# gb_trees

## General Balanced Trees

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).

#### Types

### tree(Key, Value)

A GB tree.

### tree() = tree(term(), term())

### iter(Key, Value)

A GB tree iterator.

### iter() = iter(term(), term())

#### Functions

### balance(Tree1) -> Tree2

`Tree1 = Tree2 = tree(Key, Value)`

Rebalances

. 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

`Tree1 = Tree2 = tree(Key, Value)`

Removes the node with key

from

;
returns new tree. Assumes that the key is present in the tree,
crashes otherwise.

### delete_any(Key, Tree1) -> Tree2

`Tree1 = Tree2 = tree(Key, Value)`

Removes the node with key

from

if
the key is present in the tree, otherwise does nothing;
returns new tree.

### empty() -> tree()

Returns a new empty tree

### enter(Key, Value, Tree1) -> Tree2

`Tree1 = Tree2 = tree(Key, Value)`

Inserts

with value

into

if
the key is not present in the tree, otherwise updates

to value

in

. Returns the
new tree.

### from_orddict(List) -> Tree

`List = [{Key, Value}]`

`Tree = tree(Key, Value)`

Turns an ordered list

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

### get(Key, Tree) -> Value

`Tree = tree(Key, Value)`

Retrieves the value stored with

in

.
Assumes that the key is present in the tree, crashes
otherwise.

### insert(Key, Value, Tree1) -> Tree2

`Tree1 = Tree2 = tree(Key, Value)`

Inserts

with value

into

;
returns the new tree. Assumes that the key is not present in
the tree, crashes otherwise.

### is_defined(Key, Tree) -> boolean()

`Tree = tree(Key, Value :: term())`

Returns `true`

if

is present in

,
otherwise `false`

.

### is_empty(Tree) -> boolean()

`Tree = tree()`

Returns `true`

if

is an empty tree, and
`false`

otherwise.

### iterator(Tree) -> Iter

Returns an iterator that can be used for traversing the
entries of

; 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.

### iterator_from(Key, Tree) -> Iter

Returns an iterator that can be used for traversing the
entries of

; see `next/1`

.
The difference as compared to the iterator returned by
`iterator/1`

is that the first key greater than
or equal to

is returned.

### largest(Tree) -> {Key, Value}

`Tree = tree(Key, Value)`

Returns `{`

, where

is the largest
key in

, and

is the value associated
with this key. Assumes that the tree is nonempty.

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

`Tree = tree(Key, Value)`

Looks up

in

; returns
`{value, `

, or `none`

if

is not
present.

### map(Function, Tree1) -> Tree2

`Function = fun((K :: Key, V1 :: Value1) -> V2 :: Value2)`

`Tree1 = tree(Key, Value1)`

`Tree2 = tree(Key, Value2)`

Maps the function F(

and returns a new tree

with the same set of keys
as

and the new set of values

.

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

`Iter1 = Iter2 = iter(Key, Value)`

Returns `{`

where

is the
smallest key referred to by the iterator

, and

is the new iterator to be used for
traversing the remaining nodes, or the atom `none`

if no
nodes remain.

### smallest(Tree) -> {Key, Value}

`Tree = tree(Key, Value)`

Returns `{`

, where

is the smallest
key in

, and

is the value associated
with this key. Assumes that the tree is nonempty.

### take_largest(Tree1) -> {Key, Value, Tree2}

`Tree1 = Tree2 = tree(Key, Value)`

Returns `{`

, where

is the
largest key in

,

is the value
associated with this key, and

is this tree with
the corresponding node deleted. Assumes that the tree is
nonempty.

### take_smallest(Tree1) -> {Key, Value, Tree2}

`Tree1 = Tree2 = tree(Key, Value)`

Returns `{`

, where

is the
smallest key in

,

is the value
associated with this key, and

is this tree with
the corresponding node deleted. Assumes that the tree is
nonempty.

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

`Tree = tree(Key, Value)`

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

### update(Key, Value, Tree1) -> Tree2

`Tree1 = Tree2 = tree(Key, Value)`

Updates

to value

in

;
returns the new tree. Assumes that the key is present in the
tree.

### values(Tree) -> [Value]

`Tree = tree(Key :: term(), Value)`

Returns the values in

as an ordered list, sorted
by their corresponding keys. Duplicates are not removed.