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: https://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 } 598