gb_sets(3)

NAME
gb_sets - Sets represented by general balanced trees.

DESCRIPTION

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.

The data representing a set as used by this module is to be regarded as opaque by other modules. In abstract terms, the representation is a composite type of existing Erlang terms. See note on data types. Any code assuming knowledge of the format is running on thin ice.

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

  Complexity

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

See the Compatibility Section in the sets module for information about the compatibility of the different implementations of sets in the Standard Library.

See Also

gb_trees, ordsets, sets

DATA TYPES

iter() :: iter(_).
-opaque iter(Element)

    A general balanced set iterator.
set() :: set(_).
-opaque set(Element)

    A general balanced set.

FUNCTIONS

add_element(Element, Set1) -> Set2 when Set1 :: set(Element), Set2 :: set(Element).

    Returns a new set formed from Set1 with Element inserted.

    If Element is already an element in Set1, nothing is changed.

  Examples

    1> S0 = gb_sets:new().
    2> S1 = gb_sets:add_element(7, S0).
    3> gb_sets:to_list(S1).
    [7]
    4> S2 = gb_sets:add_element(42, S1).
    5> S2 = gb_sets:add_element(42, S1).
    6> gb_sets:to_list(S2).
    [7,42]
balance(Set1) -> Set2 when Set1 :: set(Element), Set2 :: set(Element).

    Rebalances the tree representation of Set1.

    This is rarely necessary, but can be motivated when a large number of elements have been deleted from the tree without further insertions. Forcing rebalancing can minimize lookup times, as deletion does not rebalance the tree.

  Examples

    1> S0 = gb_sets:from_list(lists:seq(1, 100)).
    2> Delete = fun(E, Set) -> gb_sets:delete(E, Set) end.
    3> S1 = lists:foldl(Delete, S0, lists:seq(1, 50)).
    4> gb_sets:size(S1).
    50
    5> S2 = gb_sets:balance(S1).
delete(Element, Set1) -> Set2 when Set1 :: set(Element), Set2 :: set(Element).

    Returns a new set formed from Set1 with Element removed, assuming Element is present in Set1.

    Use delete_any/2 when deleting from a set where Element is potentially missing.

  Examples

    1> S = gb_sets:from_list([a,b]).
    2> gb_sets:to_list(gb_sets:delete(b, S)).
    [a]
delete_any(Element, Set1) -> Set2 when Set1 :: set(Element), Set2 :: set(Element).

    Returns a new set formed from Set1 with Element removed.

    If Element is not an element in Set1, nothing is changed.

  Examples

    1> S = gb_sets:from_list([a,b]).
    2> gb_sets:to_list(gb_sets:delete_any(b, S)).
    [a]
    3> S = gb_sets:delete_any(x, S).
empty() -> Set when Set :: set(none()).

    Returns a new empty set.

  Examples

    1> gb_sets:to_list(gb_sets:empty()).
    []
filter(Pred, Set1) -> Set2 when Pred :: fun((Element) -> boolean()), Set1 :: set(Element), Set2 :: set(Element).

    Filters elements in Set1 using predicate function Pred.

  Examples

    1> S = gb_sets:from_list([1,2,3,4,5,6,7]).
    2> IsEven = fun(N) -> N rem 2 =:= 0 end.
    3> Filtered = gb_sets:filter(IsEven, S).
    4> gb_sets:to_list(Filtered).
    [2,4,6]
filtermap(Fun, Set1) -> Set2 when Fun :: fun((Element1) -> boolean() | {true, Element2}), Set1 :: set(Element1), Set2 :: set(Element1 | Element2).

    Calls Fun(Elem) for each Elem of Set1 to update or remove elements from Set1.

    Fun/1 must return either a Boolean or a tuple {true, Value}. The function returns the set of elements for which Fun returns a new value, with true being equivalent to {true, Elem}.

    gb_sets:filtermap/2 behaves as if it were defined as follows:

        filtermap(Fun, Set1) ->
            gb_sets:from_list(lists:filtermap(Fun, gb_sets:to_list(Set1))).

  Examples

    1> S = gb_sets:from_list([2,4,5,6,8,9]).
    2> F = fun(X) ->
               case X rem 2 of
                   0 -> {true, X div 2};
                   1 -> false
               end
            end.
    3> Set = gb_sets:filtermap(F, S).
    4> gb_sets:to_list(Set).
    [1,2,3,4]
fold(Function, Acc0, Set) -> Acc1 when Function :: fun((Element, AccIn) -> AccOut), Acc0 :: Acc, Acc1 :: Acc, AccIn :: Acc, AccOut :: Acc, Set :: set(Element).

    Folds Function over every element in Set and returns the final value of the accumulator.

  Examples

    1> S = gb_sets:from_list([1,2,3,4]).
    2> Plus = fun erlang:'+'/2.
    3> gb_sets:fold(Plus, 0, S).
    10
from_list(List) -> Set when List :: [Element], Set :: set(Element).

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

  Examples

    1> Unordered = [x,y,a,x,y,b,b,z].
    2> gb_sets:to_list(gb_sets:from_list(Unordered)).
    [a,b,x,y,z]
from_ordset(List) -> Set when List :: [Element], Set :: set(Element).

    Turns an ordered list without duplicates List into a set.

    See from_list/1 for a function that accepts unordered lists with duplicates.

  Examples

    1> Ordset = [1,2,3].
    2> gb_sets:to_list(gb_sets:from_ordset(Ordset)).
    [1,2,3]
insert(Element, Set1) -> Set2 when Set1 :: set(Element), Set2 :: set(Element).

    Returns a new set formed from Set1 with Element inserted, assuming Element is not already present.

    Use add/2 for inserting into a set where Element is potentially already present.

  Examples

    1> S0 = gb_sets:new().
    2> S1 = gb_sets:insert(7, S0).
    3> gb_sets:to_list(S1).
    [7]
    4> S2 = gb_sets:insert(42, S1).
    5> gb_sets:to_list(S2).
    [7,42]
intersection(SetList) -> Set when SetList :: [set(Element), ...], Set :: set(Element).

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

    The intersection of multiple sets is a new set that contains only the elements that are present in all sets.

  Examples

    1> S0 = gb_sets:from_list([a,b,c,d]).
    2> S1 = gb_sets:from_list([d,e,f]).
    3> S2 = gb_sets:from_list([q,r]).
    4> Sets = [S0, S1, S2].
    5> gb_sets:to_list(gb_sets:intersection([S0, S1, S2])).
    []
    6> gb_sets:to_list(gb_sets:intersection([S0, S1])).
    [d]
    7> gb_sets:intersection([]).
    ** exception error: no function clause matching gb_sets:intersection([])
intersection(Set1, Set2) -> Set3 when Set1 :: set(Element), Set2 :: set(Element), Set3 :: set(Element).

    Returns the intersection of Set1 and Set2.

    The intersection of two sets is a new set that contains only the elements that are present in both sets.

  Examples

    1> S0 = gb_sets:from_list([a,b,c,d]).
    2> S1 = gb_sets:from_list([c,d,e,f]).
    3> S2 = gb_sets:from_list([q,r]).
    4> gb_sets:to_list(gb_sets:intersection(S0, S1)).
    [c,d]
    5> gb_sets:to_list(gb_sets:intersection(S1, S2)).
    []
is_disjoint(Set1, Set2) -> boolean() when Set1 :: set(Element), Set2 :: set(Element).

    Returns true if Set1 and Set2 are disjoint; otherwise, returns false.

    Two sets are disjoint if they have no elements in common.

    This function is equivalent to gb_sets:is_empty(gb_sets:intersection(Set1, Set2)), but faster.

  Examples

    1> S0 = gb_sets:from_list([a,b,c,d]).
    2> S1 = gb_sets:from_list([d,e,f]).
    3> S2 = gb_sets:from_list([q,r]).
    4> gb_sets:is_disjoint(S0, S1).
    false
    5> gb_sets:is_disjoint(S1, S2).
    true
is_empty(Set) -> boolean() when Set :: set().

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

  Examples

    1> gb_sets:is_empty(gb_sets:new()).
    true
    2> gb_sets:is_empty(gb_sets:singleton(1)).
    false
is_equal(Set1, Set2) -> boolean() when Set1 :: set(), Set2 :: set().

    Returns true if Set1 and Set2 are equal, that is, if every element of one set is also a member of the other set; otherwise, returns false.

  Examples

    1> Empty = gb_sets:new().
    2> S = gb_sets:from_list([a,b]).
    3> gb_sets:is_equal(S, S).
    true
    4> gb_sets:is_equal(S, Empty).
    false
is_member(Element, Set) -> boolean() when Set :: set(Element).

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

  Examples

    1> S = gb_sets:from_list([a,b,c]).
    2> gb_sets:is_member(42, S).
    false
    3> gb_sets:is_member(b, S).
    true
is_set(Term) -> boolean() when Term :: term().

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

    NOTE: This function will return true for any term that coincides with the representation of a gb_set, while not really being a gb_set, thus it might return false positive results. See also note on data types. Furthermore, since gb_sets are opaque, calling this function on terms that are not gb_sets could result in dialyzer warnings.

  Examples

    1> gb_sets:is_set(gb_sets:new()).
    true
    2> gb_sets:is_set(gb_sets:singleton(42)).
    true
    3> gb_sets:is_set(0).
    false
is_subset(Set1, Set2) -> boolean() when Set1 :: set(Element), Set2 :: set(Element).

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

  Examples

    1> S0 = gb_sets:from_list([a,b,c,d]).
    2> S1 = gb_sets:from_list([c,d]).
    3> gb_sets:is_subset(S1, S0).
    true
    4> gb_sets:is_subset(S0, S1).
    false
    5> gb_sets:is_subset(S0, S0).
    true
iterator(Set, Order) -> Iter when Set :: set(Element), Iter :: iter(Element), Order :: ordered | reversed.

    Returns an iterator that can be used for traversing the entries of Set in either ordered or reversed direction; see next/1.

    The implementation 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 avoids building the complete list of all elements in memory at once.

        1> S = gb_sets:from_list([1,2,3,4,5]).
        2> Iter0 = gb_sets:iterator(S, ordered).
        3> element(1, gb_sets:next(Iter0)).
        1
        4> Iter1 = gb_sets:iterator(S, reversed).
        5> element(1, gb_sets:next(Iter1)).
        5
iterator_from(Element, Set, Order) -> Iter when Set :: set(Element), Iter :: iter(Element), Order :: ordered | reversed.

    Returns an iterator over members of Set in the given Order, starting from Element or, if absent, the first member that follows in the iteration order, if any; see next/1.

  Examples

    1> S = gb_sets:from_list([10,20,30,40,50]).
    2> Iter1 = gb_sets:iterator_from(17, S, ordered).
    3> element(1, gb_sets:next(Iter1)).
    20
    4> Iter2 = gb_sets:iterator_from(17, S, reversed).
    5> element(1, gb_sets:next(Iter2)).
    10
larger(Element1, Set) -> none | {found, Element2} when Element1 :: Element, Element2 :: Element, Set :: set(Element).

    Returns {found, Element2}, where Element2 is the least element strictly greater than Element1.

    Returns none if no such element exists.

  Examples

    1> S = gb_sets:from_list([10,20,30]).
    2> gb_sets:larger(1, S).
    {found,10}
    3> gb_sets:larger(10, S).
    {found,20}
    4> gb_sets:larger(19, S).
    {found,20}
    5> gb_sets:larger(30, S).
    none
largest(Set) -> Element when Set :: set(Element).

    Returns the largest element in Set.

    Assumes that Set is not empty.

  Examples

    1> S = gb_sets:from_list([a,b,c]).
    2> gb_sets:largest(S).
    c
map(Fun, Set1) -> Set2 when Fun :: fun((Element1) -> Element2), Set1 :: set(Element1), Set2 :: set(Element2).

    Maps elements in Set1 with mapping function Fun.

  Examples

    1> S = gb_sets:from_list([1,2,3,4,5,6,7]).
    2> F = fun(N) -> N div 2 end.
    3> Mapped = gb_sets:map(F, S).
    4> gb_sets:to_list(Mapped).
    [0,1,2,3]
new() -> Set when Set :: set(none()).

    Returns a new empty set.

  Examples

    1> gb_sets:to_list(gb_sets:new()).
    []
next(Iter1) -> {Element, Iter2} | none when Iter1 :: iter(Element), Iter2 :: iter(Element).

    Returns {Element, Iter2}, where Element is the first 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.

        1> S = gb_sets:from_list([1,2,3,4,5]).
        2> Iter0 = gb_sets:iterator(S).
        3> {Element0, Iter1} = gb_sets:next(Iter0).
        4> Element0.
        1
        5> {Element1, Iter2} = gb_sets:next(Iter1).
        6> Element1.
        2
singleton(Element) -> set(Element).

    Returns a set containing only element Element.

  Examples

    1> S = gb_sets:singleton(42).
    2> gb_sets:to_list(S).
    [42]
size(Set) -> non_neg_integer() when Set :: set().

    Returns the number of elements in Set.

  Examples

    1> gb_sets:size(gb_sets:new()).
    0
    2> gb_sets:size(gb_sets:from_list([4,5,6])).
    3
smaller(Element1, Set) -> none | {found, Element2} when Element1 :: Element, Element2 :: Element, Set :: set(Element).

    Returns {found, Element2}, where Element2 is the greatest element strictly less than Element1.

    Returns none if no such element exists.

  Examples

    1> S = gb_sets:from_list([a,b,c]).
    2> gb_sets:smaller(b, S).
    {found,a}
    3> gb_sets:smaller(z, S).
    {found,c}
    4> gb_sets:smaller(a, S).
    none
smallest(Set) -> Element when Set :: set(Element).

    Returns the smallest element in Set.

    Assumes that Set is not empty.

  Examples

    1> S = gb_sets:from_list([a,b,c]).
    2> gb_sets:smallest(S).
    a
subtract(Set1, Set2) -> Set3 when Set1 :: set(Element), Set2 :: set(Element), Set3 :: set(Element).

    Returns a new set containing the elements of Set1 that are not elements in Set2.

  Examples

    1> S0 = gb_sets:from_list([a,b,c,d]).
    2> S1 = gb_sets:from_list([c,d,e,f]).
    3> gb_sets:to_list(gb_sets:subtract(S0, S1)).
    [a,b]
    4> gb_sets:to_list(gb_sets:subtract(S1, S0)).
    [e,f]
take_largest(Set1) -> {Element, Set2} when Set1 :: set(Element), 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.

  Examples

    1> S0 = gb_sets:from_list([a,b,c]).
    2> {Largest,S1} = gb_sets:take_largest(S0).
    3> Largest.
    c
    4> gb_sets:to_list(S1).
    [a,b]
take_smallest(Set1) -> {Element, Set2} when Set1 :: set(Element), 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.

  Examples

    1> S0 = gb_sets:from_list([a,b,c]).
    2> {Smallest,S1} = gb_sets:take_smallest(S0).
    3> Smallest.
    a
    4> gb_sets:to_list(S1).
    [b,c]
to_list(Set) -> List when Set :: set(Element), List :: [Element].

    Returns the elements of Set as an ordered list.

        1> gb_sets:to_list(gb_sets:from_list([4,3,5,1,2])).
        [1,2,3,4,5]
union(SetList) -> Set when SetList :: [set(Element)], Set :: set(Element).

    Returns the union of a list of sets.

    The union of multiple sets is a new set that contains all the elements from all sets, without duplicates.

  Examples

    1> S0 = gb_sets:from_list([a,b,c,d]).
    2> S1 = gb_sets:from_list([d,e,f]).
    3> S2 = gb_sets:from_list([q,r]).
    4> Sets = [S0, S1, S2].
    5> Union = gb_sets:union(Sets).
    6> gb_sets:to_list(Union).
    [a,b,c,d,e,f,q,r]
union(Set1, Set2) -> Set3 when Set1 :: set(Element), Set2 :: set(Element), Set3 :: set(Element).

    Returns the union of Set1 and Set2.

    The union of two sets is a new set that contains all the elements from both sets, without duplicates.

  Examples

    1> S0 = gb_sets:from_list([a,b,c,d]).
    2> S1 = gb_sets:from_list([c,d,e,f]).
    3> Union = gb_sets:union(S0, S1).
    4> gb_sets:to_list(Union).
    [a,b,c,d,e,f]