1 /* Copyright (C) 1995, 1996, 1997, 2000, 2006 Free Software Foundation, Inc.
2 Contributed by Bernd Schmidt <crux@Pool.Informatik.RWTH-Aachen.DE>, 1997.
4 NOTE: The canonical source of this file is maintained with the GNU C
5 Library. Bugs can be reported to bug-glibc@gnu.org.
7 This program is free software; you can redistribute it and/or modify it
8 under the terms of the GNU General Public License as published by the
9 Free Software Foundation; either version 2, or (at your option) any
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.
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 Foundation,
19 Inc., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301, USA. */
21 /* Tree search for red/black trees.
22 The algorithm for adding nodes is taken from one of the many "Algorithms"
23 books by Robert Sedgewick, although the implementation differs.
24 The algorithm for deleting nodes can probably be found in a book named
25 "Introduction to Algorithms" by Cormen/Leiserson/Rivest. At least that's
26 the book that my professor took most algorithms from during the "Data
29 Totally public domain. */
31 /* Red/black trees are binary trees in which the edges are colored either red
32 or black. They have the following properties:
33 1. The number of black edges on every path from the root to a leaf is
35 2. No two red edges are adjacent.
36 Therefore there is an upper bound on the length of every path, it's
37 O(log n) where n is the number of nodes in the tree. No path can be longer
38 than 1+2*P where P is the length of the shortest path in the tree.
39 Useful for the implementation:
40 3. If one of the children of a node is NULL, then the other one is red
43 In the implementation, not the edges are colored, but the nodes. The color
44 interpreted as the color of the edge leading to this node. The color is
45 meaningless for the root node, but we color the root node black for
46 convenience. All added nodes are red initially.
48 Adding to a red/black tree is rather easy. The right place is searched
49 with a usual binary tree search. Additionally, whenever a node N is
50 reached that has two red successors, the successors are colored black and
51 the node itself colored red. This moves red edges up the tree where they
52 pose less of a problem once we get to really insert the new node. Changing
53 N's color to red may violate rule 2, however, so rotations may become
54 necessary to restore the invariants. Adding a new red leaf may violate
55 the same rule, so afterwards an additional check is run and the tree
58 Deleting is hairy. There are mainly two nodes involved: the node to be
59 deleted (n1), and another node that is to be unchained from the tree (n2).
60 If n1 has a successor (the node with a smallest key that is larger than
61 n1), then the successor becomes n2 and its contents are copied into n1,
62 otherwise n1 becomes n2.
63 Unchaining a node may violate rule 1: if n2 is black, one subtree is
64 missing one black edge afterwards. The algorithm must try to move this
65 error upwards towards the root, so that the subtree that does not have
66 enough black edges becomes the whole tree. Once that happens, the error
67 has disappeared. It may not be necessary to go all the way up, since it
68 is possible that rotations and recoloring can fix the error before that.
70 Although the deletion algorithm must walk upwards through the tree, we
71 do not store parent pointers in the nodes. Instead, delete allocates a
72 small array of parent pointers and fills it while descending the tree.
73 Since we know that the length of a path is O(log n), where n is the number
74 of nodes, this is likely to use less memory. */
76 /* Tree rotations look like this:
85 In this case, A has been rotated left. This preserves the ordering of the
95 typedef int (*__compar_fn_t) (const void *, const void *);
96 typedef void (*__action_fn_t) (const void *, VISIT, int);
99 # define __tsearch tsearch
100 # define __tfind tfind
101 # define __tdelete tdelete
102 # define __twalk twalk
105 #ifndef internal_function
106 /* Inside GNU libc we mark some function in a special way. In other
107 environments simply ignore the marking. */
108 # define internal_function
111 typedef struct node_t
113 /* Callers expect this to be the first element in the structure - do not
117 struct node_t *right;
120 typedef const struct node_t *const_node;
126 /* Routines to check tree invariants. */
130 #define CHECK_TREE(a) check_tree(a)
133 check_tree_recurse (node p, int d_sofar, int d_total)
137 assert (d_sofar == d_total);
141 check_tree_recurse (p->left, d_sofar + (p->left && !p->left->red), d_total);
142 check_tree_recurse (p->right, d_sofar + (p->right && !p->right->red), d_total);
144 assert (!(p->left->red && p->red));
146 assert (!(p->right->red && p->red));
150 check_tree (node root)
157 for(p = root->left; p; p = p->left)
159 check_tree_recurse (root, 0, cnt);
165 #define CHECK_TREE(a)
169 /* Possibly "split" a node with two red successors, and/or fix up two red
170 edges in a row. ROOTP is a pointer to the lowest node we visited, PARENTP
171 and GPARENTP pointers to its parent/grandparent. P_R and GP_R contain the
172 comparison values that determined which way was taken in the tree to reach
173 ROOTP. MODE is 1 if we need not do the split, but must check for two red
174 edges between GPARENTP and ROOTP. */
176 maybe_split_for_insert (node *rootp, node *parentp, node *gparentp,
177 int p_r, int gp_r, int mode)
181 rp = &(*rootp)->right;
182 lp = &(*rootp)->left;
184 /* See if we have to split this node (both successors red). */
186 || ((*rp) != NULL && (*lp) != NULL && (*rp)->red && (*lp)->red))
188 /* This node becomes red, its successors black. */
195 /* If the parent of this node is also red, we have to do
197 if (parentp != NULL && (*parentp)->red)
201 /* There are two main cases:
202 1. The edge types (left or right) of the two red edges differ.
203 2. Both red edges are of the same type.
204 There exist two symmetries of each case, so there is a total of
206 if ((p_r > 0) != (gp_r > 0))
208 /* Put the child at the top of the tree, with its parent
209 and grandparent as successors. */
215 /* Child is left of parent. */
223 /* Child is right of parent. */
233 *gparentp = *parentp;
234 /* Parent becomes the top of the tree, grandparent and
235 child are its successors. */
255 /* Find or insert datum into search tree.
256 KEY is the key to be located, ROOTP is the address of tree root,
257 COMPAR the ordering function. */
259 __tsearch (const void *key, void **vrootp, __compar_fn_t compar)
262 node *parentp = NULL, *gparentp = NULL;
263 node *rootp = (node *) vrootp;
265 int r = 0, p_r = 0, gp_r = 0; /* No they might not, Mr Compiler. */
270 /* This saves some additional tests below. */
277 while (*nextp != NULL)
280 r = (*compar) (key, root->key);
284 maybe_split_for_insert (rootp, parentp, gparentp, p_r, gp_r, 0);
285 /* If that did any rotations, parentp and gparentp are now garbage.
286 That doesn't matter, because the values they contain are never
287 used again in that case. */
289 nextp = r < 0 ? &root->left : &root->right;
301 q = (struct node_t *) malloc (sizeof (struct node_t));
304 *nextp = q; /* link new node to old */
305 q->key = key; /* initialize new node */
307 q->left = q->right = NULL;
310 /* There may be two red edges in a row now, which we must avoid by
311 rotating the tree. */
312 maybe_split_for_insert (nextp, rootp, parentp, r, p_r, 1);
318 weak_alias (__tsearch, tsearch)
322 /* Find datum in search tree.
323 KEY is the key to be located, ROOTP is the address of tree root,
324 COMPAR the ordering function. */
326 __tfind (key, vrootp, compar)
329 __compar_fn_t compar;
331 node *rootp = (node *) vrootp;
338 while (*rootp != NULL)
343 r = (*compar) (key, root->key);
347 rootp = r < 0 ? &root->left : &root->right;
352 weak_alias (__tfind, tfind)
356 /* Delete node with given key.
357 KEY is the key to be deleted, ROOTP is the address of the root of tree,
358 COMPAR the comparison function. */
360 __tdelete (const void *key, void **vrootp, __compar_fn_t compar)
362 node p, q, r, retval;
364 node *rootp = (node *) vrootp;
365 node root, unchained;
366 /* Stack of nodes so we remember the parents without recursion. It's
367 _very_ unlikely that there are paths longer than 40 nodes. The tree
368 would need to have around 250.000 nodes. */
371 node *nodestack[100];
381 while ((cmp = (*compar) (key, (*rootp)->key)) != 0)
386 nodestack[sp++] = rootp;
395 /* This is bogus if the node to be deleted is the root... this routine
396 really should return an integer with 0 for success, -1 for failure
397 and errno = ESRCH or something. */
400 /* We don't unchain the node we want to delete. Instead, we overwrite
401 it with its successor and unchain the successor. If there is no
402 successor, we really unchain the node to be deleted. */
409 if (q == NULL || r == NULL)
413 node *parent = rootp, *up = &root->right;
418 nodestack[sp++] = parent;
420 if ((*up)->left == NULL)
427 /* We know that either the left or right successor of UNCHAINED is NULL.
428 R becomes the other one, it is chained into the parent of UNCHAINED. */
431 r = unchained->right;
436 q = *nodestack[sp-1];
437 if (unchained == q->right)
443 if (unchained != root)
444 root->key = unchained->key;
447 /* Now we lost a black edge, which means that the number of black
448 edges on every path is no longer constant. We must balance the
450 /* NODESTACK now contains all parents of R. R is likely to be NULL
451 in the first iteration. */
452 /* NULL nodes are considered black throughout - this is necessary for
454 while (sp > 0 && (r == NULL || !r->red))
456 node *pp = nodestack[sp - 1];
458 /* Two symmetric cases. */
461 /* Q is R's brother, P is R's parent. The subtree with root
462 R has one black edge less than the subtree with root Q. */
466 /* If Q is red, we know that P is black. We rotate P left
467 so that Q becomes the top node in the tree, with P below
468 it. P is colored red, Q is colored black.
469 This action does not change the black edge count for any
470 leaf in the tree, but we will be able to recognize one
471 of the following situations, which all require that Q
479 /* Make sure pp is right if the case below tries to use
481 nodestack[sp++] = pp = &q->left;
484 /* We know that Q can't be NULL here. We also know that Q is
486 if ((q->left == NULL || !q->left->red)
487 && (q->right == NULL || !q->right->red))
489 /* Q has two black successors. We can simply color Q red.
490 The whole subtree with root P is now missing one black
491 edge. Note that this action can temporarily make the
492 tree invalid (if P is red). But we will exit the loop
493 in that case and set P black, which both makes the tree
494 valid and also makes the black edge count come out
495 right. If P is black, we are at least one step closer
496 to the root and we'll try again the next iteration. */
502 /* Q is black, one of Q's successors is red. We can
503 repair the tree with one operation and will exit the
505 if (q->right == NULL || !q->right->red)
507 /* The left one is red. We perform the same action as
508 in maybe_split_for_insert where two red edges are
509 adjacent but point in different directions:
510 Q's left successor (let's call it Q2) becomes the
511 top of the subtree we are looking at, its parent (Q)
512 and grandparent (P) become its successors. The former
513 successors of Q2 are placed below P and Q.
514 P becomes black, and Q2 gets the color that P had.
515 This changes the black edge count only for node R and
528 /* It's the right one. Rotate P left. P becomes black,
529 and Q gets the color that P had. Q's right successor
530 also becomes black. This changes the black edge
531 count only for node R and its successors. */
550 /* Comments: see above. */
559 nodestack[sp++] = pp = &q->right;
562 if ((q->right == NULL || !q->right->red)
563 && (q->left == NULL || !q->left->red))
570 if (q->left == NULL || !q->left->red)
604 weak_alias (__tdelete, tdelete)
608 /* Walk the nodes of a tree.
609 ROOT is the root of the tree to be walked, ACTION the function to be
610 called at each node. LEVEL is the level of ROOT in the whole tree. */
613 trecurse (const void *vroot, __action_fn_t action, int level)
615 const_node root = (const_node) vroot;
617 if (root->left == NULL && root->right == NULL)
618 (*action) (root, leaf, level);
621 (*action) (root, preorder, level);
622 if (root->left != NULL)
623 trecurse (root->left, action, level + 1);
624 (*action) (root, postorder, level);
625 if (root->right != NULL)
626 trecurse (root->right, action, level + 1);
627 (*action) (root, endorder, level);
632 /* Walk the nodes of a tree.
633 ROOT is the root of the tree to be walked, ACTION the function to be
634 called at each node. */
636 __twalk (const void *vroot, __action_fn_t action)
638 const_node root = (const_node) vroot;
642 if (root != NULL && action != NULL)
643 trecurse (root, action, 0);
646 weak_alias (__twalk, twalk)
652 /* The standardized functions miss an important functionality: the
653 tree cannot be removed easily. We provide a function to do this. */
656 tdestroy_recurse (node root, __free_fn_t freefct)
658 if (root->left != NULL)
659 tdestroy_recurse (root->left, freefct);
660 if (root->right != NULL)
661 tdestroy_recurse (root->right, freefct);
662 (*freefct) ((void *) root->key);
663 /* Free the node itself. */
668 __tdestroy (void *vroot, __free_fn_t freefct)
670 node root = (node) vroot;
675 tdestroy_recurse (root, freefct);
677 weak_alias (__tdestroy, tdestroy)