Power set
You are encouraged to solve this task according to the task description, using any language you may know.
A set is a collection (container) of certain values, without any particular order, and no repeated values.
It corresponds with a finite set in mathematics.
A set can be implemented as an associative array (partial mapping) in which the value of each key-value pair is ignored.
Given a set S, the power set (or powerset) of S, written P(S), or 2S, is the set of all subsets of S.
- Task
By using a library or built-in set type, or by defining a set type with necessary operations, write a function with a set S as input that yields the power set 2S of S.
For example, the power set of {1,2,3,4} is
- {{}, {1}, {2}, {1,2}, {3}, {1,3}, {2,3}, {1,2,3}, {4}, {1,4}, {2,4}, {1,2,4}, {3,4}, {1,3,4}, {2,3,4}, {1,2,3,4}}.
For a set which contains n elements, the corresponding power set has 2n elements, including the edge cases of empty set.
The power set of the empty set is the set which contains itself (20 = 1):
- () = { }
- () = { }
And the power set of the set which contains only the empty set, has two subsets, the empty set and the set which contains the empty set (21 = 2):
- ({}) = { , { } }
- ({}) = { , { } }
Extra credit: Demonstrate that your language supports these last two powersets.
11l
F list_powerset(lst)
V result = [[Int]()]
L(x) lst
result.extend(result.map(subset -> subset [+] [@x]))
R result
print(list_powerset([1, 2, 3]))- Output:
[[], [1], [2], [1, 2], [3], [1, 3], [2, 3], [1, 2, 3]]
ABAP
This works for ABAP Version 7.40 and above
report z_powerset.
interface set.
methods:
add_element
importing
element_to_be_added type any
returning
value(new_set) type ref to set,
remove_element
importing
element_to_be_removed type any
returning
value(new_set) type ref to set,
contains_element
importing
element_to_be_found type any
returning
value(contains) type abap_bool,
get_size
returning
value(size) type int4,
is_equal
importing
set_to_be_compared_with type ref to set
returning
value(equal) type abap_bool,
get_elements
exporting
elements type any table,
stringify
returning
value(stringified_set) type string.
endinterface.
class string_set definition.
public section.
interfaces:
set.
methods:
constructor
importing
elements type stringtab optional,
build_powerset
returning
value(powerset) type ref to string_set.
private section.
data elements type stringtab.
endclass.
class string_set implementation.
method constructor.
loop at elements into data(element).
me->set~add_element( element ).
endloop.
endmethod.
method set~add_element.
if not line_exists( me->elements[ table_line = element_to_be_added ] ).
append element_to_be_added to me->elements.
endif.
new_set = me.
endmethod.
method set~remove_element.
if line_exists( me->elements[ table_line = element_to_be_removed ] ).
delete me->elements where table_line = element_to_be_removed.
endif.
new_set = me.
endmethod.
method set~contains_element.
contains = cond abap_bool(
when line_exists( me->elements[ table_line = element_to_be_found ] )
then abap_true
else abap_false ).
endmethod.
method set~get_size.
size = lines( me->elements ).
endmethod.
method set~is_equal.
if set_to_be_compared_with->get_size( ) ne me->set~get_size( ).
equal = abap_false.
return.
endif.
loop at me->elements into data(element).
if not set_to_be_compared_with->contains_element( element ).
equal = abap_false.
return.
endif.
endloop.
equal = abap_true.
endmethod.
method set~get_elements.
elements = me->elements.
endmethod.
method set~stringify.
stringified_set = cond string(
when me->elements is initial
then `∅`
when me->elements eq value stringtab( ( `∅` ) )
then `{ ∅ }`
else reduce string(
init result = `{ `
for element in me->elements
next result = cond string(
when element eq ``
then |{ result }∅, |
when strlen( element ) eq 1 and element ne `∅`
then |{ result }{ element }, |
else |{ result }\{{ element }\}, | ) ) ).
stringified_set = replace(
val = stringified_set
regex = `, $`
with = ` }`).
endmethod.
method build_powerset.
data(powerset_elements) = value stringtab( ( `` ) ).
loop at me->elements into data(element).
do lines( powerset_elements ) times.
if powerset_elements[ sy-index ] ne `∅`.
append |{ powerset_elements[ sy-index ] }{ element }| to powerset_elements.
else.
append element to powerset_elements.
endif.
enddo.
endloop.
powerset = new string_set( powerset_elements ).
endmethod.
endclass.
start-of-selection.
data(set1) = new string_set( ).
data(set2) = new string_set( ).
data(set3) = new string_set( ).
write: |𝑷( { set1->set~stringify( ) } ) -> { set1->build_powerset( )->set~stringify( ) }|, /.
set2->set~add_element( `∅` ).
write: |𝑷( { set2->set~stringify( ) } ) -> { set2->build_powerset( )->set~stringify( ) }|, /.
set3->set~add_element( `1` )->add_element( `2` )->add_element( `3` )->add_element( `3` )->add_element( `4`
)->add_element( `4` )->add_element( `4` ).
write: |𝑷( { set3->set~stringify( ) } ) -> { set3->build_powerset( )->set~stringify( ) }|, /.
- Output:
𝑷( ∅ ) -> { ∅ }
𝑷( { ∅ } ) -> { ∅, {∅} }
𝑷( { 1, 2, 3, 4 } ) -> { ∅, 1, 2, {12}, 3, {13}, {23}, {123}, 4, {14}, {24}, {124}, {34}, {134}, {234}, {1234} }
Ada
A solution (without recursion) that prints the power set of the n arguments passed by the command line. The idea is that the i'th bit of a natural between 0 and indicates whether or not we should put the i'th element of the command line inside the set.
with Ada.Text_IO, Ada.Command_Line;
use Ada.Text_IO, Ada.Command_Line;
procedure powerset is
begin
for set in 0..2**Argument_Count-1 loop
Put ("{");
declare
k : natural := set;
first : boolean := true;
begin
for i in 1..Argument_Count loop
if k mod 2 = 1 then
Put ((if first then "" else ",") & Argument (i));
first := false;
end if;
k := k / 2; -- we go to the next bit of "set"
end loop;
end;
Put_Line("}");
end loop;
end powerset;
- Output:
>./powerset a b c d
{}
{a}
{b}
{a,b}
{c}
{a,c}
{b,c}
{a,b,c}
{d}
{a,d}
{b,d}
{a,b,d}
{c,d}
{a,c,d}
{b,c,d}
{a,b,c,d}
ALGOL 68
Requires: ALGOL 68g mk14.1+
MODE MEMBER = INT;
PROC power set = ([]MEMBER s)[][]MEMBER:(
[2**UPB s]FLEX[1:0]MEMBER r;
INT upb r := 0;
r[upb r +:= 1] := []MEMBER(());
FOR i TO UPB s DO
MEMBER e = s[i];
FOR j TO upb r DO
[UPB r[j] + 1]MEMBER x;
x[:UPB x-1] := r[j];
x[UPB x] := e; # append to the end of x #
r[upb r +:= 1] := x # append to end of r #
OD
OD;
r[upb r] := s;
r
);
# Example: #
test:(
[][]MEMBER set = power set((1, 2, 4));
FOR member TO UPB set DO
INT upb = UPB set[member];
FORMAT repr set = $"("f( upb=0 | $$ | $n(upb-1)(d", ")d$ )");"$;
printf(($"set["d"] = "$,member, repr set, set[member],$l$))
OD
)- Output:
set[1] = (); set[2] = (1); set[3] = (2); set[4] = (1, 2); set[5] = (4); set[6] = (1, 4); set[7] = (2, 4); set[8] = (1, 2, 4);
APL
ps←(↓∘⍉(2/⍨≢)⊤(⍳2*≢))(/¨)⊂
- Output:
ps 1 2 3 4 ┌─┬─┬───┬─┬───┬───┬─────┬─┬───┬───┬─────┬───┬─────┬─────┬───────┬┐ │4│3│3 4│2│2 4│2 3│2 3 4│1│1 4│1 3│1 3 4│1 2│1 2 4│1 2 3│1 2 3 4││ └─┴─┴───┴─┴───┴───┴─────┴─┴───┴───┴─────┴───┴─────┴─────┴───────┴┘
ps ⍬ ┌┐ ││ └┘
ps ,⊂⍬ ┌──┬┐ │┌┐││ │││││ │└┘││ └──┴┘
AppleScript
(functional composition examples)
-- POWER SET -----------------------------------------------------------------
-- powerset :: [a] -> [[a]]
on powerset(xs)
script subSet
on |λ|(acc, x)
script cons
on |λ|(y)
{x} & y
end |λ|
end script
acc & map(cons, acc)
end |λ|
end script
foldr(subSet, {{}}, xs)
end powerset
-- TEST ----------------------------------------------------------------------
on run
script test
on |λ|(x)
set {setName, setMembers} to x
{setName, powerset(setMembers)}
end |λ|
end script
map(test, [¬
["Set [1,2,3]", {1, 2, 3}], ¬
["Empty set", {}], ¬
["Set containing only empty set", {{}}]])
--> {{"Set [1,2,3]", {{}, {3}, {2}, {2, 3}, {1}, {1, 3}, {1, 2}, {1, 2, 3}}},
--> {"Empty set", {{}}},
--> {"Set containing only empty set", {{}, {{}}}}}
end run
-- GENERIC FUNCTIONS ---------------------------------------------------------
-- foldr :: (a -> b -> a) -> a -> [b] -> a
on foldr(f, startValue, xs)
tell mReturn(f)
set v to startValue
set lng to length of xs
repeat with i from lng to 1 by -1
set v to |λ|(v, item i of xs, i, xs)
end repeat
return v
end tell
end foldr
-- map :: (a -> b) -> [a] -> [b]
on map(f, xs)
tell mReturn(f)
set lng to length of xs
set lst to {}
repeat with i from 1 to lng
set end of lst to |λ|(item i of xs, i, xs)
end repeat
return lst
end tell
end map
-- Lift 2nd class handler function into 1st class script wrapper
-- mReturn :: Handler -> Script
on mReturn(f)
if class of f is script then
f
else
script
property |λ| : f
end script
end if
end mReturn
- Output:
{{"Set [1,2,3]", {{}, {3}, {2}, {2, 3}, {1}, {1, 3}, {1, 2}, {1, 2, 3}}},
{"Empty set", {{}}},
{"Set containing only empty set", {{}, {{}}}}}
Arturo
print powerset [1 2 3 4]
- Output:
[2 3 4] [] [1 2 4] [1 2 3 4] [1 3 4] [1] [2] [1 3] [3 4] [4] [1 4] [3] [1 2] [2 3] [1 2 3] [2 4]
ATS
(* ****** ****** *)
//
#include
"share/atspre_define.hats" // defines some names
#include
"share/atspre_staload.hats" // for targeting C
#include
"share/HATS/atspre_staload_libats_ML.hats" // for ...
//
(* ****** ****** *)
//
extern
fun
Power_set(xs: list0(int)): void
//
(* ****** ****** *)
// Helper: fast power function.
fun power(n: int, p: int): int =
if p = 1 then n
else if p = 0 then 1
else if p % 2 = 0 then power(n*n, p/2)
else n * power(n, p-1)
fun print_list(list: list0(int)): void =
case+ list of
| nil0() => println!(" ")
| cons0(car, crd) =>
let
val () = begin print car; print ','; end
val () = print_list(crd)
in
end
fun get_list_length(list: list0(int), length: int): int =
case+ list of
| nil0() => length
| cons0(car, crd) => get_list_length(crd, length+1)
fun get_list_from_bit_mask(mask: int, list: list0(int), result: list0(int)): list0(int) =
if mask = 0 then result
else
case+ list of
| nil0() => result
| cons0(car, crd) =>
let
val current: int = mask % 2
in
if current = 0 then
get_list_from_bit_mask(mask >> 1, crd, result)
else
get_list_from_bit_mask(mask >> 1, crd, list0_cons(car, result))
end
implement
Power_set(xs) = let
val len: int = get_list_length(xs, 0)
val pow: int = power(2, len)
fun loop(mask: int, list: list0(int)): void =
if mask > 0 && mask >= pow then ()
else
let
val () = print_list(get_list_from_bit_mask(mask, list, list0_nil()))
in
loop(mask+1, list)
end
in
loop(0, xs)
end
(* ****** ****** *)
implement
main0() =
let
val xs: list0(int) = cons0(1, list0_pair(2, 3))
in
Power_set(xs)
end (* end of [main0] *)
(* ****** ****** *)
AutoHotkey
ahk discussion
a = 1,a,-- ; elements separated by commas
StringSplit a, a, `, ; a0 = #elements, a1,a2,... = elements of the set
t = {
Loop % (1<<a0) { ; generate all 0-1 sequences
x := A_Index-1
Loop % a0
t .= (x>>A_Index-1) & 1 ? a%A_Index% "," : ""
t .= "}`n{" ; new subsets in new lines
}
MsgBox % RegExReplace(SubStr(t,1,StrLen(t)-1),",}","}")
AWK
cat power_set.awk
#!/usr/local/bin/gawk -f
# User defined function
function tochar(l,n, r) {
while (l) { n--; if (l%2 != 0) r = r sprintf(" %c ",49+n); l = int(l/2) }; return r
}
# For each input
{ for (i=0;i<=2^NF-1;i++) if (i == 0) printf("empty\n"); else printf("(%s)\n",tochar(i,NF)) }
- Output:
$ gawk -f power_set.awk 1 2 3 4 empty ( 4 ) ( 3 ) ( 4 3 ) ( 2 ) ( 4 2 ) ( 3 2 ) ( 4 3 2 ) ( 1 ) ( 4 1 ) ( 3 1 ) ( 4 3 1 ) ( 2 1 ) ( 4 2 1 ) ( 3 2 1 ) ( 4 3 2 1 )
BASIC
BBC BASIC
The elements of a set are represented as the bits in an integer (hence the maximum size of set is 32).
DIM list$(3) : list$() = "1", "2", "3", "4"
PRINT FNpowerset(list$())
END
DEF FNpowerset(list$())
IF DIM(list$(),1) > 31 ERROR 100, "Set too large to represent as integer"
LOCAL i%, j%, s$
s$ = "{"
FOR i% = 0 TO (2 << DIM(list$(),1)) - 1
s$ += "{"
FOR j% = 0 TO DIM(list$(),1)
IF i% AND (1 << j%) s$ += list$(j%) + ","
NEXT
IF RIGHT$(s$) = "," s$ = LEFT$(s$)
s$ += "},"
NEXT i%
= LEFT$(s$) + "}"
- Output:
{{},{1},{2},{1,2},{3},{1,3},{2,3},{1,2,3},{4},{1,4},{2,4},{1,2,4},{3,4},{1,3,4},{2,3,4},{1,2,3,4}}
BQN
P ← (⥊·↕2⥊˜≠)/¨<
- Output:
P 1‿2‿3‿4‿5 ⟨ ⟨⟩ ⟨ 5 ⟩ ⟨ 4 ⟩ ⟨ 4 5 ⟩ ⟨ 3 ⟩ ⟨ 3 5 ⟩ ⟨ 3 4 ⟩ ⟨ 3 4 5 ⟩ ⟨ 2 ⟩ ⟨ 2 5 ⟩ ⟨ 2 4 ⟩ ⟨ 2 4 5 ⟩ ⟨ 2 3 ⟩ ⟨ 2 3 5 ⟩ ⟨ 2 3 4 ⟩ ⟨ 2 3 4 5 ⟩ ⟨ 1 ⟩ ⟨ 1 5 ⟩ ⟨ 1 4 ⟩ ⟨ 1 4 5 ⟩ ⟨ 1 3 ⟩ ⟨ 1 3 5 ⟩ ⟨ 1 3 4 ⟩ ⟨ 1 3 4 5 ⟩ ⟨ 1 2 ⟩ ⟨ 1 2 5 ⟩ ⟨ 1 2 4 ⟩ ⟨ 1 2 4 5 ⟩ ⟨ 1 2 3 ⟩ ⟨ 1 2 3 5 ⟩ ⟨ 1 2 3 4 ⟩ ⟨ 1 2 3 4 5 ⟩ ⟩
Bracmat
( ( powerset
= done todo first
. !arg:(?done.?todo)
& ( !todo:%?first ?todo
& (powerset$(!done !first.!todo),powerset$(!done.!todo))
| !done
)
)
& out$(powerset$(.1 2 3 4))
);- Output:
1 2 3 4 , 1 2 3 , 1 2 4 , 1 2 , 1 3 4 , 1 3 , 1 4 , 1 , 2 3 4 , 2 3 , 2 4 , 2 , 3 4 , 3 , 4 ,
Burlesque
blsq ) {1 2 3 4}R@
{{} {1} {2} {1 2} {3} {1 3} {2 3} {1 2 3} {4} {1 4} {2 4} {1 2 4} {3 4} {1 3 4} {2 3 4} {1 2 3 4}}C
#include <stdio.h>
struct node {
char *s;
struct node* prev;
};
void powerset(char **v, int n, struct node *up)
{
struct node me;
if (!n) {
putchar('[');
while (up) {
printf(" %s", up->s);
up = up->prev;
}
puts(" ]");
} else {
me.s = *v;
me.prev = up;
powerset(v + 1, n - 1, up);
powerset(v + 1, n - 1, &me);
}
}
int main(int argc, char **argv)
{
powerset(argv + 1, argc - 1, 0);
return 0;
}
- Output:
% ./a.out 1 2 3 [ ] [ 3 ] [ 2 ] [ 3 2 ] [ 1 ] [ 3 1 ] [ 2 1 ] [ 3 2 1 ]
C#
public IEnumerable<IEnumerable<T>> GetPowerSet<T>(List<T> list)
{
return from m in Enumerable.Range(0, 1 << list.Count)
select
from i in Enumerable.Range(0, list.Count)
where (m & (1 << i)) != 0
select list[i];
}
public void PowerSetofColors()
{
var colors = new List<KnownColor> { KnownColor.Red, KnownColor.Green,
KnownColor.Blue, KnownColor.Yellow };
var result = GetPowerSet(colors);
Console.Write( string.Join( Environment.NewLine,
result.Select(subset =>
string.Join(",", subset.Select(clr => clr.ToString()).ToArray())).ToArray()));
}
- Output:
Red Green Red,Green Blue Red,Blue Green,Blue Red,Green,Blue Yellow Red,Yellow Green,Yellow Red,Green,Yellow Blue,Yellow Red,Blue,Yellow Green,Blue,Yellow Red,Green,Blue,Yellow
An alternative implementation for an arbitrary number of elements:
public IEnumerable<IEnumerable<T>> GetPowerSet<T>(IEnumerable<T> input) {
var seed = new List<IEnumerable<T>>() { Enumerable.Empty<T>() }
as IEnumerable<IEnumerable<T>>;
return input.Aggregate(seed, (a, b) =>
a.Concat(a.Select(x => x.Concat(new List<T>() { b }))));
}
Non-recursive version
using System;
class Powerset
{
static int count = 0, n = 4;
static int [] buf = new int [n];
static void Main()
{
int ind = 0;
int n_1 = n - 1;
for (;;)
{
for (int i = 0; i <= ind; ++i) Console.Write("{0, 2}", buf [i]);
Console.WriteLine();
count++;
if (buf [ind] < n_1) { ind++; buf [ind] = buf [ind - 1] + 1; }
else if (ind > 0) { ind--; buf [ind]++; }
else break;
}
Console.WriteLine("n=" + n + " count=" + count);
}
}
Recursive version
using System;
class Powerset
{
static int n = 4;
static int [] buf = new int [n];
static void Main()
{
rec(0, 0);
}
static void rec(int ind, int begin)
{
for (int i = begin; i < n; i++)
{
buf [ind] = i;
for (int j = 0; j <= ind; j++) Console.Write("{0, 2}", buf [j]);
Console.WriteLine();
rec(ind + 1, buf [ind] + 1);
}
}
}
C++
Non-recursive version
#include <iostream>
#include <set>
#include <vector>
#include <iterator>
#include <algorithm>
typedef std::set<int> set_type;
typedef std::set<set_type> powerset_type;
powerset_type powerset(set_type const& set)
{
typedef set_type::const_iterator set_iter;
typedef std::vector<set_iter> vec;
typedef vec::iterator vec_iter;
struct local
{
static int dereference(set_iter v) { return *v; }
};
powerset_type result;
vec elements;
do
{
set_type tmp;
std::transform(elements.begin(), elements.end(),
std::inserter(tmp, tmp.end()),
local::dereference);
result.insert(tmp);
if (!elements.empty() && ++elements.back() == set.end())
{
elements.pop_back();
}
else
{
set_iter iter;
if (elements.empty())
{
iter = set.begin();
}
else
{
iter = elements.back();
++iter;
}
for (; iter != set.end(); ++iter)
{
elements.push_back(iter);
}
}
} while (!elements.empty());
return result;
}
int main()
{
int values[4] = { 2, 3, 5, 7 };
set_type test_set(values, values+4);
powerset_type test_powerset = powerset(test_set);
for (powerset_type::iterator iter = test_powerset.begin();
iter != test_powerset.end();
++iter)
{
std::cout << "{ ";
char const* prefix = "";
for (set_type::iterator iter2 = iter->begin();
iter2 != iter->end();
++iter2)
{
std::cout << prefix << *iter2;
prefix = ", ";
}
std::cout << " }\n";
}
}
- Output:
{ }
{ 2 }
{ 2, 3 }
{ 2, 3, 5 }
{ 2, 3, 5, 7 }
{ 2, 3, 7 }
{ 2, 5 }
{ 2, 5, 7 }
{ 2, 7 }
{ 3 }
{ 3, 5 }
{ 3, 5, 7 }
{ 3, 7 }
{ 5 }
{ 5, 7 }
{ 7 }
C++14 version
This simplified version has identical output to the previous code.
#include <set>
#include <iostream>
template <class S>
auto powerset(const S& s)
{
std::set<S> ret;
ret.emplace();
for (auto&& e: s) {
std::set<S> rs;
for (auto x: ret) {
x.insert(e);
rs.insert(x);
}
ret.insert(begin(rs), end(rs));
}
return ret;
}
int main()
{
std::set<int> s = {2, 3, 5, 7};
auto pset = powerset(s);
for (auto&& subset: pset) {
std::cout << "{ ";
char const* prefix = "";
for (auto&& e: subset) {
std::cout << prefix << e;
prefix = ", ";
}
std::cout << " }\n";
}
}
Recursive version
#include <iostream>
#include <set>
template<typename Set> std::set<Set> powerset(const Set& s, size_t n)
{
typedef typename Set::const_iterator SetCIt;
typedef typename std::set<Set>::const_iterator PowerSetCIt;
std::set<Set> res;
if(n > 0) {
std::set<Set> ps = powerset(s, n-1);
for(PowerSetCIt ss = ps.begin(); ss != ps.end(); ss++)
for(SetCIt el = s.begin(); el != s.end(); el++) {
Set subset(*ss);
subset.insert(*el);
res.insert(subset);
}
res.insert(ps.begin(), ps.end());
} else
res.insert(Set());
return res;
}
template<typename Set> std::set<Set> powerset(const Set& s)
{
return powerset(s, s.size());
}
Clojure
(use '[clojure.math.combinatorics :only [subsets] ])
(def S #{1 2 3 4})
user> (subsets S)
(() (1) (2) (3) (4) (1 2) (1 3) (1 4) (2 3) (2 4) (3 4) (1 2 3) (1 2 4) (1 3 4) (2 3 4) (1 2 3 4))
Alternate solution, with no dependency on third-party library:
(defn powerset [coll]
(reduce (fn [a x]
(into a (map #(conj % x)) a))
#{#{}} coll))
(powerset #{1 2 3})
#{#{} #{1} #{2} #{1 2} #{3} #{1 3} #{2 3} #{1 2 3}}
Using bit-test: see: https://clojuredocs.org/clojure.core/bit-test#example-5d401face4b0ca44402ef78b
(defn powerset [coll]
(let [cnt (count coll)
bits (Math/pow 2 cnt)]
(for [i (range bits)]
(for [j (range i)
:while (< j cnt)
:when (bit-test i j)]
(nth coll j)))))
(powerset [1 2 3])
(() (1) (2) (1 2) (3) (1 3) (2 3) (1 2 3))
CoffeeScript
print_power_set = (arr) ->
console.log "POWER SET of #{arr}"
for subset in power_set(arr)
console.log subset
power_set = (arr) ->
result = []
binary = (false for elem in arr)
n = arr.length
while binary.length <= n
result.push bin_to_arr binary, arr
i = 0
while true
if binary[i]
binary[i] = false
i += 1
else
binary[i] = true
break
binary[i] = true
result
bin_to_arr = (binary, arr) ->
(arr[i] for i of binary when binary[arr.length - i - 1])
print_power_set []
print_power_set [4, 2, 1]
print_power_set ['dog', 'c', 'b', 'a']
- Output:
> coffee power_set.coffee
POWER SET of
[]
POWER SET of 4,2,1
[]
[ 1 ]
[ 2 ]
[ 2, 1 ]
[ 4 ]
[ 4, 1 ]
[ 4, 2 ]
[ 4, 2, 1 ]
POWER SET of dog,c,b,a
[]
[ 'a' ]
[ 'b' ]
[ 'b', 'a' ]
[ 'c' ]
[ 'c', 'a' ]
[ 'c', 'b' ]
[ 'c', 'b', 'a' ]
[ 'dog' ]
[ 'dog', 'a' ]
[ 'dog', 'b' ]
[ 'dog', 'b', 'a' ]
[ 'dog', 'c' ]
[ 'dog', 'c', 'a' ]
[ 'dog', 'c', 'b' ]
[ 'dog', 'c', 'b', 'a' ]
ColdFusion
Port from the JavaScript version, compatible with ColdFusion 8+ or Railo 3+
public array function powerset(required array data)
{
var ps = [""];
var d = arguments.data;
var lenData = arrayLen(d);
var lenPS = 0;
for (var i=1; i LTE lenData; i++)
{
lenPS = arrayLen(ps);
for (var j = 1; j LTE lenPS; j++)
{
arrayAppend(ps, listAppend(ps[j], d[i]));
}
}
return ps;
}
var res = powerset([1,2,3,4]);
- Output:
["","1","2","1,2","3","1,3","2,3","1,2,3","4","1,4","2,4","1,2,4","3,4","1,3,4","2,3,4","1,2,3,4"]
Common Lisp
(defun powerset (s)
(if s (mapcan (lambda (x) (list (cons (car s) x) x))
(powerset (cdr s)))
'(())))
- Output:
> (powerset '(l i s p)) ((L I S P) (I S P) (L S P) (S P) (L I P) (I P) (L P) (P) (L I S) (I S) (L S) (S) (L I) (I) (L) NIL)
(defun power-set (s)
(reduce #'(lambda (item ps)
(append (mapcar #'(lambda (e) (cons item e))
ps)
ps))
s
:from-end t
:initial-value '(())))
- Output:
>(power-set '(1 2 3)) ((1 2 3) (1 2) (1 3) (1) (2 3) (2) (3) NIL)
Alternate, more recursive (same output):
(defun powerset (l)
(if (null l)
(list nil)
(let ((prev (powerset (cdr l))))
(append (mapcar #'(lambda (elt) (cons (car l) elt)) prev)
prev))))
Imperative-style using LOOP:
(defun powerset (xs)
(loop for i below (expt 2 (length xs)) collect
(loop for j below i for x in xs if (logbitp j i) collect x)))
- Output:
>(powerset '(1 2 3) (NIL (1) (2) (1 2) (3) (1 3) (2 3) (1 2 3))
Yet another imperative solution, this time with dolist.
(defun power-set (list)
(let ((pow-set (list nil)))
(dolist (element (reverse list) pow-set)
(dolist (set pow-set)
(push (cons element set) pow-set)))))
- Output:
>(power-set '(1 2 3)) ((1) (1 3) (1 2 3) (1 2) (2) (2 3) (3) NIL)
D
This implementation defines a range which *lazily* enumerates the power set.
import std.algorithm;
import std.range;
auto powerSet(R)(R r)
{
return
(1L<<r.length)
.iota
.map!(i =>
r.enumerate
.filter!(t => (1<<t[0]) & i)
.map!(t => t[1])
);
}
unittest
{
int[] emptyArr;
assert(emptyArr.powerSet.equal!equal([emptyArr]));
assert(emptyArr.powerSet.powerSet.equal!(equal!equal)([[], [emptyArr]]));
}
void main(string[] args)
{
import std.stdio;
args[1..$].powerSet.each!writeln;
}
An alternative version, which implements the range construct from scratch:
import std.range;
struct PowerSet(R)
if (isRandomAccessRange!R)
{
R r;
size_t position;
struct PowerSetItem
{
R r;
size_t position;
private void advance()
{
while (!(position & 1))
{
r.popFront();
position >>= 1;
}
}
@property bool empty() { return position == 0; }
@property auto front()
{
advance();
return r.front;
}
void popFront()
{
advance();
r.popFront();
position >>= 1;
}
}
@property bool empty() { return position == (1 << r.length); }
@property PowerSetItem front() { return PowerSetItem(r.save, position); }
void popFront() { position++; }
}
auto powerSet(R)(R r) { return PowerSet!R(r); }
- Output:
$ rdmd powerset a b c [] ["a"] ["b"] ["a", "b"] ["c"] ["a", "c"] ["b", "c"] ["a", "b", "c"]
Alternative: using folds
An almost verbatim translation of the Haskell code in D.
Since D doesn't foldr, I've also copied Haskell's foldr implementation here.
Main difference from the Haskell:
- It isn't lazy (but it could be made so by implementing this as a generator)
Main differences from the version above:
- It isn't lazy
- It doesn't rely on integer bit fiddling, so it should work on arrays larger than size_t.
// Haskell definition:
// foldr f z [] = z
// foldr f z (x:xs) = x `f` foldr f z xs
S foldr(T, S)(S function(T, S) f, S z, T[] rest) {
return (rest.length == 0) ? z : f(rest[0], foldr(f, z, rest[1..$]));
}
// Haskell definition:
//powerSet = foldr (\x acc -> acc ++ map (x:) acc) [[]]
T[][] powerset(T)(T[] set) {
import std.algorithm;
import std.array;
// Note: The types before x and acc aren't needed, so this could be made even more concise, but I think it helps
// to make the algorithm slightly clearer.
return foldr( (T x, T[][] acc) => acc ~ acc.map!(accx => x ~ accx).array , [[]], set );
}
Déjà Vu
In Déjà Vu, sets are dictionaries with all values true and the default set to false.
powerset s:
local :out [ set{ } ]
for value in keys s:
for subset in copy out:
local :subset+1 copy subset
set-to subset+1 value true
push-to out subset+1
out
!. powerset set{ 1 2 3 4 }- Output:
[ set{ } set{ 4 } set{ 3 4 } set{ 3 } set{ 2 3 } set{ 2 3 4 } set{ 2 4 } set{ 2 } set{ 1 2 } set{ 1 2 4 } set{ 1 2 3 4 } set{ 1 2 3 } set{ 1 3 } set{ 1 3 4 } set{ 1 4 } set{ 1 } ]
Delphi
program Power_set;
{$APPTYPE CONSOLE}
uses
System.SysUtils;
const
n = 4;
var
buf: TArray<Integer>;
procedure rec(ind, bg: Integer);
begin
for var i := bg to n - 1 do
begin
buf[ind] := i;
for var j := 0 to ind do
write(buf[j]: 2);
writeln;
rec(ind + 1, buf[ind] + 1);
end;
end;
begin
SetLength(buf, n);
rec(0,0);
{$IFNDEF UNIX}readln;{$ENDIF}
end.
Dyalect
let n = 4
let buf = Array.Empty(n)
func rec(idx, begin) {
for i in begin..<n {
buf[idx] = i
for j in 0..idx {
print("{0, 2}".Format(buf[j]), terminator: "")
}
print("")
rec(idx + 1, buf[idx] + 1)
}
}
rec(0, 0)E
pragma.enable("accumulator")
def powerset(s) {
return accum [].asSet() for k in 0..!2**s.size() {
_.with(accum [].asSet() for i ? ((2**i & k) > 0) => elem in s {
_.with(elem)
})
}
}It would also be possible to define an object which is the powerset of a provided set without actually instantiating all of its members immediately.
EchoLisp
(define (set-cons a A)
(make-set (cons a A)))
(define (power-set e)
(cond ((null? e)
(make-set (list ∅)))
(else (let [(ps (power-set (cdr e)))]
(make-set
(append ps (map set-cons (circular-list (car e)) ps)))))))
(define B (make-set ' ( 🍎 🍇 🎂 🎄 )))
(power-set B)
→ { ∅ { 🍇 } { 🍇 🍎 } { 🍇 🍎 🎂 } { 🍇 🍎 🎂 🎄 } { 🍇 🍎 🎄 } { 🍇 🎂 } { 🍇 🎂 🎄 }
{ 🍇 🎄 } { 🍎 } { 🍎 🎂 } { 🍎 🎂 🎄 } { 🍎 🎄 } { 🎂 } { 🎂 🎄 } { 🎄 } }
;; The Von Neumann universe
(define V0 (power-set null)) ;; null and ∅ are the same
→ { ∅ }
(define V1 (power-set V0))
→ { ∅ { ∅ } }
(define V2 (power-set V1))
→ { ∅ { ∅ } { ∅ { ∅ } } { { ∅ } } }
(define V3 (power-set V2))
→ { ∅ { ∅ } { ∅ { ∅ } } …🔃 )
(length V3) → 16
(define V4 (power-set V3))
(length V4) → 65536
;; length V5 = 2^65536 : out of bounds
Elixir
defmodule RC do
use Bitwise
def powerset1(list) do
n = length(list)
max = round(:math.pow(2,n))
for i <- 0..max-1, do: (for pos <- 0..n-1, band(i, bsl(1, pos)) != 0, do: Enum.at(list, pos) )
end
def powerset2([]), do: [[]]
def powerset2([h|t]) do
pt = powerset2(t)
(for x <- pt, do: [h|x]) ++ pt
end
def powerset3([]), do: [[]]
def powerset3([h|t]) do
pt = powerset3(t)
powerset3(h, pt, pt)
end
defp powerset3(_, [], acc), do: acc
defp powerset3(x, [h|t], acc), do: powerset3(x, t, [[x|h] | acc])
end
IO.inspect RC.powerset1([1,2,3])
IO.inspect RC.powerset2([1,2,3])
IO.inspect RC.powerset3([1,2,3])
IO.inspect RC.powerset1([])
IO.inspect RC.powerset1(["one"])
- Output:
[[], [1], [2], [1, 2], [3], [1, 3], [2, 3], [1, 2, 3]] [[1, 2, 3], [1, 2], [1, 3], [1], [2, 3], [2], [3], []] [[1], [1, 3], [1, 2, 3], [1, 2], [2], [2, 3], [3], []] [[]] [[], ["one"]]
Erlang
Generates all subsets of a list with the help of binary:
For [1 2 3]:
[ ] | 0 0 0 | 0
[ 3] | 0 0 1 | 1
[ 2 ] | 0 1 0 | 2
[ 2 3] | 0 1 1 | 3
[1 ] | 1 0 0 | 4
[1 3] | 1 0 1 | 5
[1 2 ] | 1 1 0 | 6
[1 2 3] | 1 1 1 | 7
¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯
powerset(Lst) ->
N = length(Lst),
Max = trunc(math:pow(2,N)),
[[lists:nth(Pos+1,Lst) || Pos <- lists:seq(0,N-1), I band (1 bsl Pos) =/= 0]
|| I <- lists:seq(0,Max-1)].
- Output:
[[], [1], [2], [1,2], [3], [1,3], [2,3], [1,2,3], [4], [1,4], [2,4], [1,2,4], [3,4], [1,3,4], [2,3,4], [1,2,3,4]]
Alternate shorter and more efficient version:
powerset([]) -> [[]];
powerset([H|T]) -> PT = powerset(T),
[ [H|X] || X <- PT ] ++ PT.
or even more efficient version:
powerset([]) -> [[]];
powerset([H|T]) -> PT = powerset(T),
powerset(H, PT, PT).
powerset(_, [], Acc) -> Acc;
powerset(X, [H|T], Acc) -> powerset(X, T, [[X|H]|Acc]).
F#
almost exact copy of OCaml version
let subsets xs = List.foldBack (fun x rest -> rest @ List.map (fun ys -> x::ys) rest) xs [[]]
alternatively with list comprehension
let rec pow =
function
| [] -> [[]]
| x::xs -> [for i in pow xs do yield! [i;x::i]]
Factor
We use hash sets, denoted by HS{ } brackets, for our sets. members converts from a set to a sequence, and <hash-set> converts back.
USING: kernel prettyprint sequences arrays sets hash-sets ;
IN: powerset
: add ( set elt -- newset ) 1array <hash-set> union ;
: powerset ( set -- newset ) members { HS{ } } [ dupd [ add ] curry map append ] reduce <hash-set> ;
Usage:
( scratchpad ) HS{ 1 2 3 4 } powerset .
HS{
HS{ 1 2 3 4 }
HS{ 1 2 }
HS{ 1 3 }
HS{ 2 3 }
HS{ 1 2 3 }
HS{ 1 4 }
HS{ 2 4 }
HS{ }
HS{ 1 }
HS{ 2 }
HS{ 3 }
HS{ 4 }
HS{ 1 2 4 }
HS{ 3 4 }
HS{ 1 3 4 }
HS{ 2 3 4 }
}
Forth
.
: ?print dup 1 and if over args type space then ;
: .set begin dup while ?print >r 1+ r> 1 rshift repeat drop drop ;
: .powerset 0 do ." ( " 1 i .set ." )" cr loop ;
: check-none dup 2 < abort" Usage: powerset [val] .. [val]" ;
: check-size dup /cell 8 [*] >= abort" Set too large" ;
: powerset 1 argn check-none check-size 1- lshift .powerset ;
powerset
- Output:
$ 4th cxq powerset.4th 1 2 3 4 ( ) ( 1 ) ( 2 ) ( 1 2 ) ( 3 ) ( 1 3 ) ( 2 3 ) ( 1 2 3 ) ( 4 ) ( 1 4 ) ( 2 4 ) ( 1 2 4 ) ( 3 4 ) ( 1 3 4 ) ( 2 3 4 ) ( 1 2 3 4 )
FreeBASIC
Los elementos de un conjunto se representan como bits en un número entero (por lo tanto, el tamaño máximo del conjunto es 32).
Function ConjuntoPotencia(set() As String) As String
If Ubound(set,1) > 31 Then Print "Set demasiado grande para representarlo como un entero" : Exit Function
If Ubound(set,1) < 0 Then Print "{}": Exit Function ' Set vacío
Dim As Integer i, j
Dim As String s = "{"
For i = Lbound(set) To (2 Shl Ubound(set,1)) - 1
s += "{"
For j = Lbound(set) To Ubound(set,1)
If i And (1 Shl j) Then s += set(j) + ","
Next j
If Right(s,1) = "," Then s = Left(s,Len(s)-1)
s += "},"
Next i
Return Left(s,Len(s)-1) + "}"
End Function
Print "El power set de [1, 2, 3, 4] comprende:"
Dim As String set(3) = {"1", "2", "3", "4"}
Print ConjuntoPotencia(set())
Print !"\nEl power set de [] comprende:"
Dim As String set0()
Print ConjuntoPotencia(set0())
Print "El power set de [[]] comprende:"
Dim As String set1(0) = {""}
Print ConjuntoPotencia(set1())
Sleep
- Output:
El power set de [1, 2, 3, 4] comprende:
{{},{1},{2},{1,2},{3},{1,3},{2,3},{1,2,3},{4},{1,4},{2,4},{1,2,4},{3,4},{1,3,4},{2,3,4},{1,2,3,4}}
El power set de [] comprende:
{}
El power set de [[]] comprende:
{{},{}}
Frink
Frink's set and array classes have built-in subsets[] methods that return all subsets. If called with an array, the results are arrays. If called with a set, the results are sets.
a = new set[1,2,3,4]
a.subsets[]FunL
FunL uses Scala type scala.collection.immutable.Set as it's set type, which has a built-in method subsets returning an (Scala) iterator over subsets.
def powerset( s ) = s.subsets().toSet()The powerset function could be implemented in FunL directly as:
def
powerset( {} ) = {{}}
powerset( s ) =
acc = powerset( s.tail() )
acc + map( x -> {s.head()} + x, acc )or, alternatively as:
import lists.foldr
def powerset( s ) = foldr( \x, acc -> acc + map( a -> {x} + a, acc), {{}}, s )
println( powerset({1, 2, 3, 4}) )- Output:
{{}, {4}, {1, 2}, {1, 3}, {2, 3, 4}, {3}, {1, 2, 3, 4}, {1, 4}, {1, 2, 3}, {2}, {1, 2, 4}, {1}, {3, 4}, {2, 3}, {2, 4}, {1, 3, 4}}
Fōrmulæ
Fōrmulæ programs are not textual, visualization/edition of programs is done showing/manipulating structures but not text. Moreover, there can be multiple visual representations of the same program. Even though it is possible to have textual representation —i.e. XML, JSON— they are intended for storage and transfer purposes more than visualization and edition.
Programs in Fōrmulæ are created/edited online in its website.
In this page you can see and run the program(s) related to this task and their results. You can also change either the programs or the parameters they are called with, for experimentation, but remember that these programs were created with the main purpose of showing a clear solution of the task, and they generally lack any kind of validation.
Solution
No program needed. Power set is intrinsically supported in Fōrmulæ.
Case 1. Power set of the set {1, 2, 3, 4}
Case 2. The power set of the empty set is the set which contains itself.
Case 3. The power set of the set which contains only the empty set, has two subsets, the empty set and the set which contains the empty set
Case 4. Even when it is intrinsically supported, a program can be written:
GAP
# Built-in
Combinations([1, 2, 3]);
# [ [ ], [ 1 ], [ 1, 2 ], [ 1, 2, 3 ], [ 1, 3 ], [ 2 ], [ 2, 3 ], [ 3 ] ]
# Note that it handles duplicates
Combinations([1, 2, 3, 1]);
# [ [ ], [ 1 ], [ 1, 1 ], [ 1, 1, 2 ], [ 1, 1, 2, 3 ], [ 1, 1, 3 ], [ 1, 2 ], [ 1, 2, 3 ], [ 1, 3 ],
# [ 2 ], [ 2, 3 ], [ 3 ] ]
Go
No native set type in Go. While the associative array trick mentioned in the task description works well in Go in most situations, it does not work here because we need sets of sets, and converting a general set to a hashable value for a map key is non-trivial.
Instead, this solution uses a simple (non-associative) slice as a set representation. To ensure uniqueness, the element interface requires an equality method, which is used by the set add method. Adding elements with the add method ensures the uniqueness property.
While the "add" and "has" methods make a usable set type, the power set method implemented here computes a result directly without using the add method. The algorithm ensures that the result will be a valid set as long as the input is a valid set. This allows the more efficient append function to be used.
package main
import (
"fmt"
"strconv"
"strings"
)
// types needed to implement general purpose sets are element and set
// element is an interface, allowing different kinds of elements to be
// implemented and stored in sets.
type elem interface {
// an element must be distinguishable from other elements to satisfy
// the mathematical definition of a set. a.eq(b) must give the same
// result as b.eq(a).
Eq(elem) bool
// String result is used only for printable output. Given a, b where
// a.eq(b), it is not required that a.String() == b.String().
fmt.Stringer
}
// integer type satisfying element interface
type Int int
func (i Int) Eq(e elem) bool {
j, ok := e.(Int)
return ok && i == j
}
func (i Int) String() string {
return strconv.Itoa(int(i))
}
// a set is a slice of elem's. methods are added to implement
// the element interface, to allow nesting.
type set []elem
// uniqueness of elements can be ensured by using add method
func (s *set) add(e elem) {
if !s.has(e) {
*s = append(*s, e)
}
}
func (s *set) has(e elem) bool {
for _, ex := range *s {
if e.Eq(ex) {
return true
}
}
return false
}
func (s set) ok() bool {
for i, e0 := range s {
for _, e1 := range s[i+1:] {
if e0.Eq(e1) {
return false
}
}
}
return true
}
// elem.Eq
func (s set) Eq(e elem) bool {
t, ok := e.(set)
if !ok {
return false
}
if len(s) != len(t) {
return false
}
for _, se := range s {
if !t.has(se) {
return false
}
}
return true
}
// elem.String
func (s set) String() string {
if len(s) == 0 {
return "∅"
}
var buf strings.Builder
buf.WriteRune('{')
for i, e := range s {
if i > 0 {
buf.WriteRune(',')
}
buf.WriteString(e.String())
}
buf.WriteRune('}')
return buf.String()
}
// method required for task
func (s set) powerSet() set {
r := set{set{}}
for _, es := range s {
var u set
for _, er := range r {
er := er.(set)
u = append(u, append(er[:len(er):len(er)], es))
}
r = append(r, u...)
}
return r
}
func main() {
var s set
for _, i := range []Int{1, 2, 2, 3, 4, 4, 4} {
s.add(i)
}
fmt.Println(" s:", s, "length:", len(s))
ps := s.powerSet()
fmt.Println(" 𝑷(s):", ps, "length:", len(ps))
fmt.Println("\n(extra credit)")
var empty set
fmt.Println(" empty:", empty, "len:", len(empty))
ps = empty.powerSet()
fmt.Println(" 𝑷(∅):", ps, "len:", len(ps))
ps = ps.powerSet()
fmt.Println("𝑷(𝑷(∅)):", ps, "len:", len(ps))
fmt.Println("\n(regression test for earlier bug)")
s = set{Int(1), Int(2), Int(3), Int(4), Int(5)}
fmt.Println(" s:", s, "length:", len(s), "ok:", s.ok())
ps = s.powerSet()
fmt.Println(" 𝑷(s):", "length:", len(ps), "ok:", ps.ok())
for _, e := range ps {
if !e.(set).ok() {
panic("invalid set in ps")
}
}
}
- Output:
s: {1,2,3,4} length: 4
𝑷(s): {∅,{1},{2},{1,2},{3},{1,3},{2,3},{1,2,3},{4},{1,4},{2,4},{1,2,4},{3,4},{1,3,4},{2,3,4},{1,2,3,4}} length: 16
(extra credit)
empty: ∅ len: 0
𝑷(∅): {∅} len: 1
𝑷(𝑷(∅)): {∅,{∅}} len: 2
(regression test for earlier bug)
s: {1,2,3,4,5} length: 5 ok: true
𝑷(s): length: 32 ok: true
Groovy
Builds on the Combinations solution. Sets are not a "natural" collection type in Groovy. Lists are much more richly supported. Thus, this solution is liberally sprinkled with coercion from Set to List and from List to Set.
def powerSetRec(head, tail) {
if (!tail) return [head]
powerSetRec(head, tail.tail()) + powerSetRec(head + [tail.head()], tail.tail())
}
def powerSet(set) { powerSetRec([], set as List) as Set}
Test program:
def vocalists = [ 'C', 'S', 'N', 'Y' ] as Set
println vocalists
println powerSet(vocalists)
- Output:
[C, S, N, Y] [[], [Y], [N], [N, Y], [S], [S, Y], [S, N], [S, N, Y], [C], [C, Y], [C, N], [C, N, Y], [C, S], [C, S, Y], [C, S, N], [C, S, N, Y]]
Haskell
import Data.Set
import Control.Monad
powerset :: Ord a => Set a -> Set (Set a)
powerset = fromList . fmap fromList . listPowerset . toList
listPowerset :: [a] -> [[a]]
listPowerset = filterM (const [True, False])
listPowerset describes the result as all possible (using the list monad) filterings (using filterM) of the input list, regardless (using const) of each item's value. powerset simply converts the input and output from lists to sets.
Alternate Solution
powerset [] = [[]]
powerset (head:tail) = acc ++ map (head:) acc where acc = powerset tail
or
powerSet :: [a] -> [[a]]
powerSet = foldr (\x acc -> acc ++ map (x:) acc) [[]]
which could also be understood, in point-free terms, as:
powerSet :: [a] -> [[a]]
powerSet = foldr ((mappend <*>) . fmap . (:)) (pure [])
Examples:
*Main> listPowerset [1,2,3]
[[1,2,3],[1,2],[1,3],[1],[2,3],[2],[3],[]]
*Main> powerset (Data.Set.fromList [1,2,3])
{{},{1},{1,2},{1,2,3},{1,3},{2},{2,3},{3}}
Prelude> import Data.List Prelude Data.List> subsequences [1,2,3] [[],[1],[2],[1,2],[3],[1,3],[2,3],[1,2,3]]
Alternate solution
A method using only set operations and set mapping is also possible. Ideally, Set would be defined as a Monad, but that's impossible given the constraint that the type of inputs to Set.map (and a few other functions) be ordered.
import qualified Data.Set as Set
type Set=Set.Set
unionAll :: (Ord a) => Set (Set a) -> Set a
unionAll = Set.fold Set.union Set.empty
--slift is the analogue of liftA2 for sets.
slift :: (Ord a, Ord b, Ord c) => (a->b->c) -> Set a -> Set b -> Set c
slift f s0 s1 = unionAll (Set.map (\e->Set.map (f e) s1) s0)
--a -> {{},{a}}
makeSet :: (Ord a) => a -> Set (Set a)
makeSet = (Set.insert Set.empty) . Set.singleton.Set.singleton
powerSet :: (Ord a) => Set a -> Set (Set a)
powerSet = (Set.fold (slift Set.union) (Set.singleton Set.empty)) . Set.map makeSet
Usage:
Prelude Data.Set> powerSet fromList [1,2,3]
fromList [fromList [], fromList [1], fromList [1,2], fromList [1,2,3], fromList [1,3], fromList [2], fromList [2,3], fromList [3]]
Icon and Unicon
The two examples below show the similarities and differences between constructing an explicit representation of the solution, i.e. a set containing the powerset, and one using generators. The basic recursive algorithm is the same in each case, but wherever the first stores part of the result away, the second uses 'suspend' to immediately pass the result back to the caller. The caller may then decide to store the results in a set, a list, or dispose of each one as it appears.
Set building
The following version returns a set containing the powerset:
To test the above procedure:
- Output:
[ 3 ] [ 4 3 ] [ 2 4 ] [ 2 3 ] [ 1 3 ] [ 4 ] [ 2 ] [ 2 1 3 ] [ 2 4 1 ] [ 4 1 3 ] [ 2 4 1 3 ] [ ] [ 2 4 3 ] [ 1 ] [ 4 1 ] [ 2 1 ]
Generator
An alternative version, which generates each item in the power set in turn:
J
There are a number of ways to generate a power set in J. Here's one:
ps =: #~ 2 #:@i.@^ #
For example:
ps 'ACE'
E
C
CE
A
AE
AC
ACE
In the typical use, this operation makes sense on collections of unique elements.
~.1 2 3 2 1
1 2 3
#ps 1 2 3 2 1
32
#ps ~.1 2 3 2 1
8
In other words, the power set of a 5 element set has 32 sets where the power set of a 3 element set has 8 sets. Thus if elements of the original "set" were not unique then sets of the power "set" will also not be unique sets.
Java
Recursion
This implementation sorts each subset, but not the whole list of subsets (which would require a custom comparator). It also destroys the original set.
public static ArrayList<String> getpowerset(int a[],int n,ArrayList<String> ps)
{
if(n<0)
{
return null;
}
if(n==0)
{
if(ps==null)
ps=new ArrayList<String>();
ps.add(" ");
return ps;
}
ps=getpowerset(a, n-1, ps);
ArrayList<String> tmp=new ArrayList<String>();
for(String s:ps)
{
if(s.equals(" "))
tmp.add(""+a[n-1]);
else
tmp.add(s+a[n-1]);
}
ps.addAll(tmp);
return ps;
}
Iterative
The iterative implementation of the above idea. Each subset is in the order that the element appears in the input list. This implementation preserves the input.
public static <T> List<List<T>> powerset(Collection<T> list) {
List<List<T>> ps = new ArrayList<List<T>>();
ps.add(new ArrayList<T>()); // add the empty set
// for every item in the original list
for (T item : list) {
List<List<T>> newPs = new ArrayList<List<T>>();
for (List<T> subset : ps) {
// copy all of the current powerset's subsets
newPs.add(subset);
// plus the subsets appended with the current item
List<T> newSubset = new ArrayList<T>(subset);
newSubset.add(item);
newPs.add(newSubset);
}
// powerset is now powerset of list.subList(0, list.indexOf(item)+1)
ps = newPs;
}
return ps;
}
Binary String
This implementation works on idea that each element in the original set can either be in the power set or not in it. With n elements in the original set, each combination can be represented by a binary string of length n. To get all possible combinations, all you need is a counter from 0 to 2n - 1. If the kth bit in the binary string is 1, the kth element of the original set is in this combination.
public static <T extends Comparable<? super T>> LinkedList<LinkedList<T>> BinPowSet(
LinkedList<T> A){
LinkedList<LinkedList<T>> ans= new LinkedList<LinkedList<T>>();
int ansSize = (int)Math.pow(2, A.size());
for(int i= 0;i< ansSize;++i){
String bin= Integer.toBinaryString(i); //convert to binary
while(bin.length() < A.size()) bin = "0" + bin; //pad with 0's
LinkedList<T> thisComb = new LinkedList<T>(); //place to put one combination
for(int j= 0;j< A.size();++j){
if(bin.charAt(j) == '1')thisComb.add(A.get(j));
}
Collections.sort(thisComb); //sort it for easy checking
ans.add(thisComb); //put this set in the answer list
}
return ans;
}
JavaScript
ES5
Iteration
Uses a JSON stringifier from http://www.json.org/js.html
function powerset(ary) {
var ps = [[]];
for (var i=0; i < ary.length; i++) {
for (var j = 0, len = ps.length; j < len; j++) {
ps.push(ps[j].concat(ary[i]));
}
}
return ps;
}
var res = powerset([1,2,3,4]);
load('json2.js');
print(JSON.stringify(res));
- Output:
[[],[1],[2],[1,2],[3],[1,3],[2,3],[1,2,3],[4],[1,4],[2,4],[1,2,4],[3,4],[1,3,4],[2,3,4],[1,2,3,4]]
Functional composition
(function () {
// translating: powerset = foldr (\x acc -> acc ++ map (x:) acc) [[]]
function powerset(xs) {
return xs.reduceRight(function (a, x) {
return a.concat(a.map(function (y) {
return [x].concat(y);
}));
}, [[]]);
}
// TEST
return {
'[1,2,3] ->': powerset([1, 2, 3]),
'empty set ->': powerset([]),
'set which contains only the empty set ->': powerset([[]])
}
})();
- Output:
{
"[1,2,3] ->":[[], [3], [2], [2, 3], [1], [1, 3], [1, 2], [1, 2, 3]],
"empty set ->":[[]],
"set which contains only the empty set ->":[[], [[]]]
}
ES6
(() => {
'use strict';
// powerset :: [a] -> [[a]]
const powerset = xs =>
xs.reduceRight((a, x) => [...a, ...a.map(y => [x, ...y])], [
[]
]);
// TEST
return {
'[1,2,3] ->': powerset([1, 2, 3]),
'empty set ->': powerset([]),
'set which contains only the empty set ->': powerset([
[]
])
};
})()
- Output:
{"[1,2,3] ->":[[], [3], [2], [2, 3], [1], [1, 3], [1, 2], [1, 2, 3]],
"empty set ->":[[]],
"set which contains only the empty set ->":[[], [[]]]}
jq
def powerset:
reduce .[] as $i ([[]];
reduce .[] as $r (.; . + [$r + [$i]]));Example:
[range(0;10)]|powerset|length # => 1024
Extra credit:
# The power set of the empty set:
[] | powerset
# => [[]]
# The power set of the set which contains only the empty set:
[ [] ] | powerset
# => [[],[[]]]Recursive version
def powerset:
if length == 0 then [[]]
else .[0] as $first
| (.[1:] | powerset)
| map([$first] + . ) + .
end;Example:
[1,2,3]|powerset # => [[1,2,3],[1,2],[1,3],[1],[2,3],[2],[3],[]]
Julia
function powerset(x::Vector{T})::Vector{Vector{T}} where T
result = Vector{T}[[]]
for elem in x, j in eachindex(result)
push!(result, [result[j] ; elem])
end
result
end
- Output:
julia> show(powerset([1,2,3])) [Int64[],[1],[2],[1,2],[3],[1,3],[2,3],[1,2,3]]
Non-Mutating Solution
using Base.Iterators
function bitmask(u, max_size)
res = BitArray(undef, max_size)
res.chunks[1] = u%UInt64
res
end
function powerset(input_collection::Vector{T})::Vector{Vector{T}} where T
num_elements = length(input_collection)
bitmask_map(x) = Iterators.map(y -> bitmask(y, num_elements), x)
getindex_map(x) = Iterators.map(y -> input_collection[y], x)
UnitRange(0, (2^num_elements)-1) |>
bitmask_map |>
getindex_map |>
collect
end
- Output:
julia> show(powerset([1,2,3])) [Int64[], [1], [2], [1, 2], [3], [1, 3], [2, 3], [1, 2, 3]]
K
ps:{x@&:'+2_vs!_2^#x}
Usage:
ps "ABC"
(""
,"C"
,"B"
"BC"
,"A"
"AC"
"AB"
"ABC")
Kotlin
// purely functional & lazy version, leveraging recursion and Sequences (a.k.a. streams)
fun <T> Set<T>.subsets(): Sequence<Set<T>> =
when (size) {
0 -> sequenceOf(emptySet())
else -> {
val head = first()
val tail = this - head
tail.subsets() + tail.subsets().map { setOf(head) + it }
}
}
// if recursion is an issue, you may change it this way:
fun <T> Set<T>.subsets(): Sequence<Set<T>> = sequence {
when (size) {
0 -> yield(emptySet<T>())
else -> {
val head = first()
val tail = this@subsets - head
yieldAll(tail.subsets())
for (subset in tail.subsets()) {
yield(setOf(head) + subset)
}
}
}
}
- Output:
Power set of setOf(1, 2, 3, 4) comprises: [] [4] [3] [3, 4] [2] [2, 4] [2, 3] [2, 3, 4] [1] [1, 4] [1, 3] [1, 3, 4] [1, 2] [1, 2, 4] [1, 2, 3] [1, 2, 3, 4] Power set of emptySet<Any>() comprises: [] Power set of setOf(emptySet<Any>()) comprises: [] [[]]
Lambdatalk
{def powerset
{def powerset.r
{lambda {:ary :ps :i}
{if {= :i {A.length :ary}}
then :ps
else {powerset.r :ary
{powerset.rr :ary :ps {A.length :ps} :i 0}
{+ :i 1}} }}}
{def powerset.rr
{lambda {:ary :ps :len :i :j}
{if {= :j :len}
then :ps
else {powerset.rr :ary
{A.addlast! {A.concat {A.get :j :ps}
{A.new {A.get :i :ary}}}
:ps}
:len
:i
{+ :j 1}} }}}
{lambda {:ary}
{A.new {powerset.r :ary {A.new {A.new}} 0}}}}
-> powerset
{powerset {A.new 1 2 3 4}}
-> [[],[1],[2],[1,2],[3],[1,3],[2,3],[1,2,3],[4],[1,4],[2,4],[1,2,4],[3,4],[1,3,4],[2,3,4],[1,2,3,4]]]
Logo
to powerset :set
if empty? :set [output [[]]]
localmake "rest powerset butfirst :set
output sentence map [sentence first :set ?] :rest :rest
end
show powerset [1 2 3]
[[1 2 3] [1 2] [1 3] [1] [2 3] [2] [3] []]Logtalk
:- object(set).
:- public(powerset/2).
powerset(Set, PowerSet) :-
reverse(Set, RSet),
powerset_1(RSet, [[]], PowerSet).
powerset_1([], PowerSet, PowerSet).
powerset_1([X| Xs], Yss0, Yss) :-
powerset_2(Yss0, X, Yss1),
powerset_1(Xs, Yss1, Yss).
powerset_2([], _, []).
powerset_2([Zs| Zss], X, [Zs, [X| Zs]| Yss]) :-
powerset_2(Zss, X, Yss).
reverse(List, Reversed) :-
reverse(List, [], Reversed).
reverse([], Reversed, Reversed).
reverse([Head| Tail], List, Reversed) :-
reverse(Tail, [Head| List], Reversed).
:- end_object.
Usage example:
| ?- set::powerset([1, 2, 3, 4], PowerSet).
PowerSet = [[],[1],[2],[1,2],[3],[1,3],[2,3],[1,2,3],[4],[1,4],[2,4],[1,2,4],[3,4],[1,3,4],[2,3,4],[1,2,3,4]]
yes
Lua
--returns the powerset of s, out of order.
function powerset(s, start)
start = start or 1
if(start > #s) then return {{}} end
local ret = powerset(s, start + 1)
for i = 1, #ret do
ret[#ret + 1] = {s[start], unpack(ret[i])}
end
return ret
end
--non-recurse implementation
function powerset(s)
local t = {{}}
for i = 1, #s do
for j = 1, #t do
t[#t+1] = {s[i],unpack(t[j])}
end
end
return t
end
--alternative, copied from the Python implementation
function powerset2(s)
local ret = {{}}
for i = 1, #s do
local k = #ret
for j = 1, k do
ret[k + j] = {s[i], unpack(ret[j])}
end
end
return ret
end
M4
define(`for',
`ifelse($#, 0, ``$0'',
eval($2 <= $3), 1,
`pushdef(`$1', `$2')$4`'popdef(
`$1')$0(`$1', incr($2), $3, `$4')')')dnl
define(`nth',
`ifelse($1, 1, $2,
`nth(decr($1), shift(shift($@)))')')dnl
define(`range',
`for(`x', eval($1 + 2), eval($2 + 2),
`nth(x, $@)`'ifelse(x, eval($2+2), `', `,')')')dnl
define(`powerpart',
`{range(2, incr($1), $@)}`'ifelse(incr($1), $#, `',
`for(`x', eval($1+2), $#,
`,powerpart(incr($1), ifelse(
eval(2 <= ($1 + 1)), 1,
`range(2,incr($1), $@), ')`'nth(x, $@)`'ifelse(
eval((x + 1) <= $#),1,`,range(incr(x), $#, $@)'))')')')dnl
define(`powerset',
`{powerpart(0, substr(`$1', 1, eval(len(`$1') - 2)))}')dnl
dnl
powerset(`{a,b,c}')- Output:
{{},{a},{a,b},{a,b,c},{a,c},{b},{b,c},{c}}
Maple
combinat:-powerset({1,2,3,4});- Output:
{{}, {1}, {2}, {3}, {4}, {1, 2}, {1, 3}, {1, 4}, {2, 3}, {2, 4}, {3, 4},
{1, 2, 3}, {1, 2, 4}, {1, 3, 4}, {2, 3, 4}, {1, 2, 3, 4}}
Mathematica /Wolfram Language
Built-in function that either gives all possible subsets, subsets with at most n elements, subsets with exactly n elements or subsets containing between n and m elements. Example of all subsets:
Subsets[{a, b, c}]
gives:
{{}, {a}, {b}, {c}, {a, b}, {a, c}, {b, c}, {a, b, c}}
Subsets[list, {n, Infinity}] gives all the subsets that have n elements or more.
Subsets[list, n] gives all the subsets that have at most n elements.
Subsets[list, {n}] gives all the subsets that have exactly n elements.
Subsets[list, {m,n}] gives all the subsets that have between m and n elements.
MATLAB
Sets are not an explicit data type in MATLAB, but cell arrays can be used for the same purpose. In fact, cell arrays have the benefit of containing any kind of data structure. So, this powerset function will work on a set of any type of data structure, without the need to overload any operators.
function pset = powerset(theSet)
pset = cell(size(theSet)); %Preallocate memory
%Generate all numbers from 0 to 2^(num elements of the set)-1
for i = ( 0:(2^numel(theSet))-1 )
%Convert i into binary, convert each digit in binary to a boolean
%and store that array of booleans
indicies = logical(bitget( i,(1:numel(theSet)) ));
%Use the array of booleans to extract the members of the original
%set, and store the set containing these members in the powerset
pset(i+1) = {theSet(indicies)};
end
end
Sample Usage: Powerset of the set of the empty set.
powerset({{}})
ans =
{} {1x1 cell} %This is the same as { {},{{}} }
Powerset of { {1,2},3 }.
powerset({{1,2},3})
ans =
{1x0 cell} {1x1 cell} {1x1 cell} {1x2 cell} %This is the same as { {},{{1,2}},{3},{{1,2},3} }
Maxima
powerset({1, 2, 3, 4});
/* {{}, {1}, {1, 2}, {1, 2, 3}, {1, 2, 3, 4}, {1, 2, 4}, {1, 3}, {1, 3, 4},
{1, 4}, {2}, {2, 3}, {2, 3, 4}, {2, 4}, {3}, {3, 4}, {4}} */
Nim
import sets, hashes
proc hash(x: HashSet[int]): Hash =
var h = 0
for i in x: h = h !& hash(i)
result = !$h
proc powerset[T](inset: HashSet[T]): HashSet[HashSet[T]] =
result.incl(initHashSet[T]()) # Initialized with empty set.
for val in inset:
let previous = result
for aSet in previous:
var newSet = aSet
newSet.incl(val)
result.incl(newSet)
echo powerset([1,2,3,4].toHashSet())
- Output:
{{4, 3, 1}, {3, 2, 1}, {3}, {3, 1}, {2}, {4, 3, 2, 1}, {}, {4, 2}, {4, 2, 1}, {4, 3, 2}, {1}, {3, 2}, {4, 3}, {4}, {4, 1}, {2, 1}}
Objective-C
#import <Foundation/Foundation.h>
+ (NSArray *)powerSetForArray:(NSArray *)array {
UInt32 subsetCount = 1 << array.count;
NSMutableArray *subsets = [NSMutableArray arrayWithCapacity:subsetCount];
for(int subsetIndex = 0; subsetIndex < subsetCount; subsetIndex++) {
NSMutableArray *subset = [[NSMutableArray alloc] init];
for (int itemIndex = 0; itemIndex < array.count; itemIndex++) {
if((subsetIndex >> itemIndex) & 0x1) {
[subset addObject:array[itemIndex]];
}
}
[subsets addObject:subset];
}
return subsets;
}
OCaml
The standard library already implements a proper Set datatype. As the base type is unspecified, the powerset must be parameterized as a module. Also, the library is lacking a map operation, which we have to implement first.
module PowerSet(S: Set.S) =
struct
include Set.Make (S)
let map f s =
let work x r = add (f x) r in
fold work s empty
;;
let powerset s =
let base = singleton (S.empty) in
let work x r = union r (map (S.add x) r) in
S.fold work s base
;;
end;; (* PowerSet *)
version for lists:
let subsets xs = List.fold_right (fun x rest -> rest @ List.map (fun ys -> x::ys) rest) xs [[]]
OPL
{string} s={"A","B","C","D"};
range r=1.. ftoi(pow(2,card(s)));
{string} s2 [k in r] = {i | i in s: ((k div (ftoi(pow(2,(ord(s,i))))) mod 2) == 1)};
execute
{
writeln(s2);
}which gives
[{} {"A"} {"B"} {"A" "B"} {"C"} {"A" "C"} {"B" "C"} {"A" "B" "C"} {"D"} {"A"
"D"} {"B" "D"} {"A" "B" "D"} {"C" "D"} {"A" "C" "D"} {"B" "C" "D"}
{"A" "B" "C" "D"}]Oz
Oz has a library for finite set constraints. Creating a power set is a trivial application of that:
declare
%% Given a set as a list, returns its powerset (again as a list)
fun {Powerset Set}
proc {Describe Root}
%% Describe sets by lower bound (nil) and upper bound (Set)
Root = {FS.var.bounds nil Set}
%% enumerate all possible sets
{FS.distribute naive [Root]}
end
AllSets = {SearchAll Describe}
in
%% convert to list representation
{Map AllSets FS.reflect.lowerBoundList}
end
in
{Inspect {Powerset [1 2 3 4]}}A more convential implementation without finite set constaints:
fun {Powerset2 Set}
case Set of nil then [nil]
[] H|T thens
Acc = {Powerset2 T}
in
{Append Acc {Map Acc fun {$ A} H|A end}}
end
endPARI/GP
vector(1<<#S,i,vecextract(S,i-1))The forsubset iterator was added in version 2.10.0 to efficiently iterate over combinations and power sets.
S=["a","b","c"]
forsubset(#S,s,print1(vecextract(S,s)" "))- Output:
[] ["a"] ["b"] ["c"] ["a", "b"] ["a", "c"] ["b", "c"] ["a", "b", "c"]
PascalABC.NET
function AllSubSets<T>(a: array of T; i: integer; lst: List<T>): sequence of List<T>;
begin
if i = a.Length then
begin
yield lst;
exit;
end;
lst.Add(a[i]);
yield sequence AllSubSets(a, i + 1, lst);
lst.RemoveAt(lst.Count-1);
yield sequence AllSubSets(a, i + 1, lst);
end;
begin
AllSubSets(Arr(1..4),0,new List<integer>).Print;
end.
- Output:
[1,2,3,4] [1,2,3] [1,2,4] [1,2] [1,3,4] [1,3] [1,4] [1] [2,3,4] [2,3] [2,4] [2] [3,4] [3] [4] []
Perl
Perl does not have a built-in set data-type. However, you can...
Module: Algorithm::Combinatorics
This module has an iterator over the power set. Note that it does not enforce that the input array is a set (no duplication). If each subset is processed immediately, this has an advantage of very low memory use.
use Algorithm::Combinatorics "subsets";
my @S = ("a","b","c");
my @PS;
my $iter = subsets(\@S);
while (my $p = $iter->next) {
push @PS, "[@$p]"
}
say join(" ",@PS);
- Output:
[a b c] [b c] [a c] [c] [a b] [b] [a] []
Module: ntheory
The simplest solution is to use the one argument version of the combination iterator, which iterates over the power set.
use ntheory "forcomb";
my @S = qw/a b c/;
forcomb { print "[@S[@_]] " } scalar(@S);
print "\n";
- Output:
[] [a] [b] [c] [a b] [a c] [b c] [a b c]
Using the two argument version of the iterator gives a solution similar to the Raku and Python array versions.
use ntheory "forcomb";
my @S = qw/a b c/;
for $k (0..@S) {
# Iterate over each $#S+1,$k combination.
forcomb { print "[@S[@_]] " } @S,$k;
}
print "\n";
- Output:
[] [a] [b] [c] [a b] [a c] [b c] [a b c]
Similar to the Pari/GP solution, one can also use vecextract with an integer mask to select elements. Note that it does not enforce that the input array is a set (no duplication). This also has low memory if each subset is processed immediately and the range is applied with a loop rather than a map. A solution using vecreduce could be done identical to the array reduce solution shown later.
use ntheory "vecextract";
my @S = qw/a b c/;
my @PS = map { "[".join(" ",vecextract(\@S,$_))."]" } 0..2**scalar(@S)-1;
say join(" ",@PS);
- Output:
[] [a] [b] [a b] [c] [a c] [b c] [a b c]
Module: Set::Object
The CPAN module Set::Object provides a set implementation for sets of arbitrary objects, for which a powerset function could be defined and used like so:
use Set::Object qw(set);
sub powerset {
my $p = Set::Object->new( set() );
foreach my $i (shift->elements) {
$p->insert( map { set($_->elements, $i) } $p->elements );
}
return $p;
}
my $set = set(1, 2, 3);
my $powerset = powerset($set);
print $powerset->as_string, "\n";
- Output:
Set::Object(Set::Object() Set::Object(1 2 3) Set::Object(1 2) Set::Object(1 3) Set::Object(1) Set::Object(2 3) Set::Object(2) Set::Object(3))
Simple custom hash-based set type
It's also easy to define a custom type for sets of strings or numbers, using a hash as the underlying representation (like the task description suggests):
package Set {
sub new { bless { map {$_ => undef} @_[1..$#_] }, shift; }
sub elements { sort keys %{shift()} }
sub as_string { 'Set(' . join(' ', sort keys %{shift()}) . ')' }
# ...more set methods could be defined here...
}
(Note: For a ready-to-use module that uses this approach, and comes with all the standard set methods that you would expect, see the CPAN module Set::Tiny)
The limitation of this approach is that only primitive strings/numbers are allowed as hash keys in Perl, so a Set of Set's cannot be represented, and the return value of our powerset function will thus have to be a list of sets rather than being a Set object itself.
We could implement the function as an imperative foreach loop similar to the Set::Object based solution above, but using list folding (with the help of Perl's List::Util core module) seems a little more elegant in this case:
use List::Util qw(reduce);
sub powerset {
@{( reduce { [@$a, map { Set->new($_->elements, $b) } @$a ] }
[Set->new()], shift->elements )};
}
my $set = Set->new(1, 2, 3);
my @subsets = powerset($set);
print $_->as_string, "\n" for @subsets;
- Output:
Set() Set(1) Set(2) Set(1 2) Set(3) Set(1 3) Set(2 3) Set(1 2 3)
Arrays
If you don't actually need a proper set data-type that guarantees uniqueness of its elements, the simplest approach is to use arrays to store "sets" of items, in which case the implementation of the powerset function becomes quite short.
Recursive solution:
sub powerset {
@_ ? map { $_, [$_[0], @$_] } powerset(@_[1..$#_]) : [];
}
List folding solution:
use List::Util qw(reduce);
sub powerset {
@{( reduce { [@$a, map([@$_, $b], @$a)] } [[]], @_ )}
}Usage & output:
my @set = (1, 2, 3);
my @powerset = powerset(@set);
sub set_to_string {
"{" . join(", ", map { ref $_ ? set_to_string(@$_) : $_ } @_) . "}"
}
print set_to_string(@powerset), "\n";- Output:
{{}, {1}, {2}, {1, 2}, {3}, {1, 3}, {2, 3}, {1, 2, 3}}
Lazy evaluation
If the initial set is quite large, constructing it's powerset all at once can consume lots of memory.
If you want to iterate through all of the elements of the powerset of a set, and don't mind each element being generated immediately before you process it, and being thrown away immediately after you're done with it, you can use vastly less memory. This is similar to the earlier solutions using the Algorithm::Combinatorics and ntheory modules.
The following algorithm uses one bit of memory for every element of the original set (technically it uses several bytes per element with current versions of Perl). This is essentially doing a vecextract operation by hand.
use strict;
use warnings;
sub powerset :prototype(&@) {
my $callback = shift;
my $bitmask = '';
my $bytes = @_/8;
{
my @indices = grep vec($bitmask, $_, 1), 0..$#_;
$callback->( @_[@indices] );
++vec($bitmask, $_, 8) and last for 0 .. $bytes;
redo if @indices != @_;
}
}
print "powerset of empty set:\n";
powerset { print "[@_]\n" };
print "powerset of set {1,2,3,4}:\n";
powerset { print "[@_]\n" } 1..4;
my $i = 0;
powerset { ++$i } 1..9;
print "The powerset of a nine element set contains $i elements.\n";- Output:
powerset of empty set:
[]
powerset of set {1,2,3,4}:
[]
[1]
[2]
[1 2]
[3]
[1 3]
[2 3]
[1 2 3]
[4]
[1 4]
[2 4]
[1 2 4]
[3 4]
[1 3 4]
[2 3 4]
[1 2 3 4]
The powerset of a nine element set contains 512 elements.
The technique shown above will work with arbitrarily large sets, and uses a trivial amount of memory.
Phix
sequence powerset integer step = 1 function pst(object key, object /*data*/, object /*user_data*/) integer k = 1 while k<length(powerset) do k += step for j=1 to step do powerset[k] = append(powerset[k],key) k += 1 end for end while step *= 2 return 1 end function function power_set(integer d) powerset = repeat({},power(2,dict_size(d))) step = 1 traverse_dict(routine_id("pst"),0,d) return powerset end function integer d1234 = new_dict({{1,0},{2,0},{3,0},{4,0}}) ?power_set(d1234) integer d0 = new_dict() ?power_set(d0) setd({},0,d0) ?power_set(d0)
- Output:
{{},{1},{2},{1,2},{3},{1,3},{2,3},{1,2,3},{4},{1,4},{2,4},{1,2,4},{3,4},{1,3,4},{2,3,4},{1,2,3,4}}
{{}}
{{},{{}}}
alternative
Adapted from the one I used on Ascending_primes#powerset.
with javascript_semantics function power_set(sequence s) sequence powerset = {{}}, subset = {{{},0}} while length(subset) do sequence next = {} for i=1 to length(subset) do {sequence sub, integer k} = subset[i] for j=k+1 to length(s) do sequence ni = append(deep_copy(sub),s[j]) next = append(next,{ni,j}) powerset = append(powerset,ni) end for end for subset = next end while assert(length(powerset)=power(2,length(s))) return powerset end function ?power_set({1,2,3,4}) ?power_set({4,3,2,1}) ?power_set({}) ?power_set({{}})
- Output:
Guaranteed to be in length order, and index order within each length.
{{},{1},{2},{3},{4},{1,2},{1,3},{1,4},{2,3},{2,4},{3,4},{1,2,3},{1,2,4},{1,3,4},{2,3,4},{1,2,3,4}}
{{},{4},{3},{2},{1},{4,3},{4,2},{4,1},{3,2},{3,1},{2,1},{4,3,2},{4,3,1},{4,2,1},{3,2,1},{4,3,2,1}}
{{}}
{{},{{}}}
PHP
<?php
function get_subset($binary, $arr) {
// based on true/false values in $binary array, include/exclude
// values from $arr
$subset = array();
foreach (range(0, count($arr)-1) as $i) {
if ($binary[$i]) {
$subset[] = $arr[count($arr) - $i - 1];
}
}
return $subset;
}
function print_array($arr) {
if (count($arr) > 0) {
echo join(" ", $arr);
} else {
echo "(empty)";
}
echo '<br>';
}
function print_power_sets($arr) {
echo "POWER SET of [" . join(", ", $arr) . "]<br>";
foreach (power_set($arr) as $subset) {
print_array($subset);
}
}
function power_set($arr) {
$binary = array();
foreach (range(1, count($arr)) as $i) {
$binary[] = false;
}
$n = count($arr);
$powerset = array();
while (count($binary) <= count($arr)) {
$powerset[] = get_subset($binary, $arr);
$i = 0;
while (true) {
if ($binary[$i]) {
$binary[$i] = false;
$i += 1;
} else {
$binary[$i] = true;
break;
}
}
$binary[$i] = true;
}
return $powerset;
}
print_power_sets(array());
print_power_sets(array('singleton'));
print_power_sets(array('dog', 'c', 'b', 'a'));
?>- Output:
POWER SET of []
POWER SET of [singleton]
(empty)
singleton
POWER SET of [dog, c, b, a]
(empty)
a
b
a b
c
a c
b c
a b c
dog
a dog
b dog
a b dog
c dog
a c dog
b c dog
a b c dogPicoLisp
(de powerset (Lst)
(ifn Lst
(cons)
(let L (powerset (cdr Lst))
(conc
(mapcar '((X) (cons (car Lst) X)) L)
L ) ) ) )PL/I
*process source attributes xref or(!);
/*--------------------------------------------------------------------
* 06.01.2014 Walter Pachl translated from REXX
*-------------------------------------------------------------------*/
powerset: Proc Options(main);
Dcl (hbound,index,left,substr) Builtin;
Dcl sysprint Print;
Dcl s(4) Char(5) Var Init('one','two','three','four');
Dcl ps Char(1000) Var;
Dcl (n,chunk,p) Bin Fixed(31);
n=hbound(s); /* number of items in the list. */
ps='{} '; /* start with a null power set. */
Do chunk=1 To n; /* loop through the ... . */
ps=ps!!combn(chunk); /* a CHUNK at a time. */
End;
Do While(ps>'');
p=index(ps,' ');
Put Edit(left(ps,p-1))(Skip,a);
ps=substr(ps,p+1);
End;
combn: Proc(y) Returns(Char(1000) Var);
/*--------------------------------------------------------------------
* returns the list of subsets with y elements of set s
*-------------------------------------------------------------------*/
Dcl (y,base,bbase,ym,p,j,d,u) Bin Fixed(31);
Dcl (z,l) Char(1000) Var Init('');
Dcl a(20) Bin Fixed(31) Init((20)0);
Dcl i Bin Fixed(31);
base=hbound(s)+1;
bbase=base-y;
ym=y-1;
Do p=1 To y;
a(p)=p;
End;
Do j=1 By 1;
l='';
Do d=1 To y;
u=a(d);
l=l!!','!!s(u);
End;
z=z!!'{'!!substr(l,2)!!'} ';
a(y)=a(y)+1;
If a(y)=base Then
If combu(ym) Then
Leave;
End;
/* Put Edit('combn',y,z)(Skip,a,f(2),x(1),a); */
Return(z);
combu: Proc(d) Recursive Returns(Bin Fixed(31));
Dcl (d,u) Bin Fixed(31);
If d=0 Then
Return(1);
p=a(d);
Do u=d To y;
a(u)=p+1;
If a(u)=bbase+u Then
Return(combu(u-1));
p=a(u);
End;
Return(0);
End;
End;
End;- Output:
{}
{one}
{two}
{three}
{four}
{one,two}
{one,three}
{one,four}
{two,three}
{two,four}
{three,four}
{one,two,three}
{one,two,four}
{one,three,four}
{two,three,four}
{one,two,three,four}
PowerShell
function power-set ($array) {
if($array) {
$n = $array.Count
function state($set, $i){
if($i -gt -1) {
state $set ($i-1)
state ($set+@($array[$i])) ($i-1)
} else {
"$($set | sort)"
}
}
$set = state @() ($n-1)
$power = 0..($set.Count-1) | foreach{@(0)}
$i = 0
$set | sort | foreach{$power[$i++] = $_.Split()}
$power | sort {$_.Count}
} else {@()}
}
$OFS = " "
$setA = power-set @(1,2,3,4)
"number of sets in setA: $($setA.Count)"
"sets in setA:"
$OFS = ", "
$setA | foreach{"{"+"$_"+"}"}
$setB = @()
"number of sets in setB: $($setB.Count)"
"sets in setB:"
$setB | foreach{"{"+"$_"+"}"}
$setC = @(@(), @(@()))
"number of sets in setC: $($setC.Count)"
"sets in setC:"
$setC | foreach{"{"+"$_"+"}"}
$OFS = " "Output:
number of sets in setA: 16
sets in setA:
{}
{1}
{2}
{3}
{4}
{1, 2}
{1, 3}
{1, 4}
{2, 3}
{2, 4}
{3, 4}
{1, 2, 3}
{1, 2, 4}
{1, 3, 4}
{2, 3, 4}
{1, 2, 3, 4}
number of sets in setB: 0
sets in setB:
number of sets in setC: 2
sets in setC:
{}
{}
Prolog
Logical (cut-free) Definition
The predicate powerset(X,Y) defined here can be read as "Y is the powerset of X", it being understood that lists are used to represent sets.
The predicate subseq(X,Y) is true if and only if the list X is a subsequence of the list Y.
The definitions here are elementary, logical (cut-free), and efficient (within the class of comparably generic implementations).
powerset(X,Y) :- bagof( S, subseq(S,X), Y).
subseq( [], []).
subseq( [], [_|_]).
subseq( [X|Xs], [X|Ys] ) :- subseq(Xs, Ys).
subseq( [X|Xs], [_|Ys] ) :- append(_, [X|Zs], Ys), subseq(Xs, Zs).- Output:
?- powerset([1,2,3], X). X = [[], [1], [1, 2], [1, 2, 3], [1, 3], [2], [2, 3], [3]]. % Symbolic: ?- powerset( [X,Y], S). S = [[], [X], [X, Y], [Y]]. % In reverse: ?- powerset( [X,Y], [[], [1], [1, 2], [2]] ). X = 1, Y = 2.
Single-Functor Definition
power_set( [], [[]]).
power_set( [X|Xs], PS) :-
power_set(Xs, PS1),
maplist( append([X]), PS1, PS2 ), % i.e. prepend X to each PS1
append(PS1, PS2, PS).- Output:
?- power_set([1,2,3,4,5,6,7,8], X), length(X,N), writeln(N). 256
Constraint Handling Rules
CHR is a programming language created by Professor Thom Frühwirth.
Works with SWI-Prolog and module chr written by Tom Schrijvers and Jan Wielemaker.
:- use_module(library(chr)).
:- chr_constraint chr_power_set/2, chr_power_set/1, clean/0.
clean @ clean \ chr_power_set(_) <=> true.
clean @ clean <=> true.
only_one @ chr_power_set(A) \ chr_power_set(A) <=> true.
creation @ chr_power_set([H | T], A) <=>
append(A, [H], B),
chr_power_set(T, A),
chr_power_set(T, B),
chr_power_set(B).
empty_element @ chr_power_set([], _) <=> chr_power_set([]).- Output:
?- chr_power_set([1,2,3,4], []), findall(L, find_chr_constraint(chr_power_set(L)), LL), clean. LL = [[1],[1,2],[1,2,3],[1,2,3,4],[1,2,4],[1,3],[1,3,4],[1,4],[2],[2,3],[2,3,4],[2,4],[3],[3,4],[4],[]] .
PureBasic
This code is for console mode.
If OpenConsole()
Define argc=CountProgramParameters()
If argc>=(SizeOf(Integer)*8) Or argc<1
PrintN("Set out of range.")
End 1
Else
Define i, j, text$
Define.q bset=1<<argc
Print("{")
For i=0 To bset-1 ; check all binary combinations
If Not i: text$= "{"
Else : text$=", {"
EndIf
k=0
For j=0 To argc-1 ; step through each bit
If i&(1<<j)
If k: text$+", ": EndIf ; pad the output
text$+ProgramParameter(j): k+1 ; append each matching bit
EndIf
Next j
Print(text$+"}")
Next i
PrintN("}")
EndIf
EndIf- Output:
C:\Users\PureBasic_User\Desktop>"Power Set.exe" 1 2 3 4
{{}, {1}, {2}, {1, 2}, {3}, {1, 3}, {2, 3}, {1, 2, 3}, {4}, {1, 4},
{2, 4}, {1, 2, 4}, {3, 4}, {1, 3, 4}, {2, 3, 4}, {1, 2, 3, 4}}
Python
def list_powerset(lst):
# the power set of the empty set has one element, the empty set
result = [[]]
for x in lst:
# for every additional element in our set
# the power set consists of the subsets that don't
# contain this element (just take the previous power set)
# plus the subsets that do contain the element (use list
# comprehension to add [x] onto everything in the
# previous power set)
result.extend([subset + [x] for subset in result])
return result
# the above function in one statement
def list_powerset2(lst):
return reduce(lambda result, x: result + [subset + [x] for subset in result],
lst, [[]])
def powerset(s):
return frozenset(map(frozenset, list_powerset(list(s))))list_powerset computes the power set of a list of distinct elements. powerset simply converts the input and output from lists to sets. We use the frozenset type here for immutable sets, because unlike mutable sets, it can be put into other sets.
- Example:
>>> list_powerset([1,2,3]) [[], [1], [2], [1, 2], [3], [1, 3], [2, 3], [1, 2, 3]] >>> powerset(frozenset([1,2,3])) frozenset([frozenset([3]), frozenset([1, 2]), frozenset([]), frozenset([2, 3]), frozenset([1]), frozenset([1, 3]), frozenset([1, 2, 3]), frozenset([2])])
Further Explanation
If you take out the requirement to produce sets and produce list versions of each powerset element, then add a print to trace the execution, you get this simplified version of the program above where it is easier to trace the inner workings
def powersetlist(s):
r = [[]]
for e in s:
print "r: %-55r e: %r" % (r,e)
r += [x+[e] for x in r]
return r
s= [0,1,2,3]
print "\npowersetlist(%r) =\n %r" % (s, powersetlist(s))- Output:
r: [[]] e: 0 r: [[], [0]] e: 1 r: [[], [0], [1], [0, 1]] e: 2 r: [[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2]] e: 3 powersetlist([0, 1, 2, 3]) = [[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]]
Binary Count method
If you list the members of the set and include them according to if the corresponding bit position of a binary count is true then you generate the powerset. (Note that only frozensets can be members of a set in the second function)
def powersequence(val):
''' Generate a 'powerset' for sequence types that are indexable by integers.
Uses a binary count to enumerate the members and returns a list
Examples:
>>> powersequence('STR') # String
['', 'S', 'T', 'ST', 'R', 'SR', 'TR', 'STR']
>>> powersequence([0,1,2]) # List
[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2]]
>>> powersequence((3,4,5)) # Tuple
[(), (3,), (4,), (3, 4), (5,), (3, 5), (4, 5), (3, 4, 5)]
>>>
'''
vtype = type(val); vlen = len(val); vrange = range(vlen)
return [ reduce( lambda x,y: x+y, (val[i:i+1] for i in vrange if 2**i & n), vtype())
for n in range(2**vlen) ]
def powerset(s):
''' Generate the powerset of s
Example:
>>> powerset(set([6,7,8]))
set([frozenset([7]), frozenset([8, 6, 7]), frozenset([6]), frozenset([6, 7]), frozenset([]), frozenset([8]), frozenset([8, 7]), frozenset([8, 6])])
'''
return set( frozenset(x) for x in powersequence(list(s)) )Recursive Alternative
This is an (inefficient) recursive version that almost reflects the recursive definition of a power set as explained in http://en.wikipedia.org/wiki/Power_set#Algorithms. It does not create a sorted output.
def p(l):
if not l: return [[]]
return p(l[1:]) + [[l[0]] + x for x in p(l[1:])]Python: Standard documentation
Pythons documentation has a method that produces the groupings, but not as sets:
>>> from pprint import pprint as pp
>>> from itertools import chain, combinations
>>>
>>> def powerset(iterable):
"powerset([1,2,3]) --> () (1,) (2,) (3,) (1,2) (1,3) (2,3) (1,2,3)"
s = list(iterable)
return chain.from_iterable(combinations(s, r) for r in range(len(s)+1))
>>> pp(set(powerset({1,2,3,4})))
{(),
(1,),
(1, 2),
(1, 2, 3),
(1, 2, 3, 4),
(1, 2, 4),
(1, 3),
(1, 3, 4),
(1, 4),
(2,),
(2, 3),
(2, 3, 4),
(2, 4),
(3,),
(3, 4),
(4,)}
>>>Qi
(define powerset
[] -> [[]]
[A|As] -> (append (map (cons A) (powerset As))
(powerset As)))Quackery
Quackery is, when seen from a certain perspective, an assembly language that recognises only three types, "operators", which correspond to op-codes, "numbers" i.e. bignums, and "nests" which are ordered sequences of zero of more operator, bignums and nests. Everything else is a matter of interpretation.
Integers can be used as a set of natural numbers, with (in binary) 0 corresponding to the empty set, 1 corresponding to the set of the natural number 1, 10 corresponding to the set of the natural number 2, 11 corresponding to the set of the natural numbers 1 and 2, and so on. With this sort of set, enumerating the powerset of the numbers 0 to 4, for example simply consists of enumerating the numbers 0 to 15 inclusive. Operations on this sort of set, such as union and intersection, correspond to bitwise logic operators.
The other way of implementing sets is with nests, with each item in a nest corresponding to an item in the set. This is computationally slower and more complex to code, but has the advantage that it permits sets of sets, which are required for this task.
[ stack ] is (ps).stack
[ stack ] is (ps).items
[ stack ] is (ps).result
[ 1 - (ps).items put
0 (ps).stack put
[] (ps).result put
[ (ps).result take
(ps).stack behead
drop nested join
(ps).result put
(ps).stack take
dup (ps).items share
= iff
[ drop
(ps).stack size 1 > iff
[ 1 (ps).stack tally ] ]
else
[ dup (ps).stack put
1+ (ps).stack put ]
(ps).stack size 1 = until ]
(ps).items release
(ps).result take ] is (ps) ( n --> )
[ dup size dip
[ witheach
[ over swap peek swap ] ]
nip pack ] is arrange ( [ [ --> [ )
[ dup [] = iff
nested done
dup size (ps)
' [ [ ] ] swap join
[] unrot witheach
[ dip dup arrange
nested
rot swap join swap ]
drop ] is powerset ( [ --> [ )
' [ [ 1 2 3 4 ] [ ] [ [ ] ] ]
witheach
[ say "The powerset of "
dup echo cr
powerset witheach [ echo cr ]
cr ]- Output:
The powerset of [ 1 2 3 4 ] [ ] [ 1 ] [ 1 2 ] [ 1 2 3 ] [ 1 2 3 4 ] [ 1 2 4 ] [ 1 3 ] [ 1 3 4 ] [ 1 4 ] [ 2 ] [ 2 3 ] [ 2 3 4 ] [ 2 4 ] [ 3 ] [ 3 4 ] [ 4 ] The powerset of [ ] [ ] The powerset of [ [ ] ] [ ] [ [ ] ]
R
Non-recursive version
The conceptual basis for this algorithm is the following:
for each element in the set:
for each subset constructed so far:
new subset = (subset + element)This method is much faster than a recursive method, though the speed is still O(2^n).
powerset <- function(set){
ps <- list()
ps[[1]] <- numeric() #Start with the empty set.
for(element in set){ #For each element in the set, take all subsets
temp <- vector(mode="list",length=length(ps)) #currently in "ps" and create new subsets (in "temp")
for(subset in 1:length(ps)){ #by adding "element" to each of them.
temp[[subset]] = c(ps[[subset]],element)
}
ps <- c(ps,temp) #Add the additional subsets ("temp") to "ps".
}
ps
}
powerset(1:4)The list "temp" is a compromise between the speed costs of doing arithmetic and of creating new lists (since R lists are immutable, appending to a list means actually creating a new list object). Thus, "temp" collects new subsets that are later added to the power set. This improves the speed by 4x compared to extending the list "ps" at every step.
Recursive version
The sets package includes a recursive method to calculate the power set. However, this method takes ~100 times longer than the non-recursive method above.
library(sets)An example with a vector.
v <- (1:3)^2
sv <- as.set(v)
2^sv{{}, {1}, {4}, {9}, {1, 4}, {1, 9}, {4, 9}, {1, 4, 9}}
An example with a list.
l <- list(a=1, b="qwerty", c=list(d=TRUE, e=1:3))
sl <- as.set(l)
2^sl{{}, {1}, {"qwerty"}, {<<list(2)>>}, {1, <<list(2)>>}, {"qwerty",
1}, {"qwerty", <<list(2)>>}, {"qwerty", 1, <<list(2)>>}}
Racket
;;; Direct translation of 'functional' ruby method
(define (powerset s)
(for/fold ([outer-set (set(set))]) ([element s])
(set-union outer-set
(list->set (set-map outer-set
(λ(inner-set) (set-add inner-set element)))))))Raku
(formerly Perl 6)
sub powerset(Set $s) { $s.combinations.map(*.Set).Set }
say powerset set <a b c d>;- Output:
set(set(), set(a), set(b), set(c), set(d), set(a, b), set(a, c), set(a, d), set(b, c), set(b, d), set(c, d), set(a, b, c), set(a, b, d), set(a, c, d), set(b, c, d), set(a, b, c, d))
If you don't care about the actual Set type, the .combinations method by itself may be good enough for you:
.say for <a b c d>.combinations- Output:
a b c d a b a c a d b c b d c d a b c a b d a c d b c d a b c d
Rascal
import Set;
public set[set[&T]] PowerSet(set[&T] s) = power(s);- Output:
rascal>PowerSet({1,2,3,4})
set[set[int]]: {
{4,3},
{4,2,1},
{4,3,1},
{4,2},
{4,3,2},
{4,1},
{4,3,2,1},
{4},
{3},
{2,1},
{3,1},
{2},
{3,2},
{1},
{3,2,1},
{}
}REXX
/*REXX program displays a power set; items may be anything (but can't have blanks).*/
Parse Arg text /*allow the user specify optional set. */
If text='' Then /*Not specified? Then use the default.*/
text='one two three four'
n=words(text)
psi=0
Do k=0 To n /* loops from 0 to n elements of a set */
cc=comb(n,k) /* returns the combinations of 1 through k */
Do while pos('/',cc)>0 /* as long as there is a combination */
Parse Var cc elist '/' cc /* get i from comb's result string */
psl='' /* initialize the list of words */
psi=psi+1 /* index of this set */
Do While elist<>'' /* loop through elements */
parse var elist e elist /* get an element (a digit) */
psl=psl','word(text,e) /* add corresponding test word to set */
End
psl=substr(psl,2) /* get rid of leading comma */
Say right(psi,2) '{'psl'}' /* show this element of the power set */
End
End
Exit
comb: Procedure
/***********************************************************************
* Returns the combinations of size digits out of things digits
* e.g. comb(4,2) -> ' 1 2/1 3/1 4/2 3/2 4/3 4/'
1 2/ 1 3/ 1 4/ 2 3/ 2 4/ 3 4 /
***********************************************************************/
Parse Arg things,size
n=2**things-1
list=''
Do u=1 To n
co=combinations(u)
If co>'' Then
list=list '/' combinations(u)
End
Return substr(space(list),2) '/' /* remove leading / */
combinations: Procedure Expose things size
Parse Arg u
nc=0
bu=x2b(d2x(u))
bu1=space(translate(bu,' ',0),0)
If length(bu1)=size Then Do
ub=reverse(bu)
res=''
Do i=1 To things
c=i
If substr(ub,i,1)=1 Then res=res i
End
Return res
End
Else
Return ''- output when using the default input:
1 {}
2 {one}
3 {two}
4 {three}
5 {four}
6 {one,two}
7 {one,three}
8 {one,four}
9 {two,three}
10 {two,four}
11 {three,four}
12 {one,two,three}
13 {one,two,four}
14 {one,three,four}
15 {two,three,four}
16 {one,two,three,four}
Ring
# Project : Power set
list = ["1", "2", "3", "4"]
see powerset(list)
func powerset(list)
s = "{"
for i = 1 to (2 << len(list)) - 1 step 2
s = s + "{"
for j = 1 to len(list)
if i & (1 << j)
s = s + list[j] + ","
ok
next
if right(s,1) = ","
s = left(s,len(s)-1)
ok
s = s + "},"
next
return left(s,len(s)-1) + "}"Output:
{{},{1},{2},{1,2},{3},{1,3},{2,3},{1,2,3},{4},{1,4},{2,4},{1,2,4},{3,4},{1,3,4},{2,3,4},{1,2,3,4}}
RPL
| RPL code | Comment |
|---|---|
≪
IF DUP SIZE
THEN LAST OVER SWAP GET → last
≪ LIST→ 1 - SWAP DROP →LIST POWST
1 OVER SIZE FOR j
DUP j GET last + 1 →LIST + NEXT
≫
ELSE 1 →LIST END
≫ 'POWST' STO
|
POWST ( { set } -- { power set } )
if set is not empty
then store last item
get power set of { set } - last item
for all sets of { set } - last item power set
add last item to set, then set to power set
else return { { } }
|
{ 1 2 3 4 } POWST
{ } POWST
- Output:
2: { { } { 1 } { 2 } { 1 2 } { 3 } { 1 3 } { 2 3 } { 1 2 3 } { 4 } { 1 4 } { 2 4 } { 1 2 4 } { 3 4 } { 1 3 4 } { 2 3 4 } { 1 2 3 4 } }
1: { { } }
Ruby
# Based on http://johncarrino.net/blog/2006/08/11/powerset-in-ruby/
# See the link if you want a shorter version.
# This was intended to show the reader how the method works.
class Array
# Adds a power_set method to every array, i.e.: [1, 2].power_set
def power_set
# Injects into a blank array of arrays.
# acc is what we're injecting into
# you is each element of the array
inject([[]]) do |acc, you|
ret = [] # Set up a new array to add into
acc.each do |i| # For each array in the injected array,
ret << i # Add itself into the new array
ret << i + [you] # Merge the array with a new array of the current element
end
ret # Return the array we're looking at to inject more.
end
end
# A more functional and even clearer variant.
def func_power_set
inject([[]]) { |ps,item| # for each item in the Array
ps + # take the powerset up to now and add
ps.map { |e| e + [item] } # it again, with the item appended to each element
}
end
end
#A direct translation of the "power array" version above
require 'set'
class Set
def powerset
inject(Set[Set[]]) do |ps, item|
ps.union ps.map {|e| e.union (Set.new [item])}
end
end
end
p [1,2,3,4].power_set
p %w(one two three).func_power_set
p Set[1,2,3].powerset- Output:
[[], [4], [3], [3, 4], [2], [2, 4], [2, 3], [2, 3, 4], [1], [1, 4], [1, 3], [1, 3, 4], [1, 2], [1, 2, 4], [1, 2, 3], [1, 2, 3, 4]]
[[], ["one"], ["two"], ["one", "two"], ["three"], ["one", "three"], ["two", "three"], ["one", "two", "three"]]
#<Set: {#<Set: {}>, #<Set: {1}>, #<Set: {2}>, #<Set: {1, 2}>, #<Set: {3}>, #<Set: {1, 3}>, #<Set: {2, 3}>, #<Set: {1, 2, 3}>}>
Rust
This implementation consumes the input set, requires that the type T has a full order a.k.a implements the Ord trait and that T is clonable.
use std::collections::BTreeSet;
fn powerset<T: Ord + Clone>(mut set: BTreeSet<T>) -> BTreeSet<BTreeSet<T>> {
if set.is_empty() {
let mut powerset = BTreeSet::new();
powerset.insert(set);
return powerset;
}
// Access the first value. This could be replaced with `set.pop_first().unwrap()`
// But this is an unstable feature
let entry = set.iter().nth(0).unwrap().clone();
set.remove(&entry);
let mut powerset = powerset(set);
for mut set in powerset.clone().into_iter() {
set.insert(entry.clone());
powerset.insert(set);
}
powerset
}
fn main() {
let set = (1..5).collect();
let set = powerset(set);
println!("{:?}", set);
let set = ["a", "b", "c", "d"].iter().collect();
let set = powerset(set);
println!("{:?}", set);
}- Output:
{{}, {1}, {1, 2}, {1, 2, 3}, {1, 2, 3, 4}, {1, 2, 4}, {1, 3}, {1, 3, 4}, {1, 4}, {2}, {2, 3}, {2, 3, 4}, {2, 4}, {3}, {3, 4}, {4}}
{{}, {"a"}, {"a", "b"}, {"a", "b", "c"}, {"a", "b", "c", "d"}, {"a", "b", "d"}, {"a", "c"}, {"a", "c", "d"}, {"a", "d"}, {"b"}, {"b", "c"}, {"b", "c", "d"}, {"b", "d"}, {"c"}, {"c", "d"}, {"d"}}
SAS
options mprint mlogic symbolgen source source2;
%macro SubSets (FieldCount = );
data _NULL_;
Fields = &FieldCount;
SubSets = 2**Fields;
call symput ("NumSubSets", SubSets);
run;
%put &NumSubSets;
data inital;
%do j = 1 %to &FieldCount;
F&j. = 1;
%end;
run;
data SubSets;
set inital;
RowCount =_n_;
call symput("SetCount",RowCount);
run;
%put SetCount ;
%do %while (&SetCount < &NumSubSets);
data loop;
%do j=1 %to &FieldCount;
if rand('GAUSSIAN') > rand('GAUSSIAN') then F&j. = 1;
%end;
data SubSets_ ;
set SubSets loop;
run;
proc sort data=SubSets_ nodupkey;
by F1 - F&FieldCount.;
run;
data Subsets;
set SubSets_;
RowCount =_n_;
run;
proc sql noprint;
select max(RowCount) into :SetCount
from SubSets;
quit;
run;
%end;
%Mend SubSets;You can then call the macro as:
%SubSets(FieldCount = 5);The output will be the dataset SUBSETS and will have a 5 columns F1, F2, F3, F4, F5 and 32 columns, one with each combination of 1 and missing values.
- Output:
Obs F1 F2 F3 F4 F5 RowCount 1 . . . . . 1 2 . . . . 1 2 3 . . . 1 . 3 4 . . . 1 1 4 5 . . 1 . . 5 6 . . 1 . 1 6 7 . . 1 1 . 7 8 . . 1 1 1 8 9 . 1 . . . 9 10 . 1 . . 1 10 11 . 1 . 1 . 11 12 . 1 . 1 1 12 13 . 1 1 . . 13 14 . 1 1 . 1 14 15 . 1 1 1 . 15 16 . 1 1 1 1 16 17 1 . . . . 17 18 1 . . . 1 18 19 1 . . 1 . 19 20 1 . . 1 1 20 21 1 . 1 . . 21 22 1 . 1 . 1 22 23 1 . 1 1 . 23 24 1 . 1 1 1 24 25 1 1 . . . 25 26 1 1 . . 1 26 27 1 1 . 1 . 27 28 1 1 . 1 1 28 29 1 1 1 . . 29 30 1 1 1 . 1 30 31 1 1 1 1 . 31 32 1 1 1 1 1 32
Scala
import scala.compat.Platform.currentTime
object Powerset extends App {
def powerset[A](s: Set[A]) = s.foldLeft(Set(Set.empty[A])) { case (ss, el) => ss ++ ss.map(_ + el)}
assert(powerset(Set(1, 2, 3, 4)) == Set(Set.empty, Set(1), Set(2), Set(3), Set(4), Set(1, 2), Set(1, 3), Set(1, 4),
Set(2, 3), Set(2, 4), Set(3, 4), Set(1, 2, 3), Set(1, 3, 4), Set(1, 2, 4), Set(2, 3, 4), Set(1, 2, 3, 4)))
println(s"Successfully completed without errors. [total ${currentTime - executionStart} ms]")
}Another option that produces lazy sequence of the sets:
def powerset[A](s: Set[A]) = (0 to s.size).map(s.toSeq.combinations(_)).reduce(_ ++ _).map(_.toSet)A tail-recursive version:
def powerset[A](s: Set[A]) = {
def powerset_rec(acc: List[Set[A]], remaining: List[A]): List[Set[A]] = remaining match {
case Nil => acc
case head :: tail => powerset_rec(acc ++ acc.map(_ + head), tail)
}
powerset_rec(List(Set.empty[A]), s.toList)
}Scheme
(define (power-set set)
(if (null? set)
'(())
(let ((rest (power-set (cdr set))))
(append (map (lambda (element) (cons (car set) element))
rest)
rest))))
(display (power-set (list 1 2 3)))
(newline)
(display (power-set (list "A" "C" "E")))
(newline)- Output:
((1 2 3) (1 2) (1 3) (1) (2 3) (2) (3) ()) ((A C E) (A C) (A E) (A) (C E) (C) (E) ())
Call/cc generation:
(define (power-set lst)
(define (iter yield)
(let recur ((a '()) (b lst))
(if (null? b) (set! yield
(call-with-current-continuation
(lambda (resume)
(set! iter resume)
(yield a))))
(begin (recur (append a (list (car b))) (cdr b))
(recur a (cdr b)))))
;; signal end of generation
(yield 'end-of-seq))
(lambda () (call-with-current-continuation iter)))
(define x (power-set '(1 2 3)))
(let loop ((a (x)))
(if (eq? a 'end-of-seq) #f
(begin
(display a)
(newline)
(loop (x)))))- Output:
(1 2) (1 3) (1) (2 3) (2) (3) ()
Iterative:
(define (power_set_iter set)
(let loop ((res '(())) (s set))
(if (empty? s)
res
(loop (append (map (lambda (i) (cons (car s) i)) res) res) (cdr s)))))- Output:
'((e d c b a) (e d c b) (e d c a) (e d c) (e d b a) (e d b) (e d a) (e d) (e c b a) (e c b) (e c a) (e c) (e b a) (e b) (e a) (e) (d c b a) (d c b) (d c a) (d c) (d b a) (d b) (d a) (d) (c b a) (c b) (c a) (c) (b a) (b) (a) ())
Seed7
$ include "seed7_05.s7i";
const func array bitset: powerSet (in bitset: baseSet) is func
result
var array bitset: pwrSet is [] (bitset.value);
local
var integer: element is 0;
var integer: index is 0;
var bitset: aSet is bitset.value;
begin
for element range baseSet do
for key index range pwrSet do
aSet := pwrSet[index];
if element not in aSet then
incl(aSet, element);
pwrSet &:= aSet;
end if;
end for;
end for;
end func;
const proc: main is func
local
var bitset: aSet is bitset.value;
begin
for aSet range powerSet({1, 2, 3, 4}) do
writeln(aSet);
end for;
end func;- Output:
{}
{1}
{2}
{1, 2}
{3}
{1, 3}
{2, 3}
{1, 2, 3}
{4}
{1, 4}
{2, 4}
{1, 2, 4}
{3, 4}
{1, 3, 4}
{2, 3, 4}
{1, 2, 3, 4}
SETL
Pfour := pow({1, 2, 3, 4});
Pempty := pow({});
PPempty := pow(Pempty);
print(Pfour);
print(Pempty);
print(PPempty);- Output:
{{} {1} {2} {3} {4} {1 2} {1 3} {1 4} {2 3} {2 4} {3 4} {1 2 3} {1 2 4} {1 3 4} {2 3 4} {1 2 3 4}}
{{}}
{{} {{}}}
Sidef
var arr = %w(a b c)
for i in (0 .. arr.len) {
say arr.combinations(i)
}- Output:
[[]] [["a"], ["b"], ["c"]] [["a", "b"], ["a", "c"], ["b", "c"]] [["a", "b", "c"]]
Simula
SIMSET
BEGIN
LINK CLASS LOF_INT(N); INTEGER N;;
LINK CLASS LOF_LOF_INT(H); REF(HEAD) H;;
REF(HEAD) PROCEDURE MAP(P_LI, P_LLI);
REF(HEAD) P_LI;
REF(HEAD) P_LLI;
BEGIN
REF(HEAD) V_RESULT;
V_RESULT :- NEW HEAD;
IF NOT P_LLI.EMPTY THEN BEGIN
REF(LOF_LOF_INT) V_LLI;
V_LLI :- P_LLI.FIRST QUA LOF_LOF_INT;
WHILE V_LLI =/= NONE DO BEGIN
REF(HEAD) V_NEWLIST;
V_NEWLIST :- NEW HEAD;
! ADD THE SAME 1ST ELEMENT TO EVERY NEWLIST ;
NEW LOF_INT(P_LI.FIRST QUA LOF_INT.N).INTO(V_NEWLIST);
IF NOT V_LLI.H.EMPTY THEN BEGIN
REF(LOF_INT) V_LI;
V_LI :- V_LLI.H.FIRST QUA LOF_INT;
WHILE V_LI =/= NONE DO BEGIN
NEW LOF_INT(V_LI.N).INTO(V_NEWLIST);
V_LI :- V_LI.SUC;
END;
END;
NEW LOF_LOF_INT(V_NEWLIST).INTO(V_RESULT);
V_LLI :- V_LLI.SUC;
END;
END;
MAP :- V_RESULT;
END MAP;
REF(HEAD) PROCEDURE SUBSETS(P_LI);
REF(HEAD) P_LI;
BEGIN
REF(HEAD) V_RESULT;
IF P_LI.EMPTY THEN BEGIN
V_RESULT :- NEW HEAD;
NEW LOF_LOF_INT(NEW HEAD).INTO(V_RESULT);
END ELSE BEGIN
REF(HEAD) V_SUBSET, V_MAP;
REF(LOF_INT) V_LI;
V_SUBSET :- NEW HEAD;
V_LI :- P_LI.FIRST QUA LOF_INT;
! SKIP OVER 1ST ELEMENT ;
IF V_LI =/= NONE THEN V_LI :- V_LI.SUC;
WHILE V_LI =/= NONE DO BEGIN
NEW LOF_INT(V_LI.N).INTO(V_SUBSET);
V_LI :- V_LI.SUC;
END;
V_RESULT :- SUBSETS(V_SUBSET);
V_MAP :- MAP(P_LI, V_RESULT);
IF NOT V_MAP.EMPTY THEN BEGIN
REF(LOF_LOF_INT) V_LLI;
V_LLI :- V_MAP.FIRST QUA LOF_LOF_INT;
WHILE V_LLI =/= NONE DO BEGIN
NEW LOF_LOF_INT(V_LLI.H).INTO(V_RESULT);
V_LLI :- V_LLI.SUC;
END;
END;
END;
SUBSETS :- V_RESULT;
END SUBSETS;
PROCEDURE PRINT_LIST(P_LI); REF(HEAD) P_LI;
BEGIN
OUTTEXT("[");
IF NOT P_LI.EMPTY THEN BEGIN
INTEGER I;
REF(LOF_INT) V_LI;
I := 0;
V_LI :- P_LI.FIRST QUA LOF_INT;
WHILE V_LI =/= NONE DO BEGIN
IF I > 0 THEN OUTTEXT(",");
OUTINT(V_LI.N, 0);
V_LI :- V_LI.SUC;
I := I+1;
END;
END;
OUTTEXT("]");
END PRINT_LIST;
PROCEDURE PRINT_LIST_LIST(P_LLI); REF(HEAD) P_LLI;
BEGIN
OUTTEXT("[");
IF NOT P_LLI.EMPTY THEN BEGIN
INTEGER I;
REF(LOF_LOF_INT) V_LLI;
I := 0;
V_LLI :- P_LLI.FIRST QUA LOF_LOF_INT;
WHILE V_LLI =/= NONE DO BEGIN
IF I > 0 THEN BEGIN
OUTTEXT(",");
! OUTIMAGE;
END;
PRINT_LIST(V_LLI.H);
V_LLI :- V_LLI.SUC;
I := I+1;
END;
END;
OUTTEXT("]");
OUTIMAGE;
END PRINT_LIST_LIST;
INTEGER N;
REF(HEAD) V_RANGE;
REF(HEAD) V_LISTS;
V_RANGE :- NEW HEAD;
V_LISTS :- SUBSETS(V_RANGE);
PRINT_LIST_LIST(V_LISTS);
OUTIMAGE;
FOR N := 1 STEP 1 UNTIL 4 DO BEGIN
NEW LOF_INT(N).INTO(V_RANGE);
V_LISTS :- SUBSETS(V_RANGE);
PRINT_LIST_LIST(V_LISTS);
OUTIMAGE;
END;
END.- Output:
[[]] [[],[1]] [[],[2],[1],[1,2]] [[],[3],[2],[2,3],[1],[1,3],[1,2],[1,2,3]] [[],[4],[3],[3,4],[2],[2,4],[2,3],[2,3,4],[1],[1,4],[1,3],[1,3,4],[1,2],[1,2,4], [1,2,3],[1,2,3,4]]
Smalltalk
Code from Bonzini's blog
Collection extend [
power [
^(0 to: (1 bitShift: self size) - 1) readStream collect: [ :each || i |
i := 0.
self select: [ :elem | (each bitAt: (i := i + 1)) = 1 ] ]
]
].#(1 2 4) power do: [ :each |
each asArray printNl ].
#( 'A' 'C' 'E' ) power do: [ :each |
each asArray printNl ].Standard ML
version for lists:
fun subsets xs = foldr (fn (x, rest) => rest @ map (fn ys => x::ys) rest) [[]] xsSwift
func powersetFrom<T>(_ elements: Set<T>) -> Set<Set<T>> {
guard elements.count > 0 else {
return [[]]
}
var powerset: Set<Set<T>> = [[]]
for element in elements {
for subset in powerset {
powerset.insert(subset.union([element]))
}
}
return powerset
}
// Example:
powersetFrom([1, 2, 4])- Output:
{
{2, 4}
{4, 1}
{4},
{2, 4, 1}
{2, 1}
Set([])
{1}
{2}
}
//Example:
powersetFrom(["a", "b", "d"])- Output:
{
{"b", "d"}
{"b"}
{"d"},
{"a"}
{"b", "d", "a"}
Set([])
{"d", "a"}
{"b", "a"}
}
Tcl
proc subsets {l} {
set res [list [list]]
foreach e $l {
foreach subset $res {lappend res [lappend subset $e]}
}
return $res
}
puts [subsets {a b c d}]- Output:
{} a b {a b} c {a c} {b c} {a b c} d {a d} {b d} {a b d} {c d} {a c d} {b c d} {a b c d}
Binary Count Method
proc powersetb set {
set res {}
for {set i 0} {$i < 2**[llength $set]} {incr i} {
set pos -1
set pset {}
foreach el $set {
if {$i & 1<<[incr pos]} {lappend pset $el}
}
lappend res $pset
}
return $res
}TXR
The power set function can be written concisely like this:
(defun power-set (s)
(mappend* (op comb s) (range 0 (length s))))This generates the lists of combinations of all possible lengths, from 0 to the length of s and catenates them. The comb function generates a lazy list, so it is appropriate to use mappend* (the lazy version of mappend) to keep the behavior lazy.
A complete program which takes command line arguments and prints the power set in comma-separated brace notation:
@(do (defun power-set (s)
(mappend* (op comb s) (range 0 (length s)))))
@(bind pset @(power-set *args*))
@(output)
@ (repeat)
{@(rep)@pset, @(last)@pset@(empty)@(end)}
@ (end)
@(end)- Output:
$ txr rosetta/power-set.txr 1 2 3
{1, 2, 3}
{1, 2}
{1, 3}
{1}
{2, 3}
{2}
{3}
{}
The above power-set function
generalizes to strings and vectors.
@(do (defun power-set (s)
(mappend* (op comb s) (range 0 (length s))))
(prinl (power-set "abc"))
(prinl (power-set "b"))
(prinl (power-set ""))
(prinl (power-set #(1 2 3))))- Output:
$ txr power-set-generic.txr
("" "a" "b" "c" "ab" "ac" "bc" "abc")
("" "b")
("")
(#() #(1) #(2) #(3) #(1 2) #(1 3) #(2 3) #(1 2 3))
UNIX Shell
From here
p() { [ $# -eq 0 ] && echo || (shift; p "$@") | while read r ; do echo -e "$1 $r\n$r"; done }Usage
|p `cat` | sort | uniq
A
C
E
^DUnixPipes
| cat A
a
b
c
| cat A |\
xargs -n 1 ksh -c 'echo \{`cat A`\}' |\
xargs |\
sed -e 's; ;,;g' \
-e 's;^;echo ;g' \
-e 's;\},;}\\ ;g' |\
ksh |unfold `wc -l A` |\
xargs -n1 -I{} ksh -c 'echo {} |\
unfold 1 |sort -u |xargs' |sort -u
a
a b
a b c
a c
b
b c
cUrsala
Sets are a built in type constructor in Ursala, represented as lexically sorted lists with duplicates removed. The powerset function is a standard library function, but could be defined as shown below.
powerset = ~&NiC+ ~&i&& ~&at^?\~&aNC ~&ahPfatPRXlNrCDrTtest program:
#cast %sSS
test = powerset {'a','b','c','d'}- Output:
{
{},
{'a'},
{'a','b'},
{'a','b','c'},
{'a','b','c','d'},
{'a','b','d'},
{'a','c'},
{'a','c','d'},
{'a','d'},
{'b'},
{'b','c'},
{'b','c','d'},
{'b','d'},
{'c'},
{'c','d'},
{'d'}}
V
V has a built in called powerlist
[A C E] powerlist
=[[A C E] [A C] [A E] [A] [C E] [C] [E] []]its implementation in std.v is (like joy)
[powerlist
[null?]
[unitlist]
[uncons]
[dup swapd [cons] map popd swoncat]
linrec].VBA
Option Base 1
Private Function power_set(ByRef st As Collection) As Collection
Dim subset As Collection, pwset As New Collection
For i = 0 To 2 ^ st.Count - 1
Set subset = New Collection
For j = 1 To st.Count
If i And 2 ^ (j - 1) Then subset.Add st(j)
Next j
pwset.Add subset
Next i
Set power_set = pwset
End Function
Private Function print_set(ByRef st As Collection) As String
'assume st is a collection of collections, holding integer variables
Dim s() As String, t() As String
ReDim s(st.Count)
'Debug.Print "{";
For i = 1 To st.Count
If st(i).Count > 0 Then
ReDim t(st(i).Count)
For j = 1 To st(i).Count
Select Case TypeName(st(i)(j))
Case "Integer": t(j) = CStr(st(i)(j))
Case "Collection": t(j) = "{}" 'assumes empty
End Select
Next j
s(i) = "{" & Join(t, ", ") & "}"
Else
s(i) = "{}"
End If
Next i
print_set = "{" & Join(s, ", ") & "}"
End Function
Public Sub rc()
Dim rcset As New Collection, result As Collection
For i = 1 To 4
rcset.Add i
Next i
Debug.Print print_set(power_set(rcset))
Set rcset = New Collection
Debug.Print print_set(power_set(rcset))
Dim emptyset As New Collection
rcset.Add emptyset
Debug.Print print_set(power_set(rcset))
Debug.Print
End Sub- Output:
{{}, {1}, {2}, {1, 2}, {3}, {1, 3}, {2, 3}, {1, 2, 3}, {4}, {1, 4}, {2, 4}, {1, 2, 4}, {3, 4}, {1, 3, 4}, {2, 3, 4}, {1, 2, 3, 4}}
{{}}
{{}, {{}}}
VBScript
Function Dec2Bin(n)
q = n
Dec2Bin = ""
Do Until q = 0
Dec2Bin = CStr(q Mod 2) & Dec2Bin
q = Int(q / 2)
Loop
Dec2Bin = Right("00000" & Dec2Bin,6)
End Function
Function PowerSet(s)
arrS = Split(s,",")
PowerSet = "{"
For i = 0 To 2^(UBound(arrS)+1)-1
If i = 0 Then
PowerSet = PowerSet & "{},"
Else
binS = Dec2Bin(i)
PowerSet = PowerSet & "{"
c = 0
For j = Len(binS) To 1 Step -1
If CInt(Mid(binS,j,1)) = 1 Then
PowerSet = PowerSet & arrS(c) & ","
End If
c = c + 1
Next
PowerSet = Mid(PowerSet,1,Len(PowerSet)-1) & "},"
End If
Next
PowerSet = Mid(PowerSet,1,Len(PowerSet)-1) & "}"
End Function
WScript.StdOut.Write PowerSet("1,2,3,4")- Output:
{{},{1},{2},{1,2},{3},{1,3},{2,3},{1,2,3},{4},{1,4},{2,4},{1,2,4},{3,4},{1,3,4},{2,3,4},{1,2,3,4}}
Wren
Although we have a module for sets, they are based on maps whose keys must be value types. This means that sets of sets are technically impossible because sets themselves are not value types.
We therefore use lists to represent sets which works fine here.
import "./perm" for Powerset
var sets = [ [1, 2, 3, 4], [], [[]] ]
for (set in sets) {
System.print("The power set of %(set) is:")
System.print(Powerset.list(set))
System.print()
}- Output:
The power set of [1, 2, 3, 4] is: [[], [1], [2], [3], [4], [1, 2], [1, 3], [1, 4], [2, 3], [2, 4], [3, 4], [1, 2, 3], [1, 2, 4], [1, 3, 4], [2, 3, 4], [1, 2, 3, 4]] The power set of [] is: [[]] The power set of [[]] is: [[], [[]]]
XPL0
func PowSet(Set, Size);
int Set, Size;
int N, M, Mask, DoComma;
[ChOut(0, ^{);
for N:= 0 to 1<<Size -1 do
[if N>0 then ChOut(0, ^,);
ChOut(0, ^{);
Mask:= 1; DoComma:= false;
for M:= 0 to Size-1 do
[if Mask & N then
[if DoComma then ChOut(0, ^,);
IntOut(0, Set(M));
DoComma:= true;
];
Mask:= Mask << 1;
];
ChOut(0, ^});
];
Text(0, "}^m^j");
];
[PowSet([2, 3, 5, 7], 4);
PowSet([1], 1);
PowSet(0, 0);
]- Output:
{{},{2},{3},{2,3},{5},{2,5},{3,5},{2,3,5},{7},{2,7},{3,7},{2,3,7},{5,7},{2,5,7},{3,5,7},{2,3,5,7}}
{{},{1}}
{{}}
zkl
Using a combinations function, build the power set from combinations of 1,2,... items.
fcn pwerSet(list){
(0).pump(list.len(),List, Utils.Helpers.pickNFrom.fp1(list),
T(Void.Write,Void.Write) ) .append(list)
}foreach n in (5){
ps:=pwerSet((1).pump(n,List)); ps.println(" Size = ",ps.len());
}- Output:
L(L()) Size = 1 L(L(),L(1)) Size = 2 L(L(),L(1),L(2),L(1,2)) Size = 4 L(L(),L(1),L(2),L(3),L(1,2),L(1,3),L(2,3),L(1,2,3)) Size = 8 L(L(),L(1),L(2),L(3),L(4),L(1,2),L(1,3),L(1,4),L(2,3),L(2,4), L(3,4),L(1,2,3),L(1,2,4),L(1,3,4),L(2,3,4),L(1,2,3,4)) Size = 16
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