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# gb_sets

### MODULE

gb_sets

### MODULE SUMMARY

General Balanced Trees

### DESCRIPTION

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

### DATA TYPES

    gb_set() = a GB set

### EXPORTS

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

Types:

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

Types:

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

Types:

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

Types:

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

Types:

Set1 = Set2 = Set3 = gb_set()

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

empty() -> Set
new() -> Set

Types:

Set = gb_set()

Returns a new empty gb_set.

filter(Pred, Set1) -> Set2

Types:

Pred = fun (E) -> bool()
E = term()
Set1 = Set2 = gb_set()

Filters elements in Set1 using predicate function Pred.

fold(Function, Acc0, Set) -> Acc1

Types:

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

Types:

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

Types:

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

Types:

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

Types:

Set1 = Set2 = Set3 = gb_set()

Returns the intersection of Set1 and Set2.

intersection(SetList) -> Set

Types:

SetList = [gb_set()]
Set = gb_set()

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

is_disjoint(Set1, Set2) -> bool()

Types:

Set1 = Set2 = gb_set()

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

is_empty(Set) -> bool()

Types:

Set = gb_set()

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

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

Types:

Element = term()
Set = gb_set()

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

is_set(Term) -> bool()

Types:

Term = term()

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

is_subset(Set1, Set2) -> bool()

Types:

Set1 = Set2 = gb_set()

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

iterator(Set) -> Iter

Types:

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()

Types:

Set = gb_set()

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

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

Types:

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()

Types:

Element = term()

Returns a gb_set containing only the element Element.

size(Set) -> int()

Types:

Set = gb_set()

Returns the number of elements in Set.

smallest(Set) -> term()

Types:

Set = gb_set()

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

take_largest(Set1) -> {Element, Set2}

Types:

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}

Types:

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

Types:

Set = gb_set()
List = [term()]

Returns the elements of Set as a list.

union(Set1, Set2) -> Set3

Types:

Set1 = Set2 = Set3 = gb_set()

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

union(SetList) -> Set

Types:

SetList = [gb_set()]
Set = gb_set()

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

### SEE ALSO

gb_trees(3), ordsets(3), sets(3)