# 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.

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 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`

A GB set.

A GB set iterator.

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

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

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

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

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

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

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

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

### new() -> Set

Returns a new empty gb_set.

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

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

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

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

### from_list(List) -> Set

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

### from_ordset(List) -> Set

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

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

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

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

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

### intersection(SetList) -> Set

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

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

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

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

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

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

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

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

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

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

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

### iterator(Set) -> Iter

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()

Returns the largest element in `Set`. Assumes that `Set` is nonempty.

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

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()

Returns a gb_set containing only the element `Element`.

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

Returns the number of elements in `Set`.

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

Returns the smallest element in `Set`. Assumes that `Set` is nonempty.

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

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}

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

Returns the elements of `Set` as a list.

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

Returns the merged (union) gb_set of `Set1` and `Set2`.

### union(SetList) -> Set

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