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   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, write to the Free Software
  19   Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA  02111-1307  USA
  20
  21   linux/lib/rbtree.c
  22 */
  23
  24 #include "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 struct rb_augment_callbacks {
  47         void (*propagate)(struct rb_node *node, struct rb_node *stop);
  48         void (*copy)(struct rb_node *old, struct rb_node *new);
  49         void (*rotate)(struct rb_node *old, struct rb_node *new);
  50 };
  51
  52 #define RB_RED          0
  53 #define RB_BLACK        1
  54
  55 #define __rb_parent(pc)    ((struct rb_node *)(pc & ~3))
  56
  57 #define __rb_color(pc)     ((pc) & 1)
  58 #define __rb_is_black(pc)  __rb_color(pc)
  59 #define __rb_is_red(pc)    (!__rb_color(pc))
  60 #define rb_color(rb)       __rb_color((rb)->__rb_parent_color)
  61 #define rb_is_red(rb)      __rb_is_red((rb)->__rb_parent_color)
  62 #define rb_is_black(rb)    __rb_is_black((rb)->__rb_parent_color)
  63
  64 static inline void rb_set_parent(struct rb_node *rb, struct rb_node *p)
  65 {
  66         rb->__rb_parent_color = rb_color(rb) | (unsigned long)p;
  67 }
  68
  69 static inline void rb_set_parent_color(struct rb_node *rb,
  70                                        struct rb_node *p, int color)
  71 {
  72         rb->__rb_parent_color = (unsigned long)p | color;
  73 }
  74
  75 static inline void rb_set_black(struct rb_node *rb)
  76 {
  77         rb->__rb_parent_color |= RB_BLACK;
  78 }
  79
  80 static inline struct rb_node *rb_red_parent(struct rb_node *red)
  81 {
  82         return (struct rb_node *)red->__rb_parent_color;
  83 }
  84
  85 static inline void
  86 __rb_change_child(struct rb_node *old, struct rb_node *new,
  87                   struct rb_node *parent, struct rb_root *root)
  88 {
  89         if (parent) {
  90                 if (parent->rb_left == old)
  91                         parent->rb_left = new;
  92                 else
  93                         parent->rb_right = new;
  94         } else
  95                 root->rb_node = new;
  96 }
  97
  98 /*
  99  * Helper function for rotations:
 100  * - old's parent and color get assigned to new
 101  * - old gets assigned new as a parent and 'color' as a color.
 102  */
 103 static inline void
 104 __rb_rotate_set_parents(struct rb_node *old, struct rb_node *new,
 105                         struct rb_root *root, int color)
 106 {
 107         struct rb_node *parent = rb_parent(old);
 108         new->__rb_parent_color = old->__rb_parent_color;
 109         rb_set_parent_color(old, new, color);
 110         __rb_change_child(old, new, parent, root);
 111 }
 112
 113 static inline void
 114 __rb_erase_color(struct rb_node *parent, struct rb_root *root,
 115         void (*augment_rotate)(struct rb_node *old, struct rb_node *new))
 116 {
 117         struct rb_node *node = NULL, *sibling, *tmp1, *tmp2;
 118
 119         while (true) {
 120                 /*
 121                  * Loop invariants:
 122                  * - node is black (or NULL on first iteration)
 123                  * - node is not the root (parent is not NULL)
 124                  * - All leaf paths going through parent and node have a
 125                  *   black node count that is 1 lower than other leaf paths.
 126                  */
 127                 sibling = parent->rb_right;
 128                 if (node != sibling) {  /* node == parent->rb_left */
 129                         if (rb_is_red(sibling)) {
 130                                 /*
 131                                  * Case 1 - left rotate at parent
 132                                  *
 133                                  *     P               S
 134                                  *    / \             / \
 135                                  *   N   s    -->    p   Sr
 136                                  *      / \         / \
 137                                  *     Sl  Sr      N   Sl
 138                                  */
 139                                 parent->rb_right = tmp1 = sibling->rb_left;
 140                                 sibling->rb_left = parent;
 141                                 rb_set_parent_color(tmp1, parent, RB_BLACK);
 142                                 __rb_rotate_set_parents(parent, sibling, root,
 143                                                         RB_RED);
 144                                 augment_rotate(parent, sibling);
 145                                 sibling = tmp1;
 146                         }
 147                         tmp1 = sibling->rb_right;
 148                         if (!tmp1 || rb_is_black(tmp1)) {
 149                                 tmp2 = sibling->rb_left;
 150                                 if (!tmp2 || rb_is_black(tmp2)) {
 151                                         /*
 152                                          * Case 2 - sibling color flip
 153                                          * (p could be either color here)
 154                                          *
 155                                          *    (p)           (p)
 156                                          *    / \           / \
 157                                          *   N   S    -->  N   s
 158                                          *      / \           / \
 159                                          *     Sl  Sr        Sl  Sr
 160                                          *
 161                                          * This leaves us violating 5) which
 162                                          * can be fixed by flipping p to black
 163                                          * if it was red, or by recursing at p.
 164                                          * p is red when coming from Case 1.
 165                                          */
 166                                         rb_set_parent_color(sibling, parent,
 167                                                             RB_RED);
 168                                         if (rb_is_red(parent))
 169                                                 rb_set_black(parent);
 170                                         else {
 171                                                 node = parent;
 172                                                 parent = rb_parent(node);
 173                                                 if (parent)
 174                                                         continue;
 175                                         }
 176                                         break;
 177                                 }
 178                                 /*
 179                                  * Case 3 - right rotate at sibling
 180                                  * (p could be either color here)
 181                                  *
 182                                  *   (p)           (p)
 183                                  *   / \           / \
 184                                  *  N   S    -->  N   Sl
 185                                  *     / \             \
 186                                  *    sl  Sr            s
 187                                  *                       \
 188                                  *                        Sr
 189                                  */
 190                                 sibling->rb_left = tmp1 = tmp2->rb_right;
 191                                 tmp2->rb_right = sibling;
 192                                 parent->rb_right = tmp2;
 193                                 if (tmp1)
 194                                         rb_set_parent_color(tmp1, sibling,
 195                                                             RB_BLACK);
 196                                 augment_rotate(sibling, tmp2);
 197                                 tmp1 = sibling;
 198                                 sibling = tmp2;
 199                         }
 200                         /*
 201                          * Case 4 - left rotate at parent + color flips
 202                          * (p and sl could be either color here.
 203                          *  After rotation, p becomes black, s acquires
 204                          *  p's color, and sl keeps its color)
 205                          *
 206                          *      (p)             (s)
 207                          *      / \             / \
 208                          *     N   S     -->   P   Sr
 209                          *        / \         / \
 210                          *      (sl) sr      N  (sl)
 211                          */
 212                         parent->rb_right = tmp2 = sibling->rb_left;
 213                         sibling->rb_left = parent;
 214                         rb_set_parent_color(tmp1, sibling, RB_BLACK);
 215                         if (tmp2)
 216                                 rb_set_parent(tmp2, parent);
 217                         __rb_rotate_set_parents(parent, sibling, root,
 218                                                 RB_BLACK);
 219                         augment_rotate(parent, sibling);
 220                         break;
 221                 } else {
 222                         sibling = parent->rb_left;
 223                         if (rb_is_red(sibling)) {
 224                                 /* Case 1 - right rotate at parent */
 225                                 parent->rb_left = tmp1 = sibling->rb_right;
 226                                 sibling->rb_right = parent;
 227                                 rb_set_parent_color(tmp1, parent, RB_BLACK);
 228                                 __rb_rotate_set_parents(parent, sibling, root,
 229                                                         RB_RED);
 230                                 augment_rotate(parent, sibling);
 231                                 sibling = tmp1;
 232                         }
 233                         tmp1 = sibling->rb_left;
 234                         if (!tmp1 || rb_is_black(tmp1)) {
 235                                 tmp2 = sibling->rb_right;
 236                                 if (!tmp2 || rb_is_black(tmp2)) {
 237                                         /* Case 2 - sibling color flip */
 238                                         rb_set_parent_color(sibling, parent,
 239                                                             RB_RED);
 240                                         if (rb_is_red(parent))
 241                                                 rb_set_black(parent);
 242                                         else {
 243                                                 node = parent;
 244                                                 parent = rb_parent(node);
 245                                                 if (parent)
 246                                                         continue;
 247                                         }
 248                                         break;
 249                                 }
 250                                 /* Case 3 - right rotate at sibling */
 251                                 sibling->rb_right = tmp1 = tmp2->rb_left;
 252                                 tmp2->rb_left = sibling;
 253                                 parent->rb_left = tmp2;
 254                                 if (tmp1)
 255                                         rb_set_parent_color(tmp1, sibling,
 256                                                             RB_BLACK);
 257                                 augment_rotate(sibling, tmp2);
 258                                 tmp1 = sibling;
 259                                 sibling = tmp2;
 260                         }
 261                         /* Case 4 - left rotate at parent + color flips */
 262                         parent->rb_left = tmp2 = sibling->rb_right;
 263                         sibling->rb_right = parent;
 264                         rb_set_parent_color(tmp1, sibling, RB_BLACK);
 265                         if (tmp2)
 266                                 rb_set_parent(tmp2, parent);
 267                         __rb_rotate_set_parents(parent, sibling, root,
 268                                                 RB_BLACK);
 269                         augment_rotate(parent, sibling);
 270                         break;
 271                 }
 272         }
 273 }
 274
 275 static inline void
 276 rb_erase_augmented(struct rb_node *node, struct rb_root *root,
 277                    const struct rb_augment_callbacks *augment)
 278 {
 279         struct rb_node *child = node->rb_right, *tmp = node->rb_left;
 280         struct rb_node *parent, *rebalance;
 281         unsigned long pc;
 282
 283         if (!tmp) {
 284                 /*
 285                  * Case 1: node to erase has no more than 1 child (easy!)
 286                  *
 287                  * Note that if there is one child it must be red due to 5)
 288                  * and node must be black due to 4). We adjust colors locally
 289                  * so as to bypass __rb_erase_color() later on.
 290                  */
 291                 pc = node->__rb_parent_color;
 292                 parent = __rb_parent(pc);
 293                 __rb_change_child(node, child, parent, root);
 294                 if (child) {
 295                         child->__rb_parent_color = pc;
 296                         rebalance = NULL;
 297                 } else
 298                         rebalance = __rb_is_black(pc) ? parent : NULL;
 299                 tmp = parent;
 300         } else if (!child) {
 301                 /* Still case 1, but this time the child is node->rb_left */
 302                 tmp->__rb_parent_color = pc = node->__rb_parent_color;
 303                 parent = __rb_parent(pc);
 304                 __rb_change_child(node, tmp, parent, root);
 305                 rebalance = NULL;
 306                 tmp = parent;
 307         } else {
 308                 struct rb_node *successor = child, *child2;
 309                 tmp = child->rb_left;
 310                 if (!tmp) {
 311                         /*
 312                          * Case 2: node's successor is its right child
 313                          *
 314                          *    (n)          (s)
 315                          *    / \          / \
 316                          *  (x) (s)  ->  (x) (c)
 317                          *        \
 318                          *        (c)
 319                          */
 320                         parent = successor;
 321                         child2 = successor->rb_right;
 322                         augment->copy(node, successor);
 323                 } else {
 324                         /*
 325                          * Case 3: node's successor is leftmost under
 326                          * node's right child subtree
 327                          *
 328                          *    (n)          (s)
 329                          *    / \          / \
 330                          *  (x) (y)  ->  (x) (y)
 331                          *      /            /
 332                          *    (p)          (p)
 333                          *    /            /
 334                          *  (s)          (c)
 335                          *    \
 336                          *    (c)
 337                          */
 338                         do {
 339                                 parent = successor;
 340                                 successor = tmp;
 341                                 tmp = tmp->rb_left;
 342                         } while (tmp);
 343                         parent->rb_left = child2 = successor->rb_right;
 344                         successor->rb_right = child;
 345                         rb_set_parent(child, successor);
 346                         augment->copy(node, successor);
 347                         augment->propagate(parent, successor);
 348                 }
 349
 350                 successor->rb_left = tmp = node->rb_left;
 351                 rb_set_parent(tmp, successor);
 352
 353                 pc = node->__rb_parent_color;
 354                 tmp = __rb_parent(pc);
 355                 __rb_change_child(node, successor, tmp, root);
 356                 if (child2) {
 357                         successor->__rb_parent_color = pc;
 358                         rb_set_parent_color(child2, parent, RB_BLACK);
 359                         rebalance = NULL;
 360                 } else {
 361                         unsigned long pc2 = successor->__rb_parent_color;
 362                         successor->__rb_parent_color = pc;
 363                         rebalance = __rb_is_black(pc2) ? parent : NULL;
 364                 }
 365                 tmp = successor;
 366         }
 367
 368         augment->propagate(tmp, NULL);
 369         if (rebalance)
 370                 __rb_erase_color(rebalance, root, augment->rotate);
 371 }
 372
 373
 374 static inline void
 375 __rb_insert(struct rb_node *node, struct rb_root *root,
 376             void (*augment_rotate)(struct rb_node *old, struct rb_node *new))
 377 {
 378         struct rb_node *parent = rb_red_parent(node), *gparent, *tmp;
 379
 380         while (true) {
 381                 /*
 382                  * Loop invariant: node is red
 383                  *
 384                  * If there is a black parent, we are done.
 385                  * Otherwise, take some corrective action as we don't
 386                  * want a red root or two consecutive red nodes.
 387                  */
 388                 if (!parent) {
 389                         rb_set_parent_color(node, NULL, RB_BLACK);
 390                         break;
 391                 } else if (rb_is_black(parent))
 392                         break;
 393
 394                 gparent = rb_red_parent(parent);
 395
 396                 tmp = gparent->rb_right;
 397                 if (parent != tmp) {    /* parent == gparent->rb_left */
 398                         if (tmp && rb_is_red(tmp)) {
 399                                 /*
 400                                  * Case 1 - color flips
 401                                  *
 402                                  *       G            g
 403                                  *      / \          / \
 404                                  *     p   u  -->   P   U
 405                                  *    /            /
 406                                  *   n            N
 407                                  *
 408                                  * However, since g's parent might be red, and
 409                                  * 4) does not allow this, we need to recurse
 410                                  * at g.
 411                                  */
 412                                 rb_set_parent_color(tmp, gparent, RB_BLACK);
 413                                 rb_set_parent_color(parent, gparent, RB_BLACK);
 414                                 node = gparent;
 415                                 parent = rb_parent(node);
 416                                 rb_set_parent_color(node, parent, RB_RED);
 417                                 continue;
 418                         }
 419
 420                         tmp = parent->rb_right;
 421                         if (node == tmp) {
 422                                 /*
 423                                  * Case 2 - left rotate at parent
 424                                  *
 425                                  *      G             G
 426                                  *     / \           / \
 427                                  *    p   U  -->    n   U
 428                                  *     \           /
 429                                  *      n         p
 430                                  *
 431                                  * This still leaves us in violation of 4), the
 432                                  * continuation into Case 3 will fix that.
 433                                  */
 434                                 parent->rb_right = tmp = node->rb_left;
 435                                 node->rb_left = parent;
 436                                 if (tmp)
 437                                         rb_set_parent_color(tmp, parent,
 438                                                             RB_BLACK);
 439                                 rb_set_parent_color(parent, node, RB_RED);
 440                                 augment_rotate(parent, node);
 441                                 parent = node;
 442                                 tmp = node->rb_right;
 443                         }
 444
 445                         /*
 446                          * Case 3 - right rotate at gparent
 447                          *
 448                          *        G           P
 449                          *       / \         / \
 450                          *      p   U  -->  n   g
 451                          *     /                 \
 452                          *    n                   U
 453                          */
 454                         gparent->rb_left = tmp;  /* == parent->rb_right */
 455                         parent->rb_right = gparent;
 456                         if (tmp)
 457                                 rb_set_parent_color(tmp, gparent, RB_BLACK);
 458                         __rb_rotate_set_parents(gparent, parent, root, RB_RED);
 459                         augment_rotate(gparent, parent);
 460                         break;
 461                 } else {
 462                         tmp = gparent->rb_left;
 463                         if (tmp && rb_is_red(tmp)) {
 464                                 /* Case 1 - color flips */
 465                                 rb_set_parent_color(tmp, gparent, RB_BLACK);
 466                                 rb_set_parent_color(parent, gparent, RB_BLACK);
 467                                 node = gparent;
 468                                 parent = rb_parent(node);
 469                                 rb_set_parent_color(node, parent, RB_RED);
 470                                 continue;
 471                         }
 472
 473                         tmp = parent->rb_left;
 474                         if (node == tmp) {
 475                                 /* Case 2 - right rotate at parent */
 476                                 parent->rb_left = tmp = node->rb_right;
 477                                 node->rb_right = parent;
 478                                 if (tmp)
 479                                         rb_set_parent_color(tmp, parent,
 480                                                             RB_BLACK);
 481                                 rb_set_parent_color(parent, node, RB_RED);
 482                                 augment_rotate(parent, node);
 483                                 parent = node;
 484                                 tmp = node->rb_left;
 485                         }
 486
 487                         /* Case 3 - left rotate at gparent */
 488                         gparent->rb_right = tmp;  /* == parent->rb_left */
 489                         parent->rb_left = gparent;
 490                         if (tmp)
 491                                 rb_set_parent_color(tmp, gparent, RB_BLACK);
 492                         __rb_rotate_set_parents(gparent, parent, root, RB_RED);
 493                         augment_rotate(gparent, parent);
 494                         break;
 495                 }
 496         }
 497 }
 498
 499
 500 /*
 501  * Non-augmented rbtree manipulation functions.
 502  *
 503  * We use dummy augmented callbacks here, and have the compiler optimize them
 504  * out of the rb_insert_color() and rb_erase() function definitions.
 505  */
 506
 507 static inline void dummy_propagate(struct rb_node *node, struct rb_node *stop) {}
 508 static inline void dummy_copy(struct rb_node *old, struct rb_node *new) {}
 509 static inline void dummy_rotate(struct rb_node *old, struct rb_node *new) {}
 510
 511 static const struct rb_augment_callbacks dummy_callbacks = {
 512         dummy_propagate, dummy_copy, dummy_rotate
 513 };
 514
 515 void rb_insert_color(struct rb_node *node, struct rb_root *root)
 516 {
 517         __rb_insert(node, root, dummy_rotate);
 518 }
 519
 520 void rb_erase(struct rb_node *node, struct rb_root *root)
 521 {
 522         rb_erase_augmented(node, root, &dummy_callbacks);
 523 }