/***
This file is part of systemd. See COPYING for details.
systemd 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.
systemd 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 systemd; If not, see .
***/
/*
* RB-Tree Implementation
* This implements the insertion/removal of elements in RB-Trees. You're highly
* recommended to have an RB-Tree documentation at hand when reading this. Both
* insertion and removal can be split into a handful of situations that can
* occur. Those situations are enumerated as "Case 1" to "Case n" here, and
* follow closely the cases described in most RB-Tree documentations. This file
* does not explain why it is enough to handle just those cases, nor does it
* provide a proof of correctness. Dig out your algorithm 101 handbook if
* you're interested.
*
* This implementation is *not* straightforward. Usually, a handful of
* rotation, reparent, swap and link helpers can be used to implement the
* rebalance operations. However, those often perform unnecessary writes.
* Therefore, this implementation hard-codes all the operations. You're highly
* recommended to look at the two basic helpers before reading the code:
* c_rbtree_swap_child()
* c_rbtree_set_parent_and_color()
* Those are the only helpers used, hence, you should really know what they do
* before digging into the code.
*
* For a highlevel documentation of the API, see the header file and docbook
* comments.
*/
#include
#include
#include "c-rbtree.h"
enum {
C_RBNODE_RED = 0,
C_RBNODE_BLACK = 1,
};
static inline unsigned long c_rbnode_color(CRBNode *n) {
return (unsigned long)n->__parent_and_color & 1UL;
}
static inline _Bool c_rbnode_is_red(CRBNode *n) {
return c_rbnode_color(n) == C_RBNODE_RED;
}
static inline _Bool c_rbnode_is_black(CRBNode *n) {
return c_rbnode_color(n) == C_RBNODE_BLACK;
}
/**
* c_rbnode_leftmost() - return leftmost child
* @n: current node, or NULL
*
* This returns the leftmost child of @n. If @n is NULL, this will return NULL.
* In all other cases, this function returns a valid pointer. That is, if @n
* does not have any left children, this returns @n.
*
* Worst case runtime (n: number of elements in tree): O(log(n))
*
* Return: Pointer to leftmost child, or NULL.
*/
CRBNode *c_rbnode_leftmost(CRBNode *n) {
if (n)
while (n->left)
n = n->left;
return n;
}
/**
* c_rbnode_rightmost() - return rightmost child
* @n: current node, or NULL
*
* This returns the rightmost child of @n. If @n is NULL, this will return
* NULL. In all other cases, this function returns a valid pointer. That is, if
* @n does not have any right children, this returns @n.
*
* Worst case runtime (n: number of elements in tree): O(log(n))
*
* Return: Pointer to rightmost child, or NULL.
*/
CRBNode *c_rbnode_rightmost(CRBNode *n) {
if (n)
while (n->right)
n = n->right;
return n;
}
/**
* c_rbnode_next() - return next node
* @n: current node, or NULL
*
* An RB-Tree always defines a linear order of its elements. This function
* returns the logically next node to @n. If @n is NULL, the last node or
* unlinked, this returns NULL.
*
* Worst case runtime (n: number of elements in tree): O(log(n))
*
* Return: Pointer to next node, or NULL.
*/
CRBNode *c_rbnode_next(CRBNode *n) {
CRBNode *p;
if (!c_rbnode_is_linked(n))
return NULL;
if (n->right)
return c_rbnode_leftmost(n->right);
while ((p = c_rbnode_parent(n)) && n == p->right)
n = p;
return p;
}
/**
* c_rbnode_prev() - return previous node
* @n: current node, or NULL
*
* An RB-Tree always defines a linear order of its elements. This function
* returns the logically previous node to @n. If @n is NULL, the first node or
* unlinked, this returns NULL.
*
* Worst case runtime (n: number of elements in tree): O(log(n))
*
* Return: Pointer to previous node, or NULL.
*/
CRBNode *c_rbnode_prev(CRBNode *n) {
CRBNode *p;
if (!c_rbnode_is_linked(n))
return NULL;
if (n->left)
return c_rbnode_rightmost(n->left);
while ((p = c_rbnode_parent(n)) && n == p->left)
n = p;
return p;
}
/**
* c_rbtree_first() - return first node
* @t: tree to operate on
*
* An RB-Tree always defines a linear order of its elements. This function
* returns the logically first node in @t. If @t is empty, NULL is returned.
*
* Fixed runtime (n: number of elements in tree): O(log(n))
*
* Return: Pointer to first node, or NULL.
*/
CRBNode *c_rbtree_first(CRBTree *t) {
assert(t);
return c_rbnode_leftmost(t->root);
}
/**
* c_rbtree_last() - return last node
* @t: tree to operate on
*
* An RB-Tree always defines a linear order of its elements. This function
* returns the logically last node in @t. If @t is empty, NULL is returned.
*
* Fixed runtime (n: number of elements in tree): O(log(n))
*
* Return: Pointer to last node, or NULL.
*/
CRBNode *c_rbtree_last(CRBTree *t) {
assert(t);
return c_rbnode_rightmost(t->root);
}
/*
* Set the color and parent of a node. This should be treated as a simple
* assignment of the 'color' and 'parent' fields of the node. No other magic is
* applied. But since both fields share its backing memory, this helper
* function is provided.
*/
static inline void c_rbnode_set_parent_and_color(CRBNode *n, CRBNode *p, unsigned long c) {
assert(!((unsigned long)p & 1));
assert(c < 2);
n->__parent_and_color = (CRBNode*)((unsigned long)p | c);
}
/* same as c_rbnode_set_parent_and_color(), but keeps the current parent */
static inline void c_rbnode_set_color(CRBNode *n, unsigned long c) {
c_rbnode_set_parent_and_color(n, c_rbnode_parent(n), c);
}
/* same as c_rbnode_set_parent_and_color(), but keeps the current color */
static inline void c_rbnode_set_parent(CRBNode *n, CRBNode *p) {
c_rbnode_set_parent_and_color(n, p, c_rbnode_color(n));
}
/*
* This function partially replaces an existing child pointer to a new one. The
* existing child must be given as @old, the new child as @new. @p must be the
* parent of @old (or NULL if it has no parent).
* This function ensures that the parent of @old now points to @new. However,
* it does *NOT* change the parent pointer of @new. The caller must ensure
* this.
* If @p is NULL, this function ensures that the root-pointer is adjusted
* instead (given as @t).
*/
static inline void c_rbtree_swap_child(CRBTree *t, CRBNode *p, CRBNode *old, CRBNode *new) {
if (p) {
if (p->left == old)
p->left = new;
else
p->right = new;
} else {
t->root = new;
}
}
static inline CRBNode *c_rbtree_paint_one(CRBTree *t, CRBNode *n) {
CRBNode *p, *g, *gg, *u, *x;
/*
* Paint a single node according to RB-Tree rules. The node must
* already be linked into the tree and painted red.
* We repaint the node or rotate the tree, if required. In case a
* recursive repaint is required, the next node to be re-painted
* is returned.
* p: parent
* g: grandparent
* gg: grandgrandparent
* u: uncle
* x: temporary
*/
/* node is red, so we can access the parent directly */
p = n->__parent_and_color;
if (!p) {
/* Case 1:
* We reached the root. Mark it black and be done. As all
* leaf-paths share the root, the ratio of black nodes on each
* path stays the same. */
c_rbnode_set_parent_and_color(n, p, C_RBNODE_BLACK);
n = NULL;
} else if (c_rbnode_is_black(p)) {
/* Case 2:
* The parent is already black. As our node is red, we did not
* change the number of black nodes on any path, nor do we have
* multiple consecutive red nodes. */
n = NULL;
} else if (p == p->__parent_and_color->left) { /* parent is red, so grandparent exists */
g = p->__parent_and_color;
gg = c_rbnode_parent(g);
u = g->right;
if (u && c_rbnode_is_red(u)) {
/* Case 3:
* Parent and uncle are both red. We know the
* grandparent must be black then. Repaint parent and
* uncle black, the grandparent red and recurse into
* the grandparent. */
c_rbnode_set_parent_and_color(p, g, C_RBNODE_BLACK);
c_rbnode_set_parent_and_color(u, g, C_RBNODE_BLACK);
c_rbnode_set_parent_and_color(g, gg, C_RBNODE_RED);
n = g;
} else {
/* parent is red, uncle is black */
if (n == p->right) {
/* Case 4:
* We're the right child. Rotate on parent to
* become left child, so we can handle it the
* same as case 5. */
x = n->left;
p->right = n->left;
n->left = p;
if (x)
c_rbnode_set_parent_and_color(x, p, C_RBNODE_BLACK);
c_rbnode_set_parent_and_color(p, n, C_RBNODE_RED);
p = n;
}
/* 'n' is invalid from here on! */
n = NULL;
/* Case 5:
* We're the red left child or a red parent, black
* grandparent and uncle. Rotate on grandparent and
* switch color with parent. Number of black nodes on
* each path stays the same, but we got rid of the
* double red path. As the grandparent is still black,
* we're done. */
x = p->right;
g->left = x;
p->right = g;
if (x)
c_rbnode_set_parent_and_color(x, g, C_RBNODE_BLACK);
c_rbnode_set_parent_and_color(p, gg, C_RBNODE_BLACK);
c_rbnode_set_parent_and_color(g, p, C_RBNODE_RED);
c_rbtree_swap_child(t, gg, g, p);
}
} else /* if (p == p->__parent_and_color->left) */ { /* same as above, but mirrored */
g = p->__parent_and_color;
gg = c_rbnode_parent(g);
u = g->left;
if (u && c_rbnode_is_red(u)) {
c_rbnode_set_parent_and_color(p, g, C_RBNODE_BLACK);
c_rbnode_set_parent_and_color(u, g, C_RBNODE_BLACK);
c_rbnode_set_parent_and_color(g, gg, C_RBNODE_RED);
n = g;
} else {
if (n == p->left) {
x = n->right;
p->left = n->right;
n->right = p;
if (x)
c_rbnode_set_parent_and_color(x, p, C_RBNODE_BLACK);
c_rbnode_set_parent_and_color(p, n, C_RBNODE_RED);
p = n;
}
n = NULL;
x = p->left;
g->right = x;
p->left = g;
if (x)
c_rbnode_set_parent_and_color(x, g, C_RBNODE_BLACK);
c_rbnode_set_parent_and_color(p, gg, C_RBNODE_BLACK);
c_rbnode_set_parent_and_color(g, p, C_RBNODE_RED);
c_rbtree_swap_child(t, gg, g, p);
}
}
return n;
}
static inline void c_rbtree_paint(CRBTree *t, CRBNode *n) {
assert(t);
assert(n);
while (n)
n = c_rbtree_paint_one(t, n);
}
/**
* c_rbtree_add() - add node to tree
* @t: tree to operate one
* @p: parent node to link under, or NULL
* @l: left/right slot of @p (or root) to link at
* @n: node to add
*
* This links @n into the tree given as @t. The caller must provide the exact
* spot where to link the node. That is, the caller must traverse the tree
* based on their search order. Once they hit a leaf where to insert the node,
* call this function to link it and rebalance the tree.
*
* A typical insertion would look like this (@t is your tree, @n is your node):
*
* CRBNode **i, *p;
*
* i = &t->root;
* p = NULL;
* while (*i) {
* p = *i;
* if (compare(n, *i) < 0)
* i = &(*i)->left;
* else
* i = &(*i)->right;
* }
*
* c_rbtree_add(t, p, i, n);
*
* Once the node is linked into the tree, a simple lookup on the same tree can
* be coded like this:
*
* CRBNode *i;
*
* i = t->root;
* while (i) {
* int v = compare(n, i);
* if (v < 0)
* i = (*i)->left;
* else if (v > 0)
* i = (*i)->right;
* else
* break;
* }
*
* When you add nodes to a tree, the memory contents of the node do not matter.
* That is, there is no need to initialize the node via c_rbnode_init().
* However, if you relink nodes multiple times during their lifetime, it is
* usually very convenient to use c_rbnode_init() and c_rbtree_remove_init().
* In those cases, you should validate that a node is unlinked before you call
* c_rbtree_add().
*/
void c_rbtree_add(CRBTree *t, CRBNode *p, CRBNode **l, CRBNode *n) {
assert(t);
assert(l);
assert(n);
assert(!p || l == &p->left || l == &p->right);
assert(p || l == &t->root);
c_rbnode_set_parent_and_color(n, p, C_RBNODE_RED);
n->left = n->right = NULL;
*l = n;
c_rbtree_paint(t, n);
}
static inline CRBNode *c_rbtree_rebalance_one(CRBTree *t, CRBNode *p, CRBNode *n) {
CRBNode *s, *x, *y, *g;
/*
* Rebalance tree after a node was removed. This happens only if you
* remove a black node and one path is now left with an unbalanced
* number or black nodes.
* This function assumes all paths through p and n have one black node
* less than all other paths. If recursive fixup is required, the
* current node is returned.
*/
if (n == p->left) {
s = p->right;
if (c_rbnode_is_red(s)) {
/* Case 3:
* We have a red node as sibling. Rotate it onto our
* side so we can later on turn it black. This way, we
* gain the additional black node in our path. */
g = c_rbnode_parent(p);
x = s->left;
p->right = x;
s->left = p;
c_rbnode_set_parent_and_color(x, p, C_RBNODE_BLACK);
c_rbnode_set_parent_and_color(s, g, c_rbnode_color(p));
c_rbnode_set_parent_and_color(p, s, C_RBNODE_RED);
c_rbtree_swap_child(t, g, p, s);
s = x;
}
x = s->right;
if (!x || c_rbnode_is_black(x)) {
y = s->left;
if (!y || c_rbnode_is_black(y)) {
/* Case 4:
* Our sibling is black and has only black
* children. Flip it red and turn parent black.
* This way we gained a black node in our path,
* or we fix it recursively one layer up, which
* will rotate the red sibling as parent. */
c_rbnode_set_parent_and_color(s, p, C_RBNODE_RED);
if (c_rbnode_is_black(p))
return p;
c_rbnode_set_parent_and_color(p, c_rbnode_parent(p), C_RBNODE_BLACK);
return NULL;
}
/* Case 5:
* Left child of our sibling is red, right one is black.
* Rotate on parent so the right child of our sibling is
* now red, and we can fall through to case 6. */
x = y->right;
s->left = y->right;
y->right = s;
p->right = y;
if (x)
c_rbnode_set_parent_and_color(x, s, C_RBNODE_BLACK);
x = s;
s = y;
}
/* Case 6:
* The right child of our sibling is red. Rotate left and flip
* colors, which gains us an additional black node in our path,
* that was previously on our sibling. */
g = c_rbnode_parent(p);
y = s->left;
p->right = y;
s->left = p;
c_rbnode_set_parent_and_color(x, s, C_RBNODE_BLACK);
if (y)
c_rbnode_set_parent_and_color(y, p, c_rbnode_color(y));
c_rbnode_set_parent_and_color(s, g, c_rbnode_color(p));
c_rbnode_set_parent_and_color(p, s, C_RBNODE_BLACK);
c_rbtree_swap_child(t, g, p, s);
} else /* if (!n || n == p->right) */ { /* same as above, but mirrored */
s = p->left;
if (c_rbnode_is_red(s)) {
g = c_rbnode_parent(p);
x = s->right;
p->left = x;
s->right = p;
c_rbnode_set_parent_and_color(x, p, C_RBNODE_BLACK);
c_rbnode_set_parent_and_color(s, g, C_RBNODE_BLACK);
c_rbnode_set_parent_and_color(p, s, C_RBNODE_RED);
c_rbtree_swap_child(t, g, p, s);
s = x;
}
x = s->left;
if (!x || c_rbnode_is_black(x)) {
y = s->right;
if (!y || c_rbnode_is_black(y)) {
c_rbnode_set_parent_and_color(s, p, C_RBNODE_RED);
if (c_rbnode_is_black(p))
return p;
c_rbnode_set_parent_and_color(p, c_rbnode_parent(p), C_RBNODE_BLACK);
return NULL;
}
x = y->left;
s->right = y->left;
y->left = s;
p->left = y;
if (x)
c_rbnode_set_parent_and_color(x, s, C_RBNODE_BLACK);
x = s;
s = y;
}
g = c_rbnode_parent(p);
y = s->right;
p->left = y;
s->right = p;
c_rbnode_set_parent_and_color(x, s, C_RBNODE_BLACK);
if (y)
c_rbnode_set_parent_and_color(y, p, c_rbnode_color(y));
c_rbnode_set_parent_and_color(s, g, c_rbnode_color(p));
c_rbnode_set_parent_and_color(p, s, C_RBNODE_BLACK);
c_rbtree_swap_child(t, g, p, s);
}
return NULL;
}
static inline void c_rbtree_rebalance(CRBTree *t, CRBNode *p) {
CRBNode *n = NULL;
assert(t);
assert(p);
do {
n = c_rbtree_rebalance_one(t, p, n);
p = n ? c_rbnode_parent(n) : NULL;
} while (p);
}
/**
* c_rbtree_remove() - remove node from tree
* @t: tree to operate one
* @n: node to remove
*
* This removes the given node from its tree. Once unlinked, the tree is
* rebalanced.
* The caller *must* ensure that the given tree is actually the tree it is
* linked on. Otherwise, behavior is undefined.
*
* This does *NOT* reset @n to being unlinked (for performance reason, this
* function *never* modifies @n at all). If you need this, use
* c_rbtree_remove_init().
*/
void c_rbtree_remove(CRBTree *t, CRBNode *n) {
CRBNode *p, *s, *gc, *x, *next = NULL;
unsigned long c;
assert(t);
assert(n);
assert(c_rbnode_is_linked(n));
/*
* There are three distinct cases during node removal of a tree:
* * The node has no children, in which case it can simply be removed.
* * The node has exactly one child, in which case the child displaces
* its parent.
* * The node has two children, in which case there is guaranteed to
* be a successor to the node (successor being the node ordered
* directly after it). This successor cannot have two children by
* itself (two interior nodes can never be successive). Therefore,
* we can simply swap the node with its successor (including color)
* and have reduced this case to either of the first two.
*
* Whenever the node we removed was black, we have to rebalance the
* tree. Note that this affects the actual node we _remove_, not @n (in
* case we swap it).
*
* p: parent
* s: successor
* gc: grand-...-child
* x: temporary
* next: next node to rebalance on
*/
if (!n->left) {
/*
* Case 1:
* The node has no left child. If it neither has a right child,
* it is a leaf-node and we can simply unlink it. If it also
* was black, we have to rebalance, as always if we remove a
* black node.
* But if the node has a right child, the child *must* be red
* (otherwise, the right path has more black nodes as the
* non-existing left path), and the node to be removed must
* hence be black. We simply replace the node with its child,
* turning the red child black, and thus no rebalancing is
* required.
*/
p = c_rbnode_parent(n);
c = c_rbnode_color(n);
c_rbtree_swap_child(t, p, n, n->right);
if (n->right)
c_rbnode_set_parent_and_color(n->right, p, c);
else
next = (c == C_RBNODE_BLACK) ? p : NULL;
} else if (!n->right) {
/*
* Case 1.1:
* The node has exactly one child, and it is on the left. Treat
* it as mirrored case of Case 1 (i.e., replace the node by its
* child).
*/
p = c_rbnode_parent(n);
c = c_rbnode_color(n);
c_rbtree_swap_child(t, p, n, n->left);
c_rbnode_set_parent_and_color(n->left, p, c);
} else {
/*
* Case 2:
* We are dealing with a full interior node with a child not on
* both sides. Find its successor and swap it. Then remove the
* node similar to Case 1. For performance reasons we don't
* perform the full swap, but skip links that are about to be
* removed, anyway.
*/
s = n->right;
if (!s->left) {
/* right child is next, no need to touch grandchild */
p = s;
gc = s->right;
} else {
/* find successor and swap partially */
s = c_rbnode_leftmost(s);
p = c_rbnode_parent(s);
gc = s->right;
p->left = s->right;
s->right = n->right;
c_rbnode_set_parent(n->right, s);
}
/* node is partially swapped, now remove as in Case 1 */
s->left = n->left;
c_rbnode_set_parent(n->left, s);
x = c_rbnode_parent(n);
c = c_rbnode_color(n);
c_rbtree_swap_child(t, x, n, s);
if (gc)
c_rbnode_set_parent_and_color(gc, p, C_RBNODE_BLACK);
else
next = c_rbnode_is_black(s) ? p : NULL;
c_rbnode_set_parent_and_color(s, x, c);
}
if (next)
c_rbtree_rebalance(t, next);
}