# gb_sets

## General Balanced Trees

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

#### 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 between 10 and 100 times. This implementation is particularly suited for accumulating elements a few at a time, building up a large set (more than 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

All of the following functions in this module also exist
and do the same thing in the `sets`

and `ordsets`

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

`add_element/2`

`del_element/2`

`filter/2`

`fold/3`

`from_list/1`

`intersection/1`

`intersection/2`

`is_element/2`

`is_set/1`

`is_subset/2`

`new/0`

`size/1`

`subtract/2`

`to_list/1`

`union/1`

`union/2`

#### DATA TYPES

gb_set() = a GB set

#### Functions

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

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

`Element = term()`

`Set1 = Set2 = gb_set()`

Returns a new gb_set formed from `Set1`

with
`Element`

inserted. If `Element`

is already an
element in `Set1`

, nothing is changed.

### balance(Set1) -> Set2

`Set1 = Set2 = gb_set()`

Rebalances the tree representation of `Set1`

. Note that
this is rarely necessary, but may be motivated when a large
number of elements 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(Element, Set1) -> Set2

`Element = term()`

`Set1 = Set2 = gb_set()`

Returns a new gb_set formed from `Set1`

with
`Element`

removed. Assumes that `Element`

is present
in `Set1`

.

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

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

`Element = term()`

`Set1 = Set2 = gb_set()`

Returns a new gb_set formed from `Set1`

with
`Element`

removed. If `Element`

is not an element
in `Set1`

, nothing is changed.

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

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

`Set1 = Set2 = Set3 = gb_set()`

Returns only the elements of `Set1`

which are not also
elements of `Set2`

.

### empty() -> Set

### new() -> Set

`Set = gb_set()`

Returns a new empty gb_set.

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

`Pred = fun (E) -> bool()`

`E = term()`

`Set1 = Set2 = gb_set()`

Filters elements in `Set1`

using predicate function
`Pred`

.

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

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

`Acc0 = Acc1 = AccIn = AccOut = term()`

`E = term()`

`Set = gb_set()`

Folds `Function`

over every element in `Set`

returning the final value of the accumulator.

### from_list(List) -> Set

`List = [term()]`

`Set = gb_set()`

Returns a gb_set of the elements in `List`

, where
`List`

may be unordered and contain duplicates.

### from_ordset(List) -> Set

`List = [term()]`

`Set = gb_set()`

Turns an ordered-set list `List`

into a gb_set. The list
must not contain duplicates.

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

`Element = term()`

`Set1 = Set2 = gb_set()`

Returns a new gb_set formed from `Set1`

with
`Element`

inserted. Assumes that `Element`

is not
present in `Set1`

.

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

`Set1 = Set2 = Set3 = gb_set()`

Returns the intersection of `Set1`

and `Set2`

.

### intersection(SetList) -> Set

`SetList = [gb_set()]`

`Set = gb_set()`

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

### is_empty(Set) -> bool()

`Set = gb_set()`

Returns `true`

if `Set`

is an empty set, and
`false`

otherwise.

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

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

`Element = term()`

`Set = gb_set()`

Returns `true`

if `Element`

is an element of
`Set`

, otherwise `false`

.

### is_set(Set) -> bool()

`Set = gb_set()`

Returns `true`

if `Set`

appears to be a gb_set,
otherwise `false`

.

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

`Set1 = Set2 = gb_set()`

Returns `true`

when every element of `Set1`

is
also a member of `Set2`

, otherwise `false`

.

### iterator(Set) -> Iter

`Set = gb_set()`

`Iter = term()`

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

; 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.

### largest(Set) -> term()

`Set = gb_set()`

Returns the largest element in `Set`

. Assumes that
`Set`

is nonempty.

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

`Iter1 = Iter2 = Element = term()`

Returns `{Element, Iter2}`

where `Element`

is the
smallest element referred to by the iterator `Iter1`

,
and `Iter2`

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

if
no elements remain.

### singleton(Element) -> gb_set()

`Element = term()`

Returns a gb_set containing only the element `Element`

.

### size(Set) -> int()

`Set = gb_set()`

Returns the number of elements in `Set`

.

### smallest(Set) -> term()

`Set = gb_set()`

Returns the smallest element in `Set`

. Assumes that
`Set`

is nonempty.

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

`Set1 = Set2 = gb_set()`

`Element = term()`

Returns `{Element, Set2}`

, where `Element`

is the
largest element in `Set1`

, and `Set2`

is this set
with `Element`

deleted. Assumes that `Set1`

is
nonempty.

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

`Set1 = Set2 = gb_set()`

`Element = term()`

Returns `{Element, Set2}`

, where `Element`

is the
smallest element in `Set1`

, and `Set2`

is this set
with `Element`

deleted. Assumes that `Set1`

is
nonempty.

### to_list(Set) -> List

`Set = gb_set()`

`List = [term()]`

Returns the elements of `Set`

as a list.

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

`Set1 = Set2 = Set3 = gb_set()`

Returns the merged (union) gb_set of `Set1`

and
`Set2`

.

### union(SetList) -> Set

`SetList = [gb_set()]`

`Set = gb_set()`

Returns the merged (union) gb_set of the list of gb_sets.

#### SEE ALSO

- add/2
- add_element/2
- balance/1
- delete/2
- delete_any/2
- del_element/2
- difference/2
- subtract/2
- empty/0
- new/0
- filter/2
- fold/3
- from_list/1
- from_ordset/1
- insert/2
- intersection/2
- intersection/1
- is_empty/1
- is_member/2
- is_element/2
- is_set/1
- is_subset/2
- iterator/1
- largest/1
- next/1
- singleton/1
- size/1
- smallest/1
- take_largest/1
- take_smallest/1
- to_list/1
- union/2
- union/1