# gb_sets

## 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.

### set(Element)

A general balanced set.

### iter(Element)

A general balanced set iterator.

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

#### Functions

• `Set1 = Set2 = set(Element)`

• `Set1 = Set2 = set(Element)`

Returns a new set formed from `Set1` with `Element` inserted. If `Element` is already an element in `Set1`, nothing is changed.

### balance(Set1) -> Set2

• `Set1 = Set2 = set(Element)`

Rebalances the tree representation of `Set1`. 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 `Set1` with `Element` removed. If `Element` is not an element in `Set1`, nothing is changed.

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

• `Set1 = Set2 = set(Element)`

Returns a new set formed from `Set1` with `Element` removed. Assumes that `Element` is present in `Set1`.

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

• `Set1 = Set2 = set(Element)`

Returns a new set formed from `Set1` with `Element` removed. If `Element` is not an element in `Set1`, nothing is changed.

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

• `Set1 = Set2 = Set3 = set(Element)`

Returns only the elements of `Set1` that are not also elements of `Set2`.

### empty() -> Set

• `Set = set()`

Returns a new empty set.

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

• `Pred = fun((Element) -> boolean())`
• `Set1 = Set2 = set(Element)`

Filters elements in `Set1` using predicate function `Pred`.

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

• `Function = fun((Element, AccIn) -> AccOut)`
• `Acc0 = Acc1 = AccIn = AccOut = Acc`
• `Set = set(Element)`

Folds `Function` over every element in `Set` returning the final value of the accumulator.

### from_list(List) -> Set

• `List = [Element]`
• `Set = set(Element)`

Returns a set of the elements in `List`, where `List` can be unordered and contain duplicates.

### from_ordset(List) -> Set

• `List = [Element]`
• `Set = set(Element)`

Turns an ordered-set list `List` into a set. The list must not contain duplicates.

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

• `Set1 = Set2 = set(Element)`

Returns a new set formed from `Set1` with `Element` inserted. Assumes that `Element` is not present in `Set1`.

### intersection(SetList) -> Set

• `SetList = [set(Element), ...]`
• `Set = set(Element)`

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

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

• `Set1 = Set2 = Set3 = set(Element)`

Returns the intersection of `Set1` and `Set2`.

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

• `Set1 = Set2 = set(Element)`

Returns `true` if `Set1` and `Set2` are disjoint (have no elements in common), otherwise `false`.

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

• `Set = set(Element)`

Returns `true` if `Element` is an element of `Set`, otherwise `false`.

### is_empty(Set) -> boolean()

• `Set = set()`

Returns `true` if `Set` is an empty set, otherwise `false`.

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

• `Set = set(Element)`

Returns `true` if `Element` is an element of `Set`, otherwise `false`.

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

• `Term = term()`

Returns `true` if `Term` appears to be a set, otherwise `false`.

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

• `Set1 = Set2 = set(Element)`

Returns `true` when every element of `Set1` is also a member of `Set2`, otherwise `false`.

### iterator(Set) -> Iter

• `Set = set(Element)`
• `Iter = iter(Element)`

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.

### iterator_from(Element, Set) -> Iter

• `Set = set(Element)`
• `Iter = iter(Element)`

Returns an iterator that can be used for traversing the entries of `Set`; see `next/1`. The difference as compared to the iterator returned by `iterator/1` is that the first element greater than or equal to `Element` is returned.

### largest(Set) -> Element

• `Set = set(Element)`

Returns the largest element in `Set`. Assumes that `Set` is not empty.

### new() -> Set

• `Set = set()`

Returns a new empty set.

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

• `Iter1 = Iter2 = iter(Element)`

Returns `{Element, Iter2}`, where `Element` is the smallest element referred to by 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) -> set(Element)

Returns a set containing only element `Element`.

### size(Set) -> integer() >= 0

• `Set = set()`

Returns the number of elements in `Set`.

### smallest(Set) -> Element

• `Set = set(Element)`

Returns the smallest element in `Set`. Assumes that `Set` is not empty.

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

• `Set1 = Set2 = Set3 = set(Element)`

Returns only the elements of `Set1` that are not also elements of `Set2`.

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

• `Set1 = Set2 = set(Element)`

Returns `{Element, Set2}`, where `Element` is the largest element in `Set1`, and `Set2` is this set with `Element` deleted. Assumes that `Set1` is not empty.

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

• `Set1 = Set2 = set(Element)`

Returns `{Element, Set2}`, where `Element` is the smallest element in `Set1`, and `Set2` is this set with `Element` deleted. Assumes that `Set1` is not empty.

### to_list(Set) -> List

• `Set = set(Element)`
• `List = [Element]`

Returns the elements of `Set` as a list.

### union(SetList) -> Set

• `SetList = [set(Element), ...]`
• `Set = set(Element)`

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 `Set1` and `Set2`.