1 /* Copyright (C) 1995-1997, 2000, 2006-2007 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
99 typedef int (*__compar_fn_t) (const void *, const void *);
100 typedef void (*__action_fn_t) (const void *, VISIT, int);
103 # define __tsearch tsearch
104 # define __tfind tfind
105 # define __tdelete tdelete
106 # define __twalk twalk
109 #ifndef internal_function
110 /* Inside GNU libc we mark some function in a special way. In other
111 environments simply ignore the marking. */
112 # define internal_function
115 typedef struct node_t
117 /* Callers expect this to be the first element in the structure - do not
121 struct node_t *right;
124 typedef const struct node_t *const_node;
130 /* Routines to check tree invariants. */
134 #define CHECK_TREE(a) check_tree(a)
137 check_tree_recurse (node p, int d_sofar, int d_total)
141 assert (d_sofar == d_total);
145 check_tree_recurse (p->left, d_sofar + (p->left && !p->left->red), d_total);
146 check_tree_recurse (p->right, d_sofar + (p->right && !p->right->red), d_total);
148 assert (!(p->left->red && p->red));
150 assert (!(p->right->red && p->red));
154 check_tree (node root)
161 for(p = root->left; p; p = p->left)
163 check_tree_recurse (root, 0, cnt);
169 #define CHECK_TREE(a)
173 /* Possibly "split" a node with two red successors, and/or fix up two red
174 edges in a row. ROOTP is a pointer to the lowest node we visited, PARENTP
175 and GPARENTP pointers to its parent/grandparent. P_R and GP_R contain the
176 comparison values that determined which way was taken in the tree to reach
177 ROOTP. MODE is 1 if we need not do the split, but must check for two red
178 edges between GPARENTP and ROOTP. */
180 maybe_split_for_insert (node *rootp, node *parentp, node *gparentp,
181 int p_r, int gp_r, int mode)
185 rp = &(*rootp)->right;
186 lp = &(*rootp)->left;
188 /* See if we have to split this node (both successors red). */
190 || ((*rp) != NULL && (*lp) != NULL && (*rp)->red && (*lp)->red))
192 /* This node becomes red, its successors black. */
199 /* If the parent of this node is also red, we have to do
201 if (parentp != NULL && (*parentp)->red)
205 /* There are two main cases:
206 1. The edge types (left or right) of the two red edges differ.
207 2. Both red edges are of the same type.
208 There exist two symmetries of each case, so there is a total of
210 if ((p_r > 0) != (gp_r > 0))
212 /* Put the child at the top of the tree, with its parent
213 and grandparent as successors. */
219 /* Child is left of parent. */
227 /* Child is right of parent. */
237 *gparentp = *parentp;
238 /* Parent becomes the top of the tree, grandparent and
239 child are its successors. */
259 /* Find or insert datum into search tree.
260 KEY is the key to be located, ROOTP is the address of tree root,
261 COMPAR the ordering function. */
263 __tsearch (const void *key, void **vrootp, __compar_fn_t compar)
266 node *parentp = NULL, *gparentp = NULL;
267 node *rootp = (node *) vrootp;
269 int r = 0, p_r = 0, gp_r = 0; /* No they might not, Mr Compiler. */
274 /* This saves some additional tests below. */
281 while (*nextp != NULL)
284 r = (*compar) (key, root->key);
288 maybe_split_for_insert (rootp, parentp, gparentp, p_r, gp_r, 0);
289 /* If that did any rotations, parentp and gparentp are now garbage.
290 That doesn't matter, because the values they contain are never
291 used again in that case. */
293 nextp = r < 0 ? &root->left : &root->right;
305 q = (struct node_t *) malloc (sizeof (struct node_t));
308 *nextp = q; /* link new node to old */
309 q->key = key; /* initialize new node */
311 q->left = q->right = NULL;
314 /* There may be two red edges in a row now, which we must avoid by
315 rotating the tree. */
316 maybe_split_for_insert (nextp, rootp, parentp, r, p_r, 1);
322 weak_alias (__tsearch, tsearch)
326 /* Find datum in search tree.
327 KEY is the key to be located, ROOTP is the address of tree root,
328 COMPAR the ordering function. */
330 __tfind (key, vrootp, compar)
333 __compar_fn_t compar;
335 node *rootp = (node *) vrootp;
342 while (*rootp != NULL)
347 r = (*compar) (key, root->key);
351 rootp = r < 0 ? &root->left : &root->right;
356 weak_alias (__tfind, tfind)
360 /* Delete node with given key.
361 KEY is the key to be deleted, ROOTP is the address of the root of tree,
362 COMPAR the comparison function. */
364 __tdelete (const void *key, void **vrootp, __compar_fn_t compar)
366 node p, q, r, retval;
368 node *rootp = (node *) vrootp;
369 node root, unchained;
370 /* Stack of nodes so we remember the parents without recursion. It's
371 _very_ unlikely that there are paths longer than 40 nodes. The tree
372 would need to have around 250.000 nodes. */
375 node *nodestack[100];
385 while ((cmp = (*compar) (key, (*rootp)->key)) != 0)
390 nodestack[sp++] = rootp;
399 /* This is bogus if the node to be deleted is the root... this routine
400 really should return an integer with 0 for success, -1 for failure
401 and errno = ESRCH or something. */
404 /* We don't unchain the node we want to delete. Instead, we overwrite
405 it with its successor and unchain the successor. If there is no
406 successor, we really unchain the node to be deleted. */
413 if (q == NULL || r == NULL)
417 node *parent = rootp, *up = &root->right;
422 nodestack[sp++] = parent;
424 if ((*up)->left == NULL)
431 /* We know that either the left or right successor of UNCHAINED is NULL.
432 R becomes the other one, it is chained into the parent of UNCHAINED. */
435 r = unchained->right;
440 q = *nodestack[sp-1];
441 if (unchained == q->right)
447 if (unchained != root)
448 root->key = unchained->key;
451 /* Now we lost a black edge, which means that the number of black
452 edges on every path is no longer constant. We must balance the
454 /* NODESTACK now contains all parents of R. R is likely to be NULL
455 in the first iteration. */
456 /* NULL nodes are considered black throughout - this is necessary for
458 while (sp > 0 && (r == NULL || !r->red))
460 node *pp = nodestack[sp - 1];
462 /* Two symmetric cases. */
465 /* Q is R's brother, P is R's parent. The subtree with root
466 R has one black edge less than the subtree with root Q. */
470 /* If Q is red, we know that P is black. We rotate P left
471 so that Q becomes the top node in the tree, with P below
472 it. P is colored red, Q is colored black.
473 This action does not change the black edge count for any
474 leaf in the tree, but we will be able to recognize one
475 of the following situations, which all require that Q
483 /* Make sure pp is right if the case below tries to use
485 nodestack[sp++] = pp = &q->left;
488 /* We know that Q can't be NULL here. We also know that Q is
490 if ((q->left == NULL || !q->left->red)
491 && (q->right == NULL || !q->right->red))
493 /* Q has two black successors. We can simply color Q red.
494 The whole subtree with root P is now missing one black
495 edge. Note that this action can temporarily make the
496 tree invalid (if P is red). But we will exit the loop
497 in that case and set P black, which both makes the tree
498 valid and also makes the black edge count come out
499 right. If P is black, we are at least one step closer
500 to the root and we'll try again the next iteration. */
506 /* Q is black, one of Q's successors is red. We can
507 repair the tree with one operation and will exit the
509 if (q->right == NULL || !q->right->red)
511 /* The left one is red. We perform the same action as
512 in maybe_split_for_insert where two red edges are
513 adjacent but point in different directions:
514 Q's left successor (let's call it Q2) becomes the
515 top of the subtree we are looking at, its parent (Q)
516 and grandparent (P) become its successors. The former
517 successors of Q2 are placed below P and Q.
518 P becomes black, and Q2 gets the color that P had.
519 This changes the black edge count only for node R and
532 /* It's the right one. Rotate P left. P becomes black,
533 and Q gets the color that P had. Q's right successor
534 also becomes black. This changes the black edge
535 count only for node R and its successors. */
554 /* Comments: see above. */
563 nodestack[sp++] = pp = &q->right;
566 if ((q->right == NULL || !q->right->red)
567 && (q->left == NULL || !q->left->red))
574 if (q->left == NULL || !q->left->red)
608 weak_alias (__tdelete, tdelete)
612 /* Walk the nodes of a tree.
613 ROOT is the root of the tree to be walked, ACTION the function to be
614 called at each node. LEVEL is the level of ROOT in the whole tree. */
617 trecurse (const void *vroot, __action_fn_t action, int level)
619 const_node root = (const_node) vroot;
621 if (root->left == NULL && root->right == NULL)
622 (*action) (root, leaf, level);
625 (*action) (root, preorder, level);
626 if (root->left != NULL)
627 trecurse (root->left, action, level + 1);
628 (*action) (root, postorder, level);
629 if (root->right != NULL)
630 trecurse (root->right, action, level + 1);
631 (*action) (root, endorder, level);
636 /* Walk the nodes of a tree.
637 ROOT is the root of the tree to be walked, ACTION the function to be
638 called at each node. */
640 __twalk (const void *vroot, __action_fn_t action)
642 const_node root = (const_node) vroot;
646 if (root != NULL && action != NULL)
647 trecurse (root, action, 0);
650 weak_alias (__twalk, twalk)
656 /* The standardized functions miss an important functionality: the
657 tree cannot be removed easily. We provide a function to do this. */
660 tdestroy_recurse (node root, __free_fn_t freefct)
662 if (root->left != NULL)
663 tdestroy_recurse (root->left, freefct);
664 if (root->right != NULL)
665 tdestroy_recurse (root->right, freefct);
666 (*freefct) ((void *) root->key);
667 /* Free the node itself. */
672 __tdestroy (void *vroot, __free_fn_t freefct)
674 node root = (node) vroot;
679 tdestroy_recurse (root, freefct);
681 weak_alias (__tdestroy, tdestroy)