/* Ordered {set,map} data type implemented by a binary tree. Copyright (C) 2006-2007, 2009-2023 Free Software Foundation, Inc. Written by Bruno Haible , 2006. This file is free software: you can redistribute it and/or modify it under the terms of the GNU Lesser General Public License as published by the Free Software Foundation; either version 2.1 of the License, or (at your option) any later version. This file is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU Lesser General Public License for more details. You should have received a copy of the GNU Lesser General Public License along with this program. If not, see . */ /* A red-black tree is a binary tree where every node is colored black or red such that 1. The root is black. 2. No red node has a red parent. Or equivalently: No red node has a red child. 3. All paths from the root down to any NULL endpoint contain the same number of black nodes. Let's call this the "black-height" bh of the tree. It follows that every such path contains exactly bh black and between 0 and bh red nodes. (The extreme cases are a path containing only black nodes, and a path colored alternately black-red-black-red-...-black-red.) The height of the tree therefore is >= bh, <= 2*bh. */ /* Color of a node. */ typedef enum color { BLACK, RED } color_t; /* Tree node implementation, valid for this file only. */ struct NODE_IMPL { struct NODE_IMPL *left; /* left branch, or NULL */ struct NODE_IMPL *right; /* right branch, or NULL */ /* Parent pointer, or NULL. The parent pointer is not needed for most operations. It is needed so that a NODE_T can be returned without memory allocation, on which the functions _remove_node, _add_before, _add_after can be implemented. */ struct NODE_IMPL *parent; color_t color; /* node's color */ NODE_PAYLOAD_FIELDS }; typedef struct NODE_IMPL * NODE_T; /* Concrete CONTAINER_IMPL type, valid for this file only. */ struct CONTAINER_IMPL { struct CONTAINER_IMPL_BASE base; struct NODE_IMPL *root; /* root node or NULL */ size_t count; /* number of nodes */ }; /* A red-black tree of height h has a black-height bh >= ceil(h/2) and therefore at least 2^ceil(h/2) - 1 elements. So, h <= 116 (because a tree of height h >= 117 would have at least 2^59 - 1 elements, and because even on 64-bit machines, sizeof (NODE_IMPL) * (2^59 - 1) > 2^64 this would exceed the address space of the machine. */ #define MAXHEIGHT 116 /* Rotates left a subtree. B D / \ / \ A D --> B E / \ / \ C E A C Changes the tree structure, updates the branch sizes. The caller must update the colors and register D as child of its parent. */ static NODE_T rotate_left (NODE_T b_node, NODE_T d_node) { NODE_T c_node = d_node->left; b_node->right = c_node; d_node->left = b_node; d_node->parent = b_node->parent; b_node->parent = d_node; if (c_node != NULL) c_node->parent = b_node; return d_node; } /* Rotates right a subtree. D B / \ / \ B E --> A D / \ / \ A C C E Changes the tree structure, updates the branch sizes. The caller must update the colors and register B as child of its parent. */ static NODE_T rotate_right (NODE_T b_node, NODE_T d_node) { NODE_T c_node = b_node->right; d_node->left = c_node; b_node->right = d_node; b_node->parent = d_node->parent; d_node->parent = b_node; if (c_node != NULL) c_node->parent = d_node; return b_node; } /* Ensures the tree is balanced, after an insertion operation. Also assigns node->color. parent is the given node's parent, known to be non-NULL. */ static void rebalance_after_add (CONTAINER_T container, NODE_T node, NODE_T parent) { for (;;) { /* At this point, parent = node->parent != NULL. Think of node->color being RED (although node->color is not yet assigned.) */ NODE_T grandparent; NODE_T uncle; if (parent->color == BLACK) { /* A RED color for node is acceptable. */ node->color = RED; return; } grandparent = parent->parent; /* Since parent is RED, we know that grandparent is != NULL and colored BLACK. */ if (grandparent->left == parent) uncle = grandparent->right; else if (grandparent->right == parent) uncle = grandparent->left; else abort (); if (uncle != NULL && uncle->color == RED) { /* Change grandparent from BLACK to RED, and change parent and uncle from RED to BLACK. This makes it acceptable for node to be RED. */ node->color = RED; parent->color = uncle->color = BLACK; node = grandparent; } else { /* grandparent and uncle are BLACK. parent is RED. node wants to be RED too. In this case, recoloring is not sufficient. Need to perform one or two rotations. */ NODE_T *grandparentp; if (grandparent->parent == NULL) grandparentp = &container->root; else if (grandparent->parent->left == grandparent) grandparentp = &grandparent->parent->left; else if (grandparent->parent->right == grandparent) grandparentp = &grandparent->parent->right; else abort (); if (grandparent->left == parent) { if (parent->right == node) { /* Rotation between node and parent. */ grandparent->left = rotate_left (parent, node); node = parent; parent = grandparent->left; } /* grandparent and uncle are BLACK. parent and node want to be RED. parent = grandparent->left. node = parent->left. grandparent parent bh+1 bh+1 / \ / \ parent uncle --> node grandparent bh bh bh bh / \ / \ node C C uncle bh bh bh bh */ *grandparentp = rotate_right (parent, grandparent); parent->color = BLACK; node->color = grandparent->color = RED; } else /* grandparent->right == parent */ { if (parent->left == node) { /* Rotation between node and parent. */ grandparent->right = rotate_right (node, parent); node = parent; parent = grandparent->right; } /* grandparent and uncle are BLACK. parent and node want to be RED. parent = grandparent->right. node = parent->right. grandparent parent bh+1 bh+1 / \ / \ uncle parent --> grandparent node bh bh bh bh / \ / \ C node uncle C bh bh bh bh */ *grandparentp = rotate_left (grandparent, parent); parent->color = BLACK; node->color = grandparent->color = RED; } return; } /* Start again with a new (node, parent) pair. */ parent = node->parent; if (parent == NULL) { /* Change node's color from RED to BLACK. This increases the tree's black-height. */ node->color = BLACK; return; } } } /* Ensures the tree is balanced, after a deletion operation. CHILD was a grandchild of PARENT and is now its child. Between them, a black node was removed. CHILD is also black, or NULL. (CHILD can also be NULL. But PARENT is non-NULL.) */ static void rebalance_after_remove (CONTAINER_T container, NODE_T child, NODE_T parent) { for (;;) { /* At this point, we reduced the black-height of the CHILD subtree by 1. To make up, either look for a possibility to turn a RED to a BLACK node, or try to reduce the black-height tree of CHILD's sibling subtree as well. */ NODE_T *parentp; if (parent->parent == NULL) parentp = &container->root; else if (parent->parent->left == parent) parentp = &parent->parent->left; else if (parent->parent->right == parent) parentp = &parent->parent->right; else abort (); if (parent->left == child) { NODE_T sibling = parent->right; /* sibling's black-height is >= 1. In particular, sibling != NULL. parent / \ child sibling bh bh+1 */ if (sibling->color == RED) { /* sibling is RED, hence parent is BLACK and sibling's children are non-NULL and BLACK. parent sibling bh+2 bh+2 / \ / \ child sibling --> parent SR bh bh+1 bh+1 bh+1 / \ / \ SL SR child SL bh+1 bh+1 bh bh+1 */ *parentp = rotate_left (parent, sibling); parent->color = RED; sibling->color = BLACK; /* Concentrate on the subtree of parent. The new sibling is one of the old sibling's children, and known to be BLACK. */ parentp = &sibling->left; sibling = parent->right; } /* Now we know that sibling is BLACK. parent / \ child sibling bh bh+1 */ if (sibling->right != NULL && sibling->right->color == RED) { /* parent sibling bh+1|bh+2 bh+1|bh+2 / \ / \ child sibling --> parent SR bh bh+1 bh+1 bh+1 / \ / \ SL SR child SL bh bh bh bh */ *parentp = rotate_left (parent, sibling); sibling->color = parent->color; parent->color = BLACK; sibling->right->color = BLACK; return; } else if (sibling->left != NULL && sibling->left->color == RED) { /* parent parent bh+1|bh+2 bh+1|bh+2 / \ / \ child sibling --> child SL bh bh+1 bh bh+1 / \ / \ SL SR SLL sibling bh bh bh bh / \ / \ SLL SLR SLR SR bh bh bh bh where SLL, SLR, SR are all black. */ parent->right = rotate_right (sibling->left, sibling); /* Change sibling from BLACK to RED and SL from RED to BLACK. */ sibling->color = RED; sibling = parent->right; sibling->color = BLACK; /* Now do as in the previous case. */ *parentp = rotate_left (parent, sibling); sibling->color = parent->color; parent->color = BLACK; sibling->right->color = BLACK; return; } else { if (parent->color == BLACK) { /* Change sibling from BLACK to RED. Then the entire subtree at parent has decreased its black-height. parent parent bh+2 bh+1 / \ / \ child sibling --> child sibling bh bh+1 bh bh */ sibling->color = RED; child = parent; } else { /* Change parent from RED to BLACK, but compensate by changing sibling from BLACK to RED. parent parent bh+1 bh+1 / \ / \ child sibling --> child sibling bh bh+1 bh bh */ parent->color = BLACK; sibling->color = RED; return; } } } else if (parent->right == child) { NODE_T sibling = parent->left; /* sibling's black-height is >= 1. In particular, sibling != NULL. parent / \ sibling child bh+1 bh */ if (sibling->color == RED) { /* sibling is RED, hence parent is BLACK and sibling's children are non-NULL and BLACK. parent sibling bh+2 bh+2 / \ / \ sibling child --> SR parent bh+1 ch bh+1 bh+1 / \ / \ SL SR SL child bh+1 bh+1 bh+1 bh */ *parentp = rotate_right (sibling, parent); parent->color = RED; sibling->color = BLACK; /* Concentrate on the subtree of parent. The new sibling is one of the old sibling's children, and known to be BLACK. */ parentp = &sibling->right; sibling = parent->left; } /* Now we know that sibling is BLACK. parent / \ sibling child bh+1 bh */ if (sibling->left != NULL && sibling->left->color == RED) { /* parent sibling bh+1|bh+2 bh+1|bh+2 / \ / \ sibling child --> SL parent bh+1 bh bh+1 bh+1 / \ / \ SL SR SR child bh bh bh bh */ *parentp = rotate_right (sibling, parent); sibling->color = parent->color; parent->color = BLACK; sibling->left->color = BLACK; return; } else if (sibling->right != NULL && sibling->right->color == RED) { /* parent parent bh+1|bh+2 bh+1|bh+2 / \ / \ sibling child --> SR child bh+1 bh bh+1 bh / \ / \ SL SR sibling SRR bh bh bh bh / \ / \ SRL SRR SL SRL bh bh bh bh where SL, SRL, SRR are all black. */ parent->left = rotate_left (sibling, sibling->right); /* Change sibling from BLACK to RED and SL from RED to BLACK. */ sibling->color = RED; sibling = parent->left; sibling->color = BLACK; /* Now do as in the previous case. */ *parentp = rotate_right (sibling, parent); sibling->color = parent->color; parent->color = BLACK; sibling->left->color = BLACK; return; } else { if (parent->color == BLACK) { /* Change sibling from BLACK to RED. Then the entire subtree at parent has decreased its black-height. parent parent bh+2 bh+1 / \ / \ sibling child --> sibling child bh+1 bh bh bh */ sibling->color = RED; child = parent; } else { /* Change parent from RED to BLACK, but compensate by changing sibling from BLACK to RED. parent parent bh+1 bh+1 / \ / \ sibling child --> sibling child bh+1 bh bh bh */ parent->color = BLACK; sibling->color = RED; return; } } } else abort (); /* Start again with a new (child, parent) pair. */ parent = child->parent; #if 0 /* Already handled. */ if (child != NULL && child->color == RED) { child->color = BLACK; return; } #endif if (parent == NULL) return; } } static NODE_T gl_tree_nx_add_first (CONTAINER_T container, NODE_PAYLOAD_PARAMS) { /* Create new node. */ NODE_T new_node = (struct NODE_IMPL *) malloc (sizeof (struct NODE_IMPL)); if (new_node == NULL) return NULL; new_node->left = NULL; new_node->right = NULL; NODE_PAYLOAD_ASSIGN(new_node) /* Add it to the tree. */ if (container->root == NULL) { new_node->color = BLACK; container->root = new_node; new_node->parent = NULL; } else { NODE_T node; for (node = container->root; node->left != NULL; ) node = node->left; node->left = new_node; new_node->parent = node; /* Color and rebalance. */ rebalance_after_add (container, new_node, node); } container->count++; return new_node; } /* Adds the already allocated NEW_NODE to the tree, right before NODE. */ static void gl_tree_add_node_before (CONTAINER_T container, NODE_T node, NODE_T new_node) { new_node->left = NULL; new_node->right = NULL; /* Add it to the tree. */ if (node->left == NULL) node->left = new_node; else { for (node = node->left; node->right != NULL; ) node = node->right; node->right = new_node; } new_node->parent = node; /* Color and rebalance. */ rebalance_after_add (container, new_node, node); container->count++; } static NODE_T gl_tree_nx_add_before (CONTAINER_T container, NODE_T node, NODE_PAYLOAD_PARAMS) { /* Create new node. */ NODE_T new_node = (struct NODE_IMPL *) malloc (sizeof (struct NODE_IMPL)); if (new_node == NULL) return NULL; NODE_PAYLOAD_ASSIGN(new_node) gl_tree_add_node_before (container, node, new_node); return new_node; } /* Adds the already allocated NEW_NODE to the tree, right after NODE. */ static void gl_tree_add_node_after (CONTAINER_T container, NODE_T node, NODE_T new_node) { new_node->left = NULL; new_node->right = NULL; /* Add it to the tree. */ if (node->right == NULL) node->right = new_node; else { for (node = node->right; node->left != NULL; ) node = node->left; node->left = new_node; } new_node->parent = node; /* Color and rebalance. */ rebalance_after_add (container, new_node, node); container->count++; } static NODE_T gl_tree_nx_add_after (CONTAINER_T container, NODE_T node, NODE_PAYLOAD_PARAMS) { /* Create new node. */ NODE_T new_node = (struct NODE_IMPL *) malloc (sizeof (struct NODE_IMPL)); if (new_node == NULL) return NULL; NODE_PAYLOAD_ASSIGN(new_node) gl_tree_add_node_after (container, node, new_node); return new_node; } static void gl_tree_remove_node_no_free (CONTAINER_T container, NODE_T node) { NODE_T parent = node->parent; if (node->left == NULL) { /* Replace node with node->right. */ NODE_T child = node->right; if (child != NULL) { child->parent = parent; /* Since node->left == NULL, child must be RED and of height 1, hence node must have been BLACK. Recolor the child. */ child->color = BLACK; } if (parent == NULL) container->root = child; else { if (parent->left == node) parent->left = child; else /* parent->right == node */ parent->right = child; if (child == NULL && node->color == BLACK) rebalance_after_remove (container, child, parent); } } else if (node->right == NULL) { /* It is not absolutely necessary to treat this case. But the more general case below is more complicated, hence slower. */ /* Replace node with node->left. */ NODE_T child = node->left; child->parent = parent; /* Since node->right == NULL, child must be RED and of height 1, hence node must have been BLACK. Recolor the child. */ child->color = BLACK; if (parent == NULL) container->root = child; else { if (parent->left == node) parent->left = child; else /* parent->right == node */ parent->right = child; } } else { /* Replace node with the rightmost element of the node->left subtree. */ NODE_T subst; NODE_T subst_parent; NODE_T child; color_t removed_color; for (subst = node->left; subst->right != NULL; ) subst = subst->right; subst_parent = subst->parent; child = subst->left; removed_color = subst->color; /* The case subst_parent == node is special: If we do nothing special, we get confusion about node->left, subst->left and child->parent. subst_parent == node <==> The 'for' loop above terminated immediately. <==> subst == subst_parent->left [otherwise subst == subst_parent->right] In this case, we would need to first set child->parent = node; node->left = child; and later - when we copy subst into node's position - again child->parent = subst; subst->left = child; Altogether a no-op. */ if (subst_parent != node) { if (child != NULL) child->parent = subst_parent; subst_parent->right = child; } /* Copy subst into node's position. (This is safer than to copy subst's value into node, keep node in place, and free subst.) */ if (subst_parent != node) { subst->left = node->left; subst->left->parent = subst; } subst->right = node->right; subst->right->parent = subst; subst->color = node->color; subst->parent = parent; if (parent == NULL) container->root = subst; else if (parent->left == node) parent->left = subst; else /* parent->right == node */ parent->right = subst; if (removed_color == BLACK) { if (child != NULL && child->color == RED) /* Recolor the child. */ child->color = BLACK; else /* Rebalancing starts at child's parent, that is subst_parent - except when subst_parent == node. In this case, we need to use its replacement, subst. */ rebalance_after_remove (container, child, subst_parent != node ? subst_parent : subst); } } container->count--; } static bool gl_tree_remove_node (CONTAINER_T container, NODE_T node) { gl_tree_remove_node_no_free (container, node); NODE_PAYLOAD_DISPOSE (container, node) free (node); return true; } /* For debugging. */ static unsigned int check_invariants (NODE_T node, NODE_T parent, size_t *counterp) { unsigned int left_blackheight = (node->left != NULL ? check_invariants (node->left, node, counterp) : 0); unsigned int right_blackheight = (node->right != NULL ? check_invariants (node->right, node, counterp) : 0); if (!(node->parent == parent)) abort (); if (!(node->color == BLACK || node->color == RED)) abort (); if (parent == NULL && !(node->color == BLACK)) abort (); if (!(left_blackheight == right_blackheight)) abort (); (*counterp)++; return left_blackheight + (node->color == BLACK ? 1 : 0); }