1 /*
2 Red Black Trees
3 (C) 1999 Andrea Arcangeli <andrea@suse.de>
4 (C) 2002 David Woodhouse <dwmw2@infradead.org>
5 (C) 2012 Michel Lespinasse <walken@google.com>
6
7 This program is free software; you can redistribute it and/or modify
8 it under the terms of the GNU General Public License as published by
9 the Free Software Foundation; either version 2 of the License, or
10 (at your option) any later version.
11
12 This program is distributed in the hope that it will be useful,
13 but WITHOUT ANY WARRANTY; without even the implied warranty of
14 MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
15 GNU General Public License for more details.
16
17 You should have received a copy of the GNU General Public License
18 along with this program; If not, see <http://www.gnu.org/licenses/>.
19
20 linux/lib/rbtree.c
21 */
22
23 #include <xen/types.h>
24 #include <xen/rbtree.h>
25
26 /*
27 * red-black trees properties: http://en.wikipedia.org/wiki/Rbtree
28 *
29 * 1) A node is either red or black
30 * 2) The root is black
31 * 3) All leaves (NULL) are black
32 * 4) Both children of every red node are black
33 * 5) Every simple path from root to leaves contains the same number
34 * of black nodes.
35 *
36 * 4 and 5 give the O(log n) guarantee, since 4 implies you cannot have two
37 * consecutive red nodes in a path and every red node is therefore followed by
38 * a black. So if B is the number of black nodes on every simple path (as per
39 * 5), then the longest possible path due to 4 is 2B.
40 *
41 * We shall indicate color with case, where black nodes are uppercase and red
42 * nodes will be lowercase. Unknown color nodes shall be drawn as red within
43 * parentheses and have some accompanying text comment.
44 */
45
46 #define RB_RED 0
47 #define RB_BLACK 1
48
49 #define __rb_parent(pc) ((struct rb_node *)((pc) & ~3))
50
51 #define __rb_color(pc) ((pc) & 1)
52 #define __rb_is_black(pc) __rb_color(pc)
53 #define __rb_is_red(pc) (!__rb_color(pc))
54 #define rb_color(rb) __rb_color((rb)->__rb_parent_color)
55 #define rb_is_red(rb) __rb_is_red((rb)->__rb_parent_color)
56 #define rb_is_black(rb) __rb_is_black((rb)->__rb_parent_color)
57
rb_set_black(struct rb_node * rb)58 static inline void rb_set_black(struct rb_node *rb)
59 {
60 rb->__rb_parent_color |= RB_BLACK;
61 }
62
rb_set_parent(struct rb_node * rb,struct rb_node * p)63 static inline void rb_set_parent(struct rb_node *rb, struct rb_node *p)
64 {
65 rb->__rb_parent_color = rb_color(rb) | (unsigned long)p;
66 }
67
rb_set_parent_color(struct rb_node * rb,struct rb_node * p,int color)68 static inline void rb_set_parent_color(struct rb_node *rb,
69 struct rb_node *p, int color)
70 {
71 rb->__rb_parent_color = (unsigned long)p | color;
72 }
73
rb_red_parent(struct rb_node * red)74 static inline struct rb_node *rb_red_parent(struct rb_node *red)
75 {
76 return (struct rb_node *)red->__rb_parent_color;
77 }
78
79 static inline void
__rb_change_child(struct rb_node * old,struct rb_node * new,struct rb_node * parent,struct rb_root * root)80 __rb_change_child(struct rb_node *old, struct rb_node *new,
81 struct rb_node *parent, struct rb_root *root)
82 {
83 if (parent) {
84 if (parent->rb_left == old)
85 parent->rb_left = new;
86 else
87 parent->rb_right = new;
88 } else
89 root->rb_node = new;
90 }
91
92 /*
93 * Helper function for rotations:
94 * - old's parent and color get assigned to new
95 * - old gets assigned new as a parent and 'color' as a color.
96 */
97 static inline void
__rb_rotate_set_parents(struct rb_node * old,struct rb_node * new,struct rb_root * root,int color)98 __rb_rotate_set_parents(struct rb_node *old, struct rb_node *new,
99 struct rb_root *root, int color)
100 {
101 struct rb_node *parent = rb_parent(old);
102 new->__rb_parent_color = old->__rb_parent_color;
103 rb_set_parent_color(old, new, color);
104 __rb_change_child(old, new, parent, root);
105 }
106
rb_insert_color(struct rb_node * node,struct rb_root * root)107 void rb_insert_color(struct rb_node *node, struct rb_root *root)
108 {
109 struct rb_node *parent = rb_red_parent(node), *gparent, *tmp;
110
111 while (true) {
112 /*
113 * Loop invariant: node is red
114 *
115 * If there is a black parent, we are done.
116 * Otherwise, take some corrective action as we don't
117 * want a red root or two consecutive red nodes.
118 */
119 if (!parent) {
120 rb_set_parent_color(node, NULL, RB_BLACK);
121 break;
122 } else if (rb_is_black(parent))
123 break;
124
125 gparent = rb_red_parent(parent);
126
127 tmp = gparent->rb_right;
128 if (parent != tmp) { /* parent == gparent->rb_left */
129 if (tmp && rb_is_red(tmp)) {
130 /*
131 * Case 1 - color flips
132 *
133 * G g
134 * / \ / \
135 * p u --> P U
136 * / /
137 * n n
138 *
139 * However, since g's parent might be red, and
140 * 4) does not allow this, we need to recurse
141 * at g.
142 */
143 rb_set_parent_color(tmp, gparent, RB_BLACK);
144 rb_set_parent_color(parent, gparent, RB_BLACK);
145 node = gparent;
146 parent = rb_parent(node);
147 rb_set_parent_color(node, parent, RB_RED);
148 continue;
149 }
150
151 tmp = parent->rb_right;
152 if (node == tmp) {
153 /*
154 * Case 2 - left rotate at parent
155 *
156 * G G
157 * / \ / \
158 * p U --> n U
159 * \ /
160 * n p
161 *
162 * This still leaves us in violation of 4), the
163 * continuation into Case 3 will fix that.
164 */
165 parent->rb_right = tmp = node->rb_left;
166 node->rb_left = parent;
167 if (tmp)
168 rb_set_parent_color(tmp, parent,
169 RB_BLACK);
170 rb_set_parent_color(parent, node, RB_RED);
171 parent = node;
172 tmp = node->rb_right;
173 }
174
175 /*
176 * Case 3 - right rotate at gparent
177 *
178 * G P
179 * / \ / \
180 * p U --> n g
181 * / \
182 * n U
183 */
184 gparent->rb_left = tmp; /* == parent->rb_right */
185 parent->rb_right = gparent;
186 if (tmp)
187 rb_set_parent_color(tmp, gparent, RB_BLACK);
188 __rb_rotate_set_parents(gparent, parent, root, RB_RED);
189 break;
190 } else {
191 tmp = gparent->rb_left;
192 if (tmp && rb_is_red(tmp)) {
193 /* Case 1 - color flips */
194 rb_set_parent_color(tmp, gparent, RB_BLACK);
195 rb_set_parent_color(parent, gparent, RB_BLACK);
196 node = gparent;
197 parent = rb_parent(node);
198 rb_set_parent_color(node, parent, RB_RED);
199 continue;
200 }
201
202 tmp = parent->rb_left;
203 if (node == tmp) {
204 /* Case 2 - right rotate at parent */
205 parent->rb_left = tmp = node->rb_right;
206 node->rb_right = parent;
207 if (tmp)
208 rb_set_parent_color(tmp, parent,
209 RB_BLACK);
210 rb_set_parent_color(parent, node, RB_RED);
211 parent = node;
212 tmp = node->rb_left;
213 }
214
215 /* Case 3 - left rotate at gparent */
216 gparent->rb_right = tmp; /* == parent->rb_left */
217 parent->rb_left = gparent;
218 if (tmp)
219 rb_set_parent_color(tmp, gparent, RB_BLACK);
220 __rb_rotate_set_parents(gparent, parent, root, RB_RED);
221 break;
222 }
223 }
224 }
225
__rb_erase_color(struct rb_node * parent,struct rb_root * root)226 static void __rb_erase_color(struct rb_node *parent, struct rb_root *root)
227 {
228 struct rb_node *node = NULL, *sibling, *tmp1, *tmp2;
229
230 while (true) {
231 /*
232 * Loop invariants:
233 * - node is black (or NULL on first iteration)
234 * - node is not the root (parent is not NULL)
235 * - All leaf paths going through parent and node have a
236 * black node count that is 1 lower than other leaf paths.
237 */
238 sibling = parent->rb_right;
239 if (node != sibling) { /* node == parent->rb_left */
240 if (rb_is_red(sibling)) {
241 /*
242 * Case 1 - left rotate at parent
243 *
244 * P S
245 * / \ / \
246 * N s --> p Sr
247 * / \ / \
248 * Sl Sr N Sl
249 */
250 parent->rb_right = tmp1 = sibling->rb_left;
251 sibling->rb_left = parent;
252 rb_set_parent_color(tmp1, parent, RB_BLACK);
253 __rb_rotate_set_parents(parent, sibling, root,
254 RB_RED);
255 sibling = tmp1;
256 }
257 tmp1 = sibling->rb_right;
258 if (!tmp1 || rb_is_black(tmp1)) {
259 tmp2 = sibling->rb_left;
260 if (!tmp2 || rb_is_black(tmp2)) {
261 /*
262 * Case 2 - sibling color flip
263 * (p could be either color here)
264 *
265 * (p) (p)
266 * / \ / \
267 * N S --> N s
268 * / \ / \
269 * Sl Sr Sl Sr
270 *
271 * This leaves us violating 5) which
272 * can be fixed by flipping p to black
273 * if it was red, or by recursing at p.
274 * p is red when coming from Case 1.
275 */
276 rb_set_parent_color(sibling, parent,
277 RB_RED);
278 if (rb_is_red(parent))
279 rb_set_black(parent);
280 else {
281 node = parent;
282 parent = rb_parent(node);
283 if (parent)
284 continue;
285 }
286 break;
287 }
288 /*
289 * Case 3 - right rotate at sibling
290 * (p could be either color here)
291 *
292 * (p) (p)
293 * / \ / \
294 * N S --> N Sl
295 * / \ \
296 * sl Sr s
297 * \
298 * Sr
299 */
300 sibling->rb_left = tmp1 = tmp2->rb_right;
301 tmp2->rb_right = sibling;
302 parent->rb_right = tmp2;
303 if (tmp1)
304 rb_set_parent_color(tmp1, sibling,
305 RB_BLACK);
306 tmp1 = sibling;
307 sibling = tmp2;
308 }
309 /*
310 * Case 4 - left rotate at parent + color flips
311 * (p and sl could be either color here.
312 * After rotation, p becomes black, s acquires
313 * p's color, and sl keeps its color)
314 *
315 * (p) (s)
316 * / \ / \
317 * N S --> P Sr
318 * / \ / \
319 * (sl) sr N (sl)
320 */
321 parent->rb_right = tmp2 = sibling->rb_left;
322 sibling->rb_left = parent;
323 rb_set_parent_color(tmp1, sibling, RB_BLACK);
324 if (tmp2)
325 rb_set_parent(tmp2, parent);
326 __rb_rotate_set_parents(parent, sibling, root,
327 RB_BLACK);
328 break;
329 } else {
330 sibling = parent->rb_left;
331 if (rb_is_red(sibling)) {
332 /* Case 1 - right rotate at parent */
333 parent->rb_left = tmp1 = sibling->rb_right;
334 sibling->rb_right = parent;
335 rb_set_parent_color(tmp1, parent, RB_BLACK);
336 __rb_rotate_set_parents(parent, sibling, root,
337 RB_RED);
338 sibling = tmp1;
339 }
340 tmp1 = sibling->rb_left;
341 if (!tmp1 || rb_is_black(tmp1)) {
342 tmp2 = sibling->rb_right;
343 if (!tmp2 || rb_is_black(tmp2)) {
344 /* Case 2 - sibling color flip */
345 rb_set_parent_color(sibling, parent,
346 RB_RED);
347 if (rb_is_red(parent))
348 rb_set_black(parent);
349 else {
350 node = parent;
351 parent = rb_parent(node);
352 if (parent)
353 continue;
354 }
355 break;
356 }
357 /* Case 3 - right rotate at sibling */
358 sibling->rb_right = tmp1 = tmp2->rb_left;
359 tmp2->rb_left = sibling;
360 parent->rb_left = tmp2;
361 if (tmp1)
362 rb_set_parent_color(tmp1, sibling,
363 RB_BLACK);
364 tmp1 = sibling;
365 sibling = tmp2;
366 }
367 /* Case 4 - left rotate at parent + color flips */
368 parent->rb_left = tmp2 = sibling->rb_right;
369 sibling->rb_right = parent;
370 rb_set_parent_color(tmp1, sibling, RB_BLACK);
371 if (tmp2)
372 rb_set_parent(tmp2, parent);
373 __rb_rotate_set_parents(parent, sibling, root,
374 RB_BLACK);
375 break;
376 }
377 }
378 }
379
rb_erase(struct rb_node * node,struct rb_root * root)380 void rb_erase(struct rb_node *node, struct rb_root *root)
381 {
382 struct rb_node *child = node->rb_right, *tmp = node->rb_left;
383 struct rb_node *parent, *rebalance;
384 unsigned long pc;
385
386 if (!tmp) {
387 /*
388 * Case 1: node to erase has no more than 1 child (easy!)
389 *
390 * Note that if there is one child it must be red due to 5)
391 * and node must be black due to 4). We adjust colors locally
392 * so as to bypass __rb_erase_color() later on.
393 */
394 pc = node->__rb_parent_color;
395 parent = __rb_parent(pc);
396 __rb_change_child(node, child, parent, root);
397 if (child) {
398 child->__rb_parent_color = pc;
399 rebalance = NULL;
400 } else
401 rebalance = __rb_is_black(pc) ? parent : NULL;
402 } else if (!child) {
403 /* Still case 1, but this time the child is node->rb_left */
404 tmp->__rb_parent_color = pc = node->__rb_parent_color;
405 parent = __rb_parent(pc);
406 __rb_change_child(node, tmp, parent, root);
407 rebalance = NULL;
408 } else {
409 struct rb_node *successor = child, *child2;
410 tmp = child->rb_left;
411 if (!tmp) {
412 /*
413 * Case 2: node's successor is its right child
414 *
415 * (n) (s)
416 * / \ / \
417 * (x) (s) -> (x) (c)
418 * \
419 * (c)
420 */
421 parent = child;
422 child2 = child->rb_right;
423 } else {
424 /*
425 * Case 3: node's successor is leftmost under
426 * node's right child subtree
427 *
428 * (n) (s)
429 * / \ / \
430 * (x) (y) -> (x) (y)
431 * / /
432 * (p) (p)
433 * / /
434 * (s) (c)
435 * \
436 * (c)
437 */
438 do {
439 parent = successor;
440 successor = tmp;
441 tmp = tmp->rb_left;
442 } while (tmp);
443 parent->rb_left = child2 = successor->rb_right;
444 successor->rb_right = child;
445 rb_set_parent(child, successor);
446 }
447
448 successor->rb_left = tmp = node->rb_left;
449 rb_set_parent(tmp, successor);
450
451 pc = node->__rb_parent_color;
452 tmp = __rb_parent(pc);
453 __rb_change_child(node, successor, tmp, root);
454 if (child2) {
455 successor->__rb_parent_color = pc;
456 rb_set_parent_color(child2, parent, RB_BLACK);
457 rebalance = NULL;
458 } else {
459 unsigned long pc2 = successor->__rb_parent_color;
460 successor->__rb_parent_color = pc;
461 rebalance = __rb_is_black(pc2) ? parent : NULL;
462 }
463 }
464
465 if (rebalance)
466 __rb_erase_color(rebalance, root);
467 }
468
469 /*
470 * This function returns the first node (in sort order) of the tree.
471 */
rb_first(const struct rb_root * root)472 struct rb_node *rb_first(const struct rb_root *root)
473 {
474 struct rb_node *n;
475
476 n = root->rb_node;
477 if (!n)
478 return NULL;
479 while (n->rb_left)
480 n = n->rb_left;
481 return n;
482 }
483
rb_last(const struct rb_root * root)484 struct rb_node *rb_last(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_right)
492 n = n->rb_right;
493 return n;
494 }
495
rb_next(const struct rb_node * node)496 struct rb_node *rb_next(const struct rb_node *node)
497 {
498 struct rb_node *parent;
499
500 if (RB_EMPTY_NODE(node))
501 return NULL;
502
503 /*
504 * If we have a right-hand child, go down and then left as far
505 * as we can.
506 */
507 if (node->rb_right) {
508 node = node->rb_right;
509 while (node->rb_left)
510 node=node->rb_left;
511 return (struct rb_node *)node;
512 }
513
514 /*
515 * No right-hand children. Everything down and left is smaller than us,
516 * so any 'next' node must be in the general direction of our parent.
517 * Go up the tree; any time the ancestor is a right-hand child of its
518 * parent, keep going up. First time it's a left-hand child of its
519 * parent, said parent is our 'next' node.
520 */
521 while ((parent = rb_parent(node)) && node == parent->rb_right)
522 node = parent;
523
524 return parent;
525 }
526
rb_prev(const struct rb_node * node)527 struct rb_node *rb_prev(const struct rb_node *node)
528 {
529 struct rb_node *parent;
530
531 if (RB_EMPTY_NODE(node))
532 return NULL;
533
534 /*
535 * If we have a left-hand child, go down and then right as far
536 * as we can.
537 */
538 if (node->rb_left) {
539 node = node->rb_left;
540 while (node->rb_right)
541 node=node->rb_right;
542 return (struct rb_node *)node;
543 }
544
545 /*
546 * No left-hand children. Go up till we find an ancestor which
547 * is a right-hand child of its parent
548 */
549 while ((parent = rb_parent(node)) && node == parent->rb_left)
550 node = parent;
551
552 return parent;
553 }
554
rb_replace_node(struct rb_node * victim,struct rb_node * new,struct rb_root * root)555 void rb_replace_node(struct rb_node *victim, struct rb_node *new,
556 struct rb_root *root)
557 {
558 struct rb_node *parent = rb_parent(victim);
559
560 /* Set the surrounding nodes to point to the replacement */
561 __rb_change_child(victim, new, parent, root);
562 if (victim->rb_left)
563 rb_set_parent(victim->rb_left, new);
564 if (victim->rb_right)
565 rb_set_parent(victim->rb_right, new);
566
567 /* Copy the pointers/colour from the victim to the replacement */
568 *new = *victim;
569 }
570