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# gb_trees

### MODULE

gb_trees

### MODULE SUMMARY

General Balanced Trees

### DESCRIPTION

An efficient implementation of Prof. Arne Andersson's General Balanced Trees. These have no storage overhead compared to unbalanced binary trees, and their performance is in general better than AVL trees.

This module considers two keys as different if and only if they do not compare equal (==).

### Data structure

Data structure:

    - {Size, Tree}, where `Tree' is composed of nodes of the form:
      - {Key, Value, Smaller, Bigger}, and the "empty tree" node:
      - nil.

There is no attempt to balance trees after deletions. Since deletions do not increase the height of a tree, this should be OK.

Original balance condition h(T) <= ceil(c * log(|T|)) has been changed to the similar (but not quite equivalent) condition 2 ^ h(T) <= |T| ^ c. This should also be OK.

Performance is comparable to the AVL trees in the Erlang book (and faster in general due to less overhead); the difference is that deletion works for these trees, but not for the book's trees. Behaviour is logarithmic (as it should be).

### DATA TYPES

gb_tree()

A GB tree.

iter()

A GB tree iterator.

### EXPORTS

balance(Tree1) -> Tree2

Types:

Tree1 = Tree2 = gb_tree()

Rebalances Tree1. Note that this is rarely necessary, but may be motivated when a large number of nodes have been deleted from the tree without further insertions. Rebalancing could then be forced in order to minimise lookup times, since deletion only does not rebalance the tree.

delete(Key, Tree1) -> Tree2

Types:

Key = term()
Tree1 = Tree2 = gb_tree()

Removes the node with key Key from Tree1; returns new tree. Assumes that the key is present in the tree, crashes otherwise.

delete_any(Key, Tree1) -> Tree2

Types:

Key = term()
Tree1 = Tree2 = gb_tree()

Removes the node with key Key from Tree1 if the key is present in the tree, otherwise does nothing; returns new tree.

empty() -> gb_tree()

Returns a new empty tree

enter(Key, Val, Tree1) -> Tree2

Types:

Key = Val = term()
Tree1 = Tree2 = gb_tree()

Inserts Key with value Val into Tree1 if the key is not present in the tree, otherwise updates Key to value Val in Tree1. Returns the new tree.

from_orddict(List) -> Tree

Types:

List = [{Key :: term(), Val :: term()}]
Tree = gb_tree()

Turns an ordered list List of key-value tuples into a tree. The list must not contain duplicate keys.

get(Key, Tree) -> Val

Types:

Key = term()
Tree = gb_tree()
Val = term()

Retrieves the value stored with Key in Tree. Assumes that the key is present in the tree, crashes otherwise.

lookup(Key, Tree) -> none | {value, Val}

Types:

Key = Val = term()
Tree = gb_tree()

Looks up Key in Tree; returns {value, Val}, or none if Key is not present.

insert(Key, Val, Tree1) -> Tree2

Types:

Key = Val = term()
Tree1 = Tree2 = gb_tree()

Inserts Key with value Val into Tree1; returns the new tree. Assumes that the key is not present in the tree, crashes otherwise.

is_defined(Key, Tree) -> boolean()

Types:

Key = term()
Tree = gb_tree()

Returns true if Key is present in Tree, otherwise false.

is_empty(Tree) -> boolean()

Types:

Tree = gb_tree()

Returns true if Tree is an empty tree, and false otherwise.

iterator(Tree) -> Iter

Types:

Tree = gb_tree()
Iter = iter()

Returns an iterator that can be used for traversing the entries of Tree; see next/1. The implementation of this is very efficient; traversing the whole tree using next/1 is only slightly slower than getting the list of all elements using to_list/1 and traversing that. The main advantage of the iterator approach is that it does not require the complete list of all elements to be built in memory at one time.

keys(Tree) -> [Key]

Types:

Tree = gb_tree()
Key = term()

Returns the keys in Tree as an ordered list.

largest(Tree) -> {Key, Val}

Types:

Tree = gb_tree()
Key = Val = term()

Returns {Key, Val}, where Key is the largest key in Tree, and Val is the value associated with this key. Assumes that the tree is nonempty.

map(Function, Tree1) -> Tree2

Types:

Function = fun((K :: term(), V1 :: term()) -> V2 :: term())
Tree1 = Tree2 = gb_tree()

Maps the function F(K, V1) -> V2 to all key-value pairs of the tree Tree1 and returns a new tree Tree2 with the same set of keys as Tree1 and the new set of values V2.

next(Iter1) -> none | {Key, Val, Iter2}

Types:

Iter1 = Iter2 = iter()
Key = Val = term()

Returns {Key, Val, Iter2} where Key is the smallest key referred to by the iterator Iter1, and Iter2 is the new iterator to be used for traversing the remaining nodes, or the atom none if no nodes remain.

size(Tree) -> integer() >= 0

Types:

Tree = gb_tree()

Returns the number of nodes in Tree.

smallest(Tree) -> {Key, Val}

Types:

Tree = gb_tree()
Key = Val = term()

Returns {Key, Val}, where Key is the smallest key in Tree, and Val is the value associated with this key. Assumes that the tree is nonempty.

take_largest(Tree1) -> {Key, Val, Tree2}

Types:

Tree1 = Tree2 = gb_tree()
Key = Val = term()

Returns {Key, Val, Tree2}, where Key is the largest key in Tree1, Val is the value associated with this key, and Tree2 is this tree with the corresponding node deleted. Assumes that the tree is nonempty.

take_smallest(Tree1) -> {Key, Val, Tree2}

Types:

Tree1 = Tree2 = gb_tree()
Key = Val = term()

Returns {Key, Val, Tree2}, where Key is the smallest key in Tree1, Val is the value associated with this key, and Tree2 is this tree with the corresponding node deleted. Assumes that the tree is nonempty.

to_list(Tree) -> [{Key, Val}]

Types:

Tree = gb_tree()
Key = Val = term()

Converts a tree into an ordered list of key-value tuples.

update(Key, Val, Tree1) -> Tree2

Types:

Key = Val = term()
Tree1 = Tree2 = gb_tree()

Updates Key to value Val in Tree1; returns the new tree. Assumes that the key is present in the tree.

values(Tree) -> [Val]

Types:

Tree = gb_tree()
Val = term()

Returns the values in Tree as an ordered list, sorted by their corresponding keys. Duplicates are not removed.

### SEE ALSO

gb_sets(3), dict(3)