%%
%% %CopyrightBegin%
%%
%% Copyright Ericsson AB 2001-2009. All Rights Reserved.
%%
%% The contents of this file are subject to the Erlang Public License,
%% Version 1.1, (the "License"); you may not use this file except in
%% compliance with the License. You should have received a copy of the
%% Erlang Public License along with this software. If not, it can be
%% retrieved online at http://www.erlang.org/.
%%
%% Software distributed under the License is distributed on an "AS IS"
%% basis, WITHOUT WARRANTY OF ANY KIND, either express or implied. See
%% the License for the specific language governing rights and limitations
%% under the License.
%%
%% %CopyrightEnd%
%%
%% =====================================================================
%% Ordered Sets implemented as General Balanced Trees
%%
%% Copyright (C) 1999-2001 Richard Carlsson
%%
%% 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. See
%% notes below for details.
%% ---------------------------------------------------------------------
%% Notes:
%%
%% 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 ackumulating 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.
%%
%% Operations:
%%
%% - empty(): returns empty set.
%%
%% Alias: new(), for compatibility with `sets'.
%%
%% - is_empty(S): returns 'true' if S is an empty set, and 'false'
%% otherwise.
%%
%% - size(S): returns the number of nodes in the set as an integer.
%% Returns 0 (zero) if the set is empty.
%%
%% - singleton(X): returns a set containing only the element X.
%%
%% - is_member(X, S): returns `true' if element X is a member of set S,
%% and `false' otherwise.
%%
%% Alias: is_element(), for compatibility with `sets'.
%%
%% - insert(X, S): inserts element X into set S; returns the new set.
%% *Assumes that the element is not present in S.*
%%
%% - add(X, S): adds element X to set S; returns the new set. If X is
%% already an element in S, nothing is changed.
%%
%% Alias: add_element(), for compatibility with `sets'.
%%
%% - delete(X, S): removes element X from set S; returns new set.
%% Assumes that the element exists in the set.
%%
%% - delete_any(X, S): removes key X from set S if the key is present
%% in the set, otherwise does nothing; returns new set.
%%
%% Alias: del_element(), for compatibility with `sets'.
%%
%% - balance(S): rebalances the tree representation of S. 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.
%%
%% - union(S1, S2): returns a new set that contains each element that is
%% in either S1 or S2 or both, and no other elements.
%%
%% - union(Ss): returns a new set that contains each element that is in
%% at least one of the sets in the list Ss, and no other elements.
%%
%% - intersection(S1, S2): returns a new set that contains each element
%% that is in both S1 and S2, and no other elements.
%%
%% - intersection(Ss): returns a new set that contains each element that
%% is in all of the sets in the list Ss, and no other elements.
%%
%% - is_disjoint(S1, S2): returns `true' if none of the elements in S1
%% occurs in S2.
%%
%% - difference(S1, S2): returns a new set that contains each element in
%% S1 that is not also in S2, and no other elements.
%%
%% Alias: subtract(), for compatibility with `sets'.
%%
%% - is_subset(S1, S2): returns `true' if each element in S1 is also a
%% member of S2, and `false' otherwise.
%%
%% - to_list(S): returns an ordered list of all elements in set S. The
%% list never contains duplicates.
%%
%% - from_list(List): creates a set containing all elements in List,
%% where List may be unordered and contain duplicates.
%%
%% - from_ordset(L): turns an ordered-set list L into a set. The list
%% must not contain duplicates.
%%
%% - smallest(S): returns the smallest element in set S. Assumes that
%% the set S is nonempty.
%%
%% - largest(S): returns the largest element in set S. Assumes that the
%% set S is nonempty.
%%
%% - take_smallest(S): returns {X, S1}, where X is the smallest element
%% in set S, and S1 is the set S with element X deleted. Assumes that
%% the set S is nonempty.
%%
%% - take_largest(S): returns {X, S1}, where X is the largest element in
%% set S, and S1 is the set S with element X deleted. Assumes that the
%% set S is nonempty.
%%
%% - iterator(S): returns an iterator that can be used for traversing
%% the entries of set S; see `next'. The implementation of this is
%% very efficient; traversing the whole set using `next' is only
%% slightly slower than getting the list of all elements using
%% `to_list' 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.
%%
%% - next(T): returns {X, T1} where X is the smallest element referred
%% to by the iterator T, and T1 is the new iterator to be used for
%% traversing the remaining elements, or the atom `none' if no
%% elements remain.
%%
%% - filter(P, S): Filters set S using predicate function P. Included
%% for compatibility with `sets'.
%%
%% - fold(F, A, S): Folds function F over set S with A as the initial
%% ackumulator. Included for compatibility with `sets'.
%%
%% - is_set(S): returns 'true' if S appears to be a set, and 'false'
%% otherwise. Not recommended; included for compatibility with `sets'.
-module(gb_sets).
-export([empty/0, is_empty/1, size/1, singleton/1, is_member/2,
insert/2, add/2, delete/2, delete_any/2, balance/1, union/2,
union/1, intersection/2, intersection/1, is_disjoint/2, difference/2,
is_subset/2, to_list/1, from_list/1, from_ordset/1, smallest/1,
largest/1, take_smallest/1, take_largest/1, iterator/1, next/1,
filter/2, fold/3, is_set/1]).
%% `sets' compatibility aliases:
-export([new/0, is_element/2, add_element/2, del_element/2,
subtract/2]).
%% GB-trees adapted from Sven-Olof Nystr�m's implementation for
%% representation of sets.
%%
%% Data structures:
%% - {Size, Tree}, where `Tree' is composed of nodes of the form:
%% - {Key, Smaller, Bigger}, and the "empty tree" node:
%% - nil.
%%
%% No attempt is made to balance trees after deletions. Since deletions
%% don't increase the height of a tree, this should be OK.
%%
%% Original balance condition h(T) <= ceil(c * log(|T|)) has been
%% changed to the similar (but not quite equivalent) condition 2 ^ h(T)
%% <= |T| ^ c. This should also be OK.
%%
%% Behaviour is logarithmic (as it should be).
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% Some macros.
-define(p, 2). % It seems that p = 2 is optimal for sorted keys
-define(pow(A, _), A * A). % correct with exponent as defined above.
-define(div2(X), X bsr 1).
-define(mul2(X), X bsl 1).
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% Some types.
-type gb_set_node() :: 'nil' | {term(), _, _}.
%% A declaration equivalent to the following is currently hard-coded
%% in erl_types.erl
%%
%% -opaque gb_set() :: {non_neg_integer(), gb_set_node()}.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-spec empty() -> gb_set().
empty() ->
{0, nil}.
-spec new() -> gb_set().
new() -> empty().
-spec is_empty(gb_set()) -> boolean().
is_empty({0, nil}) ->
true;
is_empty(_) ->
false.
-spec size(gb_set()) -> non_neg_integer().
size({Size, _}) ->
Size.
-spec singleton(term()) -> gb_set().
singleton(Key) ->
{1, {Key, nil, nil}}.
-spec is_element(term(), gb_set()) -> boolean().
is_element(Key, S) ->
is_member(Key, S).
-spec is_member(term(), gb_set()) -> boolean().
is_member(Key, {_, T}) ->
is_member_1(Key, T).
is_member_1(Key, {Key1, Smaller, _}) when Key < Key1 ->
is_member_1(Key, Smaller);
is_member_1(Key, {Key1, _, Bigger}) when Key > Key1 ->
is_member_1(Key, Bigger);
is_member_1(_, {_, _, _}) ->
true;
is_member_1(_, nil) ->
false.
-spec insert(term(), gb_set()) -> gb_set().
insert(Key, {S, T}) ->
S1 = S + 1,
{S1, insert_1(Key, T, ?pow(S1, ?p))}.
insert_1(Key, {Key1, Smaller, Bigger}, S) when Key < Key1 ->
case insert_1(Key, Smaller, ?div2(S)) of
{T1, H1, S1} when is_integer(H1) ->
T = {Key1, T1, Bigger},
{H2, S2} = count(Bigger),
H = ?mul2(erlang:max(H1, H2)),
SS = S1 + S2 + 1,
P = ?pow(SS, ?p),
if
H > P ->
balance(T, SS);
true ->
{T, H, SS}
end;
T1 ->
{Key1, T1, Bigger}
end;
insert_1(Key, {Key1, Smaller, Bigger}, S) when Key > Key1 ->
case insert_1(Key, Bigger, ?div2(S)) of
{T1, H1, S1} when is_integer(H1) ->
T = {Key1, Smaller, T1},
{H2, S2} = count(Smaller),
H = ?mul2(erlang:max(H1, H2)),
SS = S1 + S2 + 1,
P = ?pow(SS, ?p),
if
H > P ->
balance(T, SS);
true ->
{T, H, SS}
end;
T1 ->
{Key1, Smaller, T1}
end;
insert_1(Key, nil, 0) ->
{{Key, nil, nil}, 1, 1};
insert_1(Key, nil, _) ->
{Key, nil, nil};
insert_1(Key, _, _) ->
erlang:error({key_exists, Key}).
count({_, nil, nil}) ->
{1, 1};
count({_, Sm, Bi}) ->
{H1, S1} = count(Sm),
{H2, S2} = count(Bi),
{?mul2(erlang:max(H1, H2)), S1 + S2 + 1};
count(nil) ->
{1, 0}.
-spec balance(gb_set()) -> gb_set().
balance({S, T}) ->
{S, balance(T, S)}.
balance(T, S) ->
balance_list(to_list_1(T), S).
balance_list(L, S) ->
{T, _} = balance_list_1(L, S),
T.
balance_list_1(L, S) when S > 1 ->
Sm = S - 1,
S2 = Sm div 2,
S1 = Sm - S2,
{T1, [K | L1]} = balance_list_1(L, S1),
{T2, L2} = balance_list_1(L1, S2),
T = {K, T1, T2},
{T, L2};
balance_list_1([Key | L], 1) ->
{{Key, nil, nil}, L};
balance_list_1(L, 0) ->
{nil, L}.
-spec add_element(term(), gb_set()) -> gb_set().
add_element(X, S) ->
add(X, S).
-spec add(term(), gb_set()) -> gb_set().
add(X, S) ->
case is_member(X, S) of
true ->
S; % we don't have to do anything here
false ->
insert(X, S)
end.
-spec from_list([term()]) -> gb_set().
from_list(L) ->
from_ordset(ordsets:from_list(L)).
-spec from_ordset([term()]) -> gb_set().
from_ordset(L) ->
S = length(L),
{S, balance_list(L, S)}.
-spec del_element(term(), gb_set()) -> gb_set().
del_element(Key, S) ->
delete_any(Key, S).
-spec delete_any(term(), gb_set()) -> gb_set().
delete_any(Key, S) ->
case is_member(Key, S) of
true ->
delete(Key, S);
false ->
S
end.
-spec delete(term(), gb_set()) -> gb_set().
delete(Key, {S, T}) ->
{S - 1, delete_1(Key, T)}.
delete_1(Key, {Key1, Smaller, Larger}) when Key < Key1 ->
Smaller1 = delete_1(Key, Smaller),
{Key1, Smaller1, Larger};
delete_1(Key, {Key1, Smaller, Bigger}) when Key > Key1 ->
Bigger1 = delete_1(Key, Bigger),
{Key1, Smaller, Bigger1};
delete_1(_, {_, Smaller, Larger}) ->
merge(Smaller, Larger).
merge(Smaller, nil) ->
Smaller;
merge(nil, Larger) ->
Larger;
merge(Smaller, Larger) ->
{Key, Larger1} = take_smallest1(Larger),
{Key, Smaller, Larger1}.
-spec take_smallest(gb_set()) -> {term(), gb_set()}.
take_smallest({S, T}) ->
{Key, Larger} = take_smallest1(T),
{Key, {S - 1, Larger}}.
take_smallest1({Key, nil, Larger}) ->
{Key, Larger};
take_smallest1({Key, Smaller, Larger}) ->
{Key1, Smaller1} = take_smallest1(Smaller),
{Key1, {Key, Smaller1, Larger}}.
-spec smallest(gb_set()) -> term().
smallest({_, T}) ->
smallest_1(T).
smallest_1({Key, nil, _Larger}) ->
Key;
smallest_1({_Key, Smaller, _Larger}) ->
smallest_1(Smaller).
-spec take_largest(gb_set()) -> {term(), gb_set()}.
take_largest({S, T}) ->
{Key, Smaller} = take_largest1(T),
{Key, {S - 1, Smaller}}.
take_largest1({Key, Smaller, nil}) ->
{Key, Smaller};
take_largest1({Key, Smaller, Larger}) ->
{Key1, Larger1} = take_largest1(Larger),
{Key1, {Key, Smaller, Larger1}}.
-spec largest(gb_set()) -> term().
largest({_, T}) ->
largest_1(T).
largest_1({Key, _Smaller, nil}) ->
Key;
largest_1({_Key, _Smaller, Larger}) ->
largest_1(Larger).
-spec to_list(gb_set()) -> [term()].
to_list({_, T}) ->
to_list(T, []).
to_list_1(T) -> to_list(T, []).
to_list({Key, Small, Big}, L) ->
to_list(Small, [Key | to_list(Big, L)]);
to_list(nil, L) -> L.
-spec iterator(gb_set()) -> [term()].
iterator({_, T}) ->
iterator(T, []).
%% The iterator structure is really just a list corresponding to the
%% call stack of an in-order traversal. This is quite fast.
iterator({_, nil, _} = T, As) ->
[T | As];
iterator({_, L, _} = T, As) ->
iterator(L, [T | As]);
iterator(nil, As) ->
As.
-spec next([term()]) -> {term(), [term()]} | 'none'.
next([{X, _, T} | As]) ->
{X, iterator(T, As)};
next([]) ->
none.
%% Set operations:
%% If |X| < |Y|, then we traverse the elements of X. The cost for
%% testing a single random element for membership in a tree S is
%% proportional to log(|S|); thus, if |Y| / |X| < c * log(|Y|), for some
%% c, it is more efficient to scan the ordered sequence of elements of Y
%% while traversing X (under the same ordering) in order to test whether
%% elements of X are already in Y. Since the `math' module does not have
%% a `log2'-function, we rewrite the condition to |X| < |Y| * c1 *
%% ln(|X|), where c1 = c / ln 2.
-define(c, 1.46). % 1 / ln 2; this appears to be best
%% If the sets are not very different in size, i.e., if |Y| / |X| >= c *
%% log(|Y|), then the fastest way to do union (and the other similar set
%% operations) is to build the lists of elements, traverse these lists
%% in parallel while building a reversed ackumulator list, and finally
%% rebuild the tree directly from the ackumulator. Other methods of
%% traversing the elements can be devised, but they all have higher
%% overhead.
-spec union(gb_set(), gb_set()) -> gb_set().
union({N1, T1}, {N2, T2}) when N2 < N1 ->
union(to_list_1(T2), N2, T1, N1);
union({N1, T1}, {N2, T2}) ->
union(to_list_1(T1), N1, T2, N2).
%% We avoid the expensive mathematical computations if there is little
%% chance at saving at least the same amount of time by making the right
%% choice of strategy. Recall that N1 < N2 here.
union(L, N1, T2, N2) when N2 < 10 ->
%% Break even is about 7 for N1 = 1 and 10 for N1 = 2
union_2(L, to_list_1(T2), N1 + N2);
union(L, N1, T2, N2) ->
X = N1 * round(?c * math:log(N2)),
if N2 < X ->
union_2(L, to_list_1(T2), N1 + N2);
true ->
union_1(L, mk_set(N2, T2))
end.
-spec mk_set(non_neg_integer(), gb_set_node()) -> gb_set().
mk_set(N, T) ->
{N, T}.
%% If the length of the list is in proportion with the size of the
%% target set, this version spends too much time doing lookups, compared
%% to the below version.
union_1([X | Xs], S) ->
union_1(Xs, add(X, S));
union_1([], S) ->
S.
%% If the length of the first list is too small in comparison with the
%% size of the target set, this version spends too much time scanning
%% the element list of the target set for possible membership, compared
%% with the above version.
%% Some notes on sequential scanning of ordered lists
%%
%% 1) We want to put the equality case last, if we can assume that the
%% probability for overlapping elements is relatively low on average.
%% Doing this also allows us to completely skip the (arithmetic)
%% equality test, since the term order is arithmetically total.
%%
%% 2) We always test for `smaller than' first, i.e., whether the head of
%% the left list is smaller than the head of the right list, and if the
%% `greater than' test should instead turn out to be true, we switch
%% left and right arguments in the recursive call under the assumption
%% that the same is likely to apply to the next element also,
%% statistically reducing the number of failed tests and automatically
%% adapting to cases of lists having very different lengths. This saves
%% 10-40% of the traversation time compared to a "fixed" strategy,
%% depending on the sizes and contents of the lists.
%%
%% 3) A tail recursive version using `lists:reverse/2' is about 5-10%
%% faster than a plain recursive version using the stack, for lists of
%% more than about 20 elements and small stack frames. For very short
%% lists, however (length < 10), the stack version can be several times
%% faster. As stack frames grow larger, the advantages of using
%% `reverse' could get greater.
union_2(Xs, Ys, S) ->
union_2(Xs, Ys, [], S). % S is the sum of the sizes here
union_2([X | Xs1], [Y | _] = Ys, As, S) when X < Y ->
union_2(Xs1, Ys, [X | As], S);
union_2([X | _] = Xs, [Y | Ys1], As, S) when X > Y ->
union_2(Ys1, Xs, [Y | As], S);
union_2([X | Xs1], [_ | Ys1], As, S) ->
union_2(Xs1, Ys1, [X | As], S - 1);
union_2([], Ys, As, S) ->
{S, balance_revlist(push(Ys, As), S)};
union_2(Xs, [], As, S) ->
{S, balance_revlist(push(Xs, As), S)}.
push([X | Xs], As) ->
push(Xs, [X | As]);
push([], As) ->
As.
balance_revlist(L, S) ->
{T, _} = balance_revlist_1(L, S),
T.
balance_revlist_1(L, S) when S > 1 ->
Sm = S - 1,
S2 = Sm div 2,
S1 = Sm - S2,
{T2, [K | L1]} = balance_revlist_1(L, S1),
{T1, L2} = balance_revlist_1(L1, S2),
T = {K, T1, T2},
{T, L2};
balance_revlist_1([Key | L], 1) ->
{{Key, nil, nil}, L};
balance_revlist_1(L, 0) ->
{nil, L}.
-spec union([gb_set()]) -> gb_set().
union([S | Ss]) ->
union_list(S, Ss);
union([]) -> empty().
union_list(S, [S1 | Ss]) ->
union_list(union(S, S1), Ss);
union_list(S, []) -> S.
%% The rest is modelled on the above.
-spec intersection(gb_set(), gb_set()) -> gb_set().
intersection({N1, T1}, {N2, T2}) when N2 < N1 ->
intersection(to_list_1(T2), N2, T1, N1);
intersection({N1, T1}, {N2, T2}) ->
intersection(to_list_1(T1), N1, T2, N2).
intersection(L, _N1, T2, N2) when N2 < 10 ->
intersection_2(L, to_list_1(T2));
intersection(L, N1, T2, N2) ->
X = N1 * round(?c * math:log(N2)),
if N2 < X ->
intersection_2(L, to_list_1(T2));
true ->
intersection_1(L, T2)
end.
%% We collect the intersecting elements in an accumulator list and count
%% them at the same time so we can balance the list afterwards.
intersection_1(Xs, T) ->
intersection_1(Xs, T, [], 0).
intersection_1([X | Xs], T, As, N) ->
case is_member_1(X, T) of
true ->
intersection_1(Xs, T, [X | As], N + 1);
false ->
intersection_1(Xs, T, As, N)
end;
intersection_1([], _, As, N) ->
{N, balance_revlist(As, N)}.
intersection_2(Xs, Ys) ->
intersection_2(Xs, Ys, [], 0).
intersection_2([X | Xs1], [Y | _] = Ys, As, S) when X < Y ->
intersection_2(Xs1, Ys, As, S);
intersection_2([X | _] = Xs, [Y | Ys1], As, S) when X > Y ->
intersection_2(Ys1, Xs, As, S);
intersection_2([X | Xs1], [_ | Ys1], As, S) ->
intersection_2(Xs1, Ys1, [X | As], S + 1);
intersection_2([], _, As, S) ->
{S, balance_revlist(As, S)};
intersection_2(_, [], As, S) ->
{S, balance_revlist(As, S)}.
-spec intersection([gb_set(),...]) -> gb_set().
intersection([S | Ss]) ->
intersection_list(S, Ss).
intersection_list(S, [S1 | Ss]) ->
intersection_list(intersection(S, S1), Ss);
intersection_list(S, []) -> S.
-spec is_disjoint(gb_set(), gb_set()) -> boolean().
is_disjoint({N1, T1}, {N2, T2}) when N1 < N2 ->
is_disjoint_1(T1, T2);
is_disjoint({_, T1}, {_, T2}) ->
is_disjoint_1(T2, T1).
is_disjoint_1({K1, Smaller1, Bigger}, {K2, Smaller2, _}=Tree) when K1 < K2 ->
not is_member_1(K1, Smaller2) andalso
is_disjoint_1(Smaller1, Smaller2) andalso
is_disjoint_1(Bigger, Tree);
is_disjoint_1({K1, Smaller, Bigger1}, {K2, _, Bigger2}=Tree) when K1 > K2 ->
not is_member_1(K1, Bigger2) andalso
is_disjoint_1(Bigger1, Bigger2) andalso
is_disjoint_1(Smaller, Tree);
is_disjoint_1({_K1, _, _}, {_K2, _, _}) -> %K1 == K2
false;
is_disjoint_1(nil, _) ->
true;
is_disjoint_1(_, nil) ->
true.
%% Note that difference is not symmetric. We don't use `delete' here,
%% since the GB-trees implementation does not rebalance after deletion
%% and so we could end up with very unbalanced trees indeed depending on
%% the sets. Therefore, we always build a new tree, and thus we need to
%% traverse the whole element list of the left operand.
-spec subtract(gb_set(), gb_set()) -> gb_set().
subtract(S1, S2) ->
difference(S1, S2).
-spec difference(gb_set(), gb_set()) -> gb_set().
difference({N1, T1}, {N2, T2}) ->
difference(to_list_1(T1), N1, T2, N2).
difference(L, N1, T2, N2) when N2 < 10 ->
difference_2(L, to_list_1(T2), N1);
difference(L, N1, T2, N2) ->
X = N1 * round(?c * math:log(N2)),
if N2 < X ->
difference_2(L, to_list_1(T2), N1);
true ->
difference_1(L, T2)
end.
difference_1(Xs, T) ->
difference_1(Xs, T, [], 0).
difference_1([X | Xs], T, As, N) ->
case is_member_1(X, T) of
true ->
difference_1(Xs, T, As, N);
false ->
difference_1(Xs, T, [X | As], N + 1)
end;
difference_1([], _, As, N) ->
{N, balance_revlist(As, N)}.
difference_2(Xs, Ys, S) ->
difference_2(Xs, Ys, [], S). % S is the size of the left set
difference_2([X | Xs1], [Y | _] = Ys, As, S) when X < Y ->
difference_2(Xs1, Ys, [X | As], S);
difference_2([X | _] = Xs, [Y | Ys1], As, S) when X > Y ->
difference_2(Xs, Ys1, As, S);
difference_2([_X | Xs1], [_Y | Ys1], As, S) ->
difference_2(Xs1, Ys1, As, S - 1);
difference_2([], _Ys, As, S) ->
{S, balance_revlist(As, S)};
difference_2(Xs, [], As, S) ->
{S, balance_revlist(push(Xs, As), S)}.
%% Subset testing is much the same thing as set difference, but
%% without the construction of a new set.
-spec is_subset(gb_set(), gb_set()) -> boolean().
is_subset({N1, T1}, {N2, T2}) ->
is_subset(to_list_1(T1), N1, T2, N2).
is_subset(L, _N1, T2, N2) when N2 < 10 ->
is_subset_2(L, to_list_1(T2));
is_subset(L, N1, T2, N2) ->
X = N1 * round(?c * math:log(N2)),
if N2 < X ->
is_subset_2(L, to_list_1(T2));
true ->
is_subset_1(L, T2)
end.
is_subset_1([X | Xs], T) ->
case is_member_1(X, T) of
true ->
is_subset_1(Xs, T);
false ->
false
end;
is_subset_1([], _) ->
true.
is_subset_2([X | _], [Y | _]) when X < Y ->
false;
is_subset_2([X | _] = Xs, [Y | Ys1]) when X > Y ->
is_subset_2(Xs, Ys1);
is_subset_2([_ | Xs1], [_ | Ys1]) ->
is_subset_2(Xs1, Ys1);
is_subset_2([], _) ->
true;
is_subset_2(_, []) ->
false.
%% For compatibility with `sets':
-spec is_set(term()) -> boolean().
is_set({0, nil}) -> true;
is_set({N, {_, _, _}}) when is_integer(N), N >= 0 -> true;
is_set(_) -> false.
-spec filter(fun((term()) -> boolean()), gb_set()) -> gb_set().
filter(F, S) ->
from_ordset([X || X <- to_list(S), F(X)]).
-spec fold(fun((term(), term()) -> term()), term(), gb_set()) -> term().
fold(F, A, {_, T}) when is_function(F, 2) ->
fold_1(F, A, T).
fold_1(F, Acc0, {Key, Small, Big}) ->
Acc1 = fold_1(F, Acc0, Small),
Acc = F(Key, Acc1),
fold_1(F, Acc, Big);
fold_1(_, Acc, _) ->
Acc.
