Adding upstream version 4.19.249.upstream/4.19.249upstream

Signed-off-by: Daniel Baumann <daniel.baumann@progress-linux.org>

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+=================================
+Red-black Trees (rbtree) in Linux
+=================================
+
+
+:Date: January 18, 2007
+:Author: Rob Landley <rob@landley.net>
+
+What are red-black trees, and what are they for?
+------------------------------------------------
+
+Red-black trees are a type of self-balancing binary search tree, used for
+storing sortable key/value data pairs.  This differs from radix trees (which
+are used to efficiently store sparse arrays and thus use long integer indexes
+to insert/access/delete nodes) and hash tables (which are not kept sorted to
+be easily traversed in order, and must be tuned for a specific size and
+hash function where rbtrees scale gracefully storing arbitrary keys).
+
+Red-black trees are similar to AVL trees, but provide faster real-time bounded
+worst case performance for insertion and deletion (at most two rotations and
+three rotations, respectively, to balance the tree), with slightly slower
+(but still O(log n)) lookup time.
+
+To quote Linux Weekly News:
+
+    There are a number of red-black trees in use in the kernel.
+    The deadline and CFQ I/O schedulers employ rbtrees to
+    track requests; the packet CD/DVD driver does the same.
+    The high-resolution timer code uses an rbtree to organize outstanding
+    timer requests.  The ext3 filesystem tracks directory entries in a
+    red-black tree.  Virtual memory areas (VMAs) are tracked with red-black
+    trees, as are epoll file descriptors, cryptographic keys, and network
+    packets in the "hierarchical token bucket" scheduler.
+
+This document covers use of the Linux rbtree implementation.  For more
+information on the nature and implementation of Red Black Trees,  see:
+
+  Linux Weekly News article on red-black trees
+    http://lwn.net/Articles/184495/
+
+  Wikipedia entry on red-black trees
+    http://en.wikipedia.org/wiki/Red-black_tree
+
+Linux implementation of red-black trees
+---------------------------------------
+
+Linux's rbtree implementation lives in the file "lib/rbtree.c".  To use it,
+"#include <linux/rbtree.h>".
+
+The Linux rbtree implementation is optimized for speed, and thus has one
+less layer of indirection (and better cache locality) than more traditional
+tree implementations.  Instead of using pointers to separate rb_node and data
+structures, each instance of struct rb_node is embedded in the data structure
+it organizes.  And instead of using a comparison callback function pointer,
+users are expected to write their own tree search and insert functions
+which call the provided rbtree functions.  Locking is also left up to the
+user of the rbtree code.
+
+Creating a new rbtree
+---------------------
+
+Data nodes in an rbtree tree are structures containing a struct rb_node member::
+
+  struct mytype {
+  	struct rb_node node;
+  	char *keystring;
+  };
+
+When dealing with a pointer to the embedded struct rb_node, the containing data
+structure may be accessed with the standard container_of() macro.  In addition,
+individual members may be accessed directly via rb_entry(node, type, member).
+
+At the root of each rbtree is an rb_root structure, which is initialized to be
+empty via:
+
+  struct rb_root mytree = RB_ROOT;
+
+Searching for a value in an rbtree
+----------------------------------
+
+Writing a search function for your tree is fairly straightforward: start at the
+root, compare each value, and follow the left or right branch as necessary.
+
+Example::
+
+  struct mytype *my_search(struct rb_root *root, char *string)
+  {
+  	struct rb_node *node = root->rb_node;
+
+  	while (node) {
+  		struct mytype *data = container_of(node, struct mytype, node);
+		int result;
+
+		result = strcmp(string, data->keystring);
+
+		if (result < 0)
+  			node = node->rb_left;
+		else if (result > 0)
+  			node = node->rb_right;
+		else
+  			return data;
+	}
+	return NULL;
+  }
+
+Inserting data into an rbtree
+-----------------------------
+
+Inserting data in the tree involves first searching for the place to insert the
+new node, then inserting the node and rebalancing ("recoloring") the tree.
+
+The search for insertion differs from the previous search by finding the
+location of the pointer on which to graft the new node.  The new node also
+needs a link to its parent node for rebalancing purposes.
+
+Example::
+
+  int my_insert(struct rb_root *root, struct mytype *data)
+  {
+  	struct rb_node **new = &(root->rb_node), *parent = NULL;
+
+  	/* Figure out where to put new node */
+  	while (*new) {
+  		struct mytype *this = container_of(*new, struct mytype, node);
+  		int result = strcmp(data->keystring, this->keystring);
+
+		parent = *new;
+  		if (result < 0)
+  			new = &((*new)->rb_left);
+  		else if (result > 0)
+  			new = &((*new)->rb_right);
+  		else
+  			return FALSE;
+  	}
+
+  	/* Add new node and rebalance tree. */
+  	rb_link_node(&data->node, parent, new);
+  	rb_insert_color(&data->node, root);
+
+	return TRUE;
+  }
+
+Removing or replacing existing data in an rbtree
+------------------------------------------------
+
+To remove an existing node from a tree, call::
+
+  void rb_erase(struct rb_node *victim, struct rb_root *tree);
+
+Example::
+
+  struct mytype *data = mysearch(&mytree, "walrus");
+
+  if (data) {
+  	rb_erase(&data->node, &mytree);
+  	myfree(data);
+  }
+
+To replace an existing node in a tree with a new one with the same key, call::
+
+  void rb_replace_node(struct rb_node *old, struct rb_node *new,
+  			struct rb_root *tree);
+
+Replacing a node this way does not re-sort the tree: If the new node doesn't
+have the same key as the old node, the rbtree will probably become corrupted.
+
+Iterating through the elements stored in an rbtree (in sort order)
+------------------------------------------------------------------
+
+Four functions are provided for iterating through an rbtree's contents in
+sorted order.  These work on arbitrary trees, and should not need to be
+modified or wrapped (except for locking purposes)::
+
+  struct rb_node *rb_first(struct rb_root *tree);
+  struct rb_node *rb_last(struct rb_root *tree);
+  struct rb_node *rb_next(struct rb_node *node);
+  struct rb_node *rb_prev(struct rb_node *node);
+
+To start iterating, call rb_first() or rb_last() with a pointer to the root
+of the tree, which will return a pointer to the node structure contained in
+the first or last element in the tree.  To continue, fetch the next or previous
+node by calling rb_next() or rb_prev() on the current node.  This will return
+NULL when there are no more nodes left.
+
+The iterator functions return a pointer to the embedded struct rb_node, from
+which the containing data structure may be accessed with the container_of()
+macro, and individual members may be accessed directly via
+rb_entry(node, type, member).
+
+Example::
+
+  struct rb_node *node;
+  for (node = rb_first(&mytree); node; node = rb_next(node))
+	printk("key=%s\n", rb_entry(node, struct mytype, node)->keystring);
+
+Cached rbtrees
+--------------
+
+Computing the leftmost (smallest) node is quite a common task for binary
+search trees, such as for traversals or users relying on a the particular
+order for their own logic. To this end, users can use 'struct rb_root_cached'
+to optimize O(logN) rb_first() calls to a simple pointer fetch avoiding
+potentially expensive tree iterations. This is done at negligible runtime
+overhead for maintanence; albeit larger memory footprint.
+
+Similar to the rb_root structure, cached rbtrees are initialized to be
+empty via:
+
+  struct rb_root_cached mytree = RB_ROOT_CACHED;
+
+Cached rbtree is simply a regular rb_root with an extra pointer to cache the
+leftmost node. This allows rb_root_cached to exist wherever rb_root does,
+which permits augmented trees to be supported as well as only a few extra
+interfaces:
+
+  struct rb_node *rb_first_cached(struct rb_root_cached *tree);
+  void rb_insert_color_cached(struct rb_node *, struct rb_root_cached *, bool);
+  void rb_erase_cached(struct rb_node *node, struct rb_root_cached *);
+
+Both insert and erase calls have their respective counterpart of augmented
+trees:
+
+  void rb_insert_augmented_cached(struct rb_node *node, struct rb_root_cached *,
+				  bool, struct rb_augment_callbacks *);
+  void rb_erase_augmented_cached(struct rb_node *, struct rb_root_cached *,
+				 struct rb_augment_callbacks *);
+
+
+Support for Augmented rbtrees
+-----------------------------
+
+Augmented rbtree is an rbtree with "some" additional data stored in
+each node, where the additional data for node N must be a function of
+the contents of all nodes in the subtree rooted at N. This data can
+be used to augment some new functionality to rbtree. Augmented rbtree
+is an optional feature built on top of basic rbtree infrastructure.
+An rbtree user who wants this feature will have to call the augmentation
+functions with the user provided augmentation callback when inserting
+and erasing nodes.
+
+C files implementing augmented rbtree manipulation must include
+<linux/rbtree_augmented.h> instead of <linux/rbtree.h>. Note that
+linux/rbtree_augmented.h exposes some rbtree implementations details
+you are not expected to rely on; please stick to the documented APIs
+there and do not include <linux/rbtree_augmented.h> from header files
+either so as to minimize chances of your users accidentally relying on
+such implementation details.
+
+On insertion, the user must update the augmented information on the path
+leading to the inserted node, then call rb_link_node() as usual and
+rb_augment_inserted() instead of the usual rb_insert_color() call.
+If rb_augment_inserted() rebalances the rbtree, it will callback into
+a user provided function to update the augmented information on the
+affected subtrees.
+
+When erasing a node, the user must call rb_erase_augmented() instead of
+rb_erase(). rb_erase_augmented() calls back into user provided functions
+to updated the augmented information on affected subtrees.
+
+In both cases, the callbacks are provided through struct rb_augment_callbacks.
+3 callbacks must be defined:
+
+- A propagation callback, which updates the augmented value for a given
+  node and its ancestors, up to a given stop point (or NULL to update
+  all the way to the root).
+
+- A copy callback, which copies the augmented value for a given subtree
+  to a newly assigned subtree root.
+
+- A tree rotation callback, which copies the augmented value for a given
+  subtree to a newly assigned subtree root AND recomputes the augmented
+  information for the former subtree root.
+
+The compiled code for rb_erase_augmented() may inline the propagation and
+copy callbacks, which results in a large function, so each augmented rbtree
+user should have a single rb_erase_augmented() call site in order to limit
+compiled code size.
+
+
+Sample usage
+^^^^^^^^^^^^
+
+Interval tree is an example of augmented rb tree. Reference -
+"Introduction to Algorithms" by Cormen, Leiserson, Rivest and Stein.
+More details about interval trees:
+
+Classical rbtree has a single key and it cannot be directly used to store
+interval ranges like [lo:hi] and do a quick lookup for any overlap with a new
+lo:hi or to find whether there is an exact match for a new lo:hi.
+
+However, rbtree can be augmented to store such interval ranges in a structured
+way making it possible to do efficient lookup and exact match.
+
+This "extra information" stored in each node is the maximum hi
+(max_hi) value among all the nodes that are its descendants. This
+information can be maintained at each node just be looking at the node
+and its immediate children. And this will be used in O(log n) lookup
+for lowest match (lowest start address among all possible matches)
+with something like::
+
+  struct interval_tree_node *
+  interval_tree_first_match(struct rb_root *root,
+			    unsigned long start, unsigned long last)
+  {
+	struct interval_tree_node *node;
+
+	if (!root->rb_node)
+		return NULL;
+	node = rb_entry(root->rb_node, struct interval_tree_node, rb);
+
+	while (true) {
+		if (node->rb.rb_left) {
+			struct interval_tree_node *left =
+				rb_entry(node->rb.rb_left,
+					 struct interval_tree_node, rb);
+			if (left->__subtree_last >= start) {
+				/*
+				 * Some nodes in left subtree satisfy Cond2.
+				 * Iterate to find the leftmost such node N.
+				 * If it also satisfies Cond1, that's the match
+				 * we are looking for. Otherwise, there is no
+				 * matching interval as nodes to the right of N
+				 * can't satisfy Cond1 either.
+				 */
+				node = left;
+				continue;
+			}
+		}
+		if (node->start <= last) {		/* Cond1 */
+			if (node->last >= start)	/* Cond2 */
+				return node;	/* node is leftmost match */
+			if (node->rb.rb_right) {
+				node = rb_entry(node->rb.rb_right,
+					struct interval_tree_node, rb);
+				if (node->__subtree_last >= start)
+					continue;
+			}
+		}
+		return NULL;	/* No match */
+	}
+  }
+
+Insertion/removal are defined using the following augmented callbacks::
+
+  static inline unsigned long
+  compute_subtree_last(struct interval_tree_node *node)
+  {
+	unsigned long max = node->last, subtree_last;
+	if (node->rb.rb_left) {
+		subtree_last = rb_entry(node->rb.rb_left,
+			struct interval_tree_node, rb)->__subtree_last;
+		if (max < subtree_last)
+			max = subtree_last;
+	}
+	if (node->rb.rb_right) {
+		subtree_last = rb_entry(node->rb.rb_right,
+			struct interval_tree_node, rb)->__subtree_last;
+		if (max < subtree_last)
+			max = subtree_last;
+	}
+	return max;
+  }
+
+  static void augment_propagate(struct rb_node *rb, struct rb_node *stop)
+  {
+	while (rb != stop) {
+		struct interval_tree_node *node =
+			rb_entry(rb, struct interval_tree_node, rb);
+		unsigned long subtree_last = compute_subtree_last(node);
+		if (node->__subtree_last == subtree_last)
+			break;
+		node->__subtree_last = subtree_last;
+		rb = rb_parent(&node->rb);
+	}
+  }
+
+  static void augment_copy(struct rb_node *rb_old, struct rb_node *rb_new)
+  {
+	struct interval_tree_node *old =
+		rb_entry(rb_old, struct interval_tree_node, rb);
+	struct interval_tree_node *new =
+		rb_entry(rb_new, struct interval_tree_node, rb);
+
+	new->__subtree_last = old->__subtree_last;
+  }
+
+  static void augment_rotate(struct rb_node *rb_old, struct rb_node *rb_new)
+  {
+	struct interval_tree_node *old =
+		rb_entry(rb_old, struct interval_tree_node, rb);
+	struct interval_tree_node *new =
+		rb_entry(rb_new, struct interval_tree_node, rb);
+
+	new->__subtree_last = old->__subtree_last;
+	old->__subtree_last = compute_subtree_last(old);
+  }
+
+  static const struct rb_augment_callbacks augment_callbacks = {
+	augment_propagate, augment_copy, augment_rotate
+  };
+
+  void interval_tree_insert(struct interval_tree_node *node,
+			    struct rb_root *root)
+  {
+	struct rb_node **link = &root->rb_node, *rb_parent = NULL;
+	unsigned long start = node->start, last = node->last;
+	struct interval_tree_node *parent;
+
+	while (*link) {
+		rb_parent = *link;
+		parent = rb_entry(rb_parent, struct interval_tree_node, rb);
+		if (parent->__subtree_last < last)
+			parent->__subtree_last = last;
+		if (start < parent->start)
+			link = &parent->rb.rb_left;
+		else
+			link = &parent->rb.rb_right;
+	}
+
+	node->__subtree_last = last;
+	rb_link_node(&node->rb, rb_parent, link);
+	rb_insert_augmented(&node->rb, root, &augment_callbacks);
+  }
+
+  void interval_tree_remove(struct interval_tree_node *node,
+			    struct rb_root *root)
+  {
+	rb_erase_augmented(&node->rb, root, &augment_callbacks);
+  }