## General balanced trees.

This module provides ordered sets using Prof. Arne Andersson's General Balanced Trees. Ordered sets can be much more efficient than using ordered lists, for larger sets, but depends on the application.

This module considers two elements as different if and only if
they do not compare equal (`==`

).

#### Complexity Note

The complexity on set operations is bounded by either *O(|S|)* or
*O(|T| * log(|S|))*, where S is the largest given set, depending
on which is fastest for any particular function call. For
operating on sets of almost equal size, this implementation is
about 3 times slower than using ordered-list sets directly. For
sets of very different sizes, however, this solution can be
arbitrarily much faster; in practical cases, often
10-100 times. This implementation is particularly suited for
accumulating elements a few at a time, building up a large set
(> 100-200 elements), and repeatedly testing for
membership in the current set.

As with normal tree structures, lookup (membership testing), insertion, and deletion have logarithmic complexity.

#### Compatibility

The following functions in this module also exist and provides
the same functionality in the
`sets(3)`

and
`ordsets(3)`

modules. That is, by only changing the module name for each call,
you can try out different set representations.

#### Types

### set(Element)

A general balanced set.

### set() = set(term())

### iter(Element)

A general balanced set iterator.

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

#### Functions

### add(Element, Set1) -> Set2

`Set1 = Set2 = set(Element)`

### add_element(Element, Set1) -> Set2

`Set1 = Set2 = set(Element)`

Returns a new set formed from

with

inserted. If

is already an
element in

, nothing is changed.

### balance(Set1) -> Set2

`Set1 = Set2 = set(Element)`

Rebalances the tree representation of

.
Notice that
this is rarely necessary, but can be motivated when a large
number of elements have been deleted from the tree without
further insertions. Rebalancing can then be forced
to minimise lookup times, as deletion does not
rebalance the tree.

### del_element(Element, Set1) -> Set2

`Set1 = Set2 = set(Element)`

Returns a new set formed from

with

removed. If

is not an element
in

, nothing is changed.

### delete(Element, Set1) -> Set2

`Set1 = Set2 = set(Element)`

Returns a new set formed from

with

removed. Assumes that

is present
in

.

### delete_any(Element, Set1) -> Set2

`Set1 = Set2 = set(Element)`

Returns a new set formed from

with

removed. If

is not an element
in

, nothing is changed.

### difference(Set1, Set2) -> Set3

`Set1 = Set2 = Set3 = set(Element)`

Returns only the elements of

that are not
also elements of

.

### filter(Pred, Set1) -> Set2

`Pred = fun((Element) -> boolean())`

`Set1 = Set2 = set(Element)`

Filters elements in

using predicate function

.

### fold(Function, Acc0, Set) -> Acc1

`Function = fun((Element, AccIn) -> AccOut)`

`Acc0 = Acc1 = AccIn = AccOut = Acc`

`Set = set(Element)`

Folds

over every element in

returning the final value of the accumulator.

### from_list(List) -> Set

`List = [Element]`

`Set = set(Element)`

Returns a set of the elements in

, where

can be unordered and contain duplicates.

### from_ordset(List) -> Set

`List = [Element]`

`Set = set(Element)`

Turns an ordered-set list

into a set.
The list must not contain duplicates.

### insert(Element, Set1) -> Set2

`Set1 = Set2 = set(Element)`

Returns a new set formed from

with

inserted. Assumes that

is not
present in

.

### intersection(SetList) -> Set

Returns the intersection of the non-empty list of sets.

### intersection(Set1, Set2) -> Set3

`Set1 = Set2 = Set3 = set(Element)`

Returns the intersection of

and

.

### is_disjoint(Set1, Set2) -> boolean()

`Set1 = Set2 = set(Element)`

Returns `true`

if

and

are disjoint (have no elements in common),
otherwise `false`

.

### is_element(Element, Set) -> boolean()

`Set = set(Element)`

Returns `true`

if

is an element of

, otherwise `false`

.

### is_member(Element, Set) -> boolean()

`Set = set(Element)`

Returns `true`

if

is an element of

, otherwise `false`

.

### is_set(Term) -> boolean()

`Term = term()`

Returns `true`

if

appears to be a set,
otherwise `false`

.

### is_subset(Set1, Set2) -> boolean()

`Set1 = Set2 = set(Element)`

Returns `true`

when every element of

is
also a member of

, otherwise `false`

.

### iterator(Set) -> 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 set 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(Element, Set) -> 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 element greater than
or equal to

is returned.

### largest(Set) -> Element

`Set = set(Element)`

Returns the largest element in

. Assumes that

is not empty.

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

`Iter1 = Iter2 = iter(Element)`

Returns `{`

, where

is the smallest element referred to by
iterator

,
and

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

if
no elements remain.

### singleton(Element) -> set(Element)

Returns a set containing only element

.

### smallest(Set) -> Element

`Set = set(Element)`

Returns the smallest element in

. Assumes that

is not empty.

### subtract(Set1, Set2) -> Set3

`Set1 = Set2 = Set3 = set(Element)`

Returns only the elements of

that are not
also elements of

.

### take_largest(Set1) -> {Element, Set2}

`Set1 = Set2 = set(Element)`

Returns `{`

, where

is the largest element in

, and

is this set
with

deleted. Assumes that

is not empty.

### take_smallest(Set1) -> {Element, Set2}

`Set1 = Set2 = set(Element)`

Returns `{`

, where

is the smallest element in

, and

is this set
with

deleted. Assumes that

is not empty.

### union(SetList) -> Set

Returns the merged (union) set of the list of sets.

### union(Set1, Set2) -> Set3

`Set1 = Set2 = Set3 = set(Element)`

Returns the merged (union) set of

and

.