
435 __rb_insert(node, root, dummy_rotate);
436 }
437
438 void rb_erase(struct rb_node *node, struct rb_root *root)
439 {
440 struct rb_node *rebalance;
441 rebalance = __rb_erase_augmented(node, root, &dummy_callbacks);
442 if (rebalance)
443 ____rb_erase_color(rebalance, root, dummy_rotate);
444 }
445
446 /*
447 * Augmented rbtree manipulation functions.
448 *
449 * This instantiates the same __always_inline functions as in the non-augmented
450 * case, but this time with user-defined callbacks.
451 */
452
453 void __rb_insert_augmented(struct rb_node *node, struct rb_root *root,
454 void (*augment_rotate)(struct rb_node *old, struct rb_node *new))
455 {
456 __rb_insert(node, root, augment_rotate);
457 }
458
459 /*
460 * This function returns the first node (in sort order) of the tree.
461 */
462 struct rb_node *rb_first(const struct rb_root *root)
463 {
464 struct rb_node *n;
465
466 n = root->rb_node;
467 if (!n)
468 return NULL;
469 while (n->rb_left)
470 n = n->rb_left;
471 return n;
472 }
473
474 struct rb_node *rb_last(const struct rb_root *root)
475 {
476 struct rb_node *n;
477
478 n = root->rb_node;
479 if (!n)
480 return NULL;
481 while (n->rb_right)
482 n = n->rb_right;
483 return n;
484 }
485
486 struct rb_node *rb_next(const struct rb_node *node)
487 {
488 struct rb_node *parent;
489
490 if (RB_EMPTY_NODE(node))
491 return NULL;
492
493 /*
494 * If we have a right-hand child, go down and then left as far
495 * as we can.
496 */
497 if (node->rb_right) {
498 node = node->rb_right;
499 while (node->rb_left)
500 node=node->rb_left;
501 return (struct rb_node *)node;
502 }
503
504 /*
505 * No right-hand children. Everything down and left is smaller than us,
506 * so any 'next' node must be in the general direction of our parent.
507 * Go up the tree; any time the ancestor is a right-hand child of its
508 * parent, keep going up. First time it's a left-hand child of its
509 * parent, said parent is our 'next' node.
510 */
511 while ((parent = rb_parent(node)) && node == parent->rb_right)
512 node = parent;
513
514 return parent;
515 }
516
517 struct rb_node *rb_prev(const struct rb_node *node)
518 {
519 struct rb_node *parent;
520
521 if (RB_EMPTY_NODE(node))
522 return NULL;
523
524 /*
525 * If we have a left-hand child, go down and then right as far
526 * as we can.
527 */
528 if (node->rb_left) {
529 node = node->rb_left;
530 while (node->rb_right)
531 node=node->rb_right;
532 return (struct rb_node *)node;
533 }
534
535 /*
536 * No left-hand children. Go up till we find an ancestor which
537 * is a right-hand child of its parent.
538 */
539 while ((parent = rb_parent(node)) && node == parent->rb_left)
540 node = parent;
541
542 return parent;
543 }
544
545 void rb_replace_node(struct rb_node *victim, struct rb_node *new,
546 struct rb_root *root)
547 {
548 struct rb_node *parent = rb_parent(victim);
549
550 /* Copy the pointers/colour from the victim to the replacement */
551 *new = *victim;
552
553 /* Set the surrounding nodes to point to the replacement */
554 if (victim->rb_left)
555 rb_set_parent(victim->rb_left, new);
556 if (victim->rb_right)
557 rb_set_parent(victim->rb_right, new);
558 __rb_change_child(victim, new, parent, root);
559 }
560
561 static struct rb_node *rb_left_deepest_node(const struct rb_node *node)
562 {
563 for (;;) {
564 if (node->rb_left)
565 node = node->rb_left;
566 else if (node->rb_right)
567 node = node->rb_right;
568 else
569 return (struct rb_node *)node;
570 }
571 }
572
573 struct rb_node *rb_next_postorder(const struct rb_node *node)
574 {
575 const struct rb_node *parent;
576 if (!node)
577 return NULL;
578 parent = rb_parent(node);
579
580 /* If we're sitting on node, we've already seen our children */
581 if (parent && node == parent->rb_left && parent->rb_right) {
582 /* If we are the parent's left node, go to the parent's right
583 * node then all the way down to the left */
584 return rb_left_deepest_node(parent->rb_right);
585 } else
586 /* Otherwise we are the parent's right node, and the parent
587 * should be next */
588 return (struct rb_node *)parent;
589 }
590
591 struct rb_node *rb_first_postorder(const struct rb_root *root)
592 {
593 if (!root->rb_node)
594 return NULL;
595
596 return rb_left_deepest_node(root->rb_node);
597 }