Cartesian product of two or more lists
You are encouraged to solve this task according to the task description, using any language you may know.
- Task
Show one or more idiomatic ways of generating the Cartesian product of two arbitrary lists in your language.
Demonstrate that your function/method correctly returns:
- {1, 2} × {3, 4} = {(1, 3), (1, 4), (2, 3), (2, 4)}
and, in contrast:
- {3, 4} × {1, 2} = {(3, 1), (3, 2), (4, 1), (4, 2)}
Also demonstrate, using your function/method, that the product of an empty list with any other list is empty.
- {1, 2} × {} = {}
- {} × {1, 2} = {}
For extra credit, show or write a function returning the n-ary product of an arbitrary number of lists, each of arbitrary length. Your function might, for example, accept a single argument which is itself a list of lists, and return the n-ary product of those lists.
Use your n-ary Cartesian product function to show the following products:
- {1776, 1789} × {7, 12} × {4, 14, 23} × {0, 1}
- {1, 2, 3} × {30} × {500, 100}
- {1, 2, 3} × {} × {500, 100}
11l
F cart_prod(a, b)
V p = [(0, 0)] * (a.len * b.len)
V i = 0
L(aa) a
L(bb) b
p[i++] = (aa, bb)
R p
print(cart_prod([1, 2], [3, 4]))
print(cart_prod([3, 4], [1, 2]))
[Int] empty_array
print(cart_prod([1, 2], empty_array))
print(cart_prod(empty_array, [1, 2]))Alternative version
F cart_prod(a, b)
R multiloop(a, b, (aa, bb) -> (aa, bb))- Output:
[(1, 3), (1, 4), (2, 3), (2, 4)] [(3, 1), (3, 2), (4, 1), (4, 2)] [] []
Action!
DEFINE MAX_COUNT="10"
DEFINE MAX_RESULT="100"
DEFINE PTR="CARD"
PROC PrintInput(PTR ARRAY a INT count)
INT i,j,n
INT ARRAY tmp
FOR i=0 TO count-1
DO
tmp=a(i) n=tmp(0)
Put('[)
FOR j=1 TO n
DO
PrintI(tmp(j))
IF j<n THEN Put(',) FI
OD
Put('])
IF i<count-1 THEN Put('x) FI
OD
RETURN
PROC PrintOutput(INT ARRAY a INT groups,count)
INT i,j,k
Put('[)
k=0
FOR i=0 TO groups-1
DO
Put('()
FOR j=0 TO count-1
DO
PrintI(a(k)) k==+1
IF j<count-1 THEN Put(',) FI
OD
Put('))
IF i<groups-1 THEN Put(',) FI
OD
Put('])
RETURN
PROC Product(PTR ARRAY a INT count
INT ARRAY r INT POINTER groups)
INT ARRAY ind(MAX_COUNT),tmp
INT i,j,k
IF count>MAX_COUNT THEN Break() FI
groups^=1
FOR i=0 TO count-1
DO
ind(i)=1 tmp=a(i)
groups^==*tmp(0)
OD
IF groups^=0 THEN RETURN FI
j=count-1 k=0
DO
FOR i=0 TO count-1
DO
tmp=a(i)
r(k)=tmp(ind(i)) k==+1
OD
DO
tmp=a(j)
IF ind(j)<tmp(0) THEN
ind(j)==+1
FOR i=j+1 TO count-1
DO
ind(i)=1
OD
j=count-1
EXIT
ELSE
IF j=0 THEN RETURN FI
j==-1
FI
OD
OD
RETURN
PROC Test(PTR ARRAY a INT count)
INT ARRAY r(MAX_RESULT)
INT groups
IF count<2 THEN Break() FI
Product(a,count,r,@groups)
PrintInput(a,count)
Put('=)
PrintOutput(r,groups,count)
PutE()
RETURN
PROC Main()
INT ARRAY
a1=[2 1 2],a2=[2 3 4],a3=[0],
a4=[2 1776 1789],a5=[2 7 12],
a6=[3 4 14 23],a7=[2 0 1],
a8=[3 1 2 3],a9=[1 30],a10=[2 500 100]
PTR ARRAY a(4)
a(0)=a1 a(1)=a2 Test(a,2)
a(0)=a2 a(1)=a1 Test(a,2)
a(0)=a1 a(1)=a3 Test(a,2)
a(0)=a3 a(1)=a1 Test(a,2) PutE()
a(0)=a4 a(1)=a5 a(2)=a6 a(3)=a7 Test(a,4) PutE()
a(0)=a8 a(1)=a9 a(2)=a10 Test(a,3) PutE()
a(0)=a8 a(1)=a3 a(2)=a10 Test(a,3)
RETURN- Output:
Screenshot from Atari 8-bit computer
[1,2]x[3,4]=[(1,3),(1,4),(2,3),(2,4)] [3,4]x[1,2]=[(3,1),(3,2),(4,1),(4,2)] [1,2]x[]=[] []x[1,2]=[] [1776,1789]x[7,12]x[4,14,23]x[0,1]=[(1776,7,4,0),(1776,7,4,1),(1776,7,14,0),(1776,7,14,1),(1776,7,23,0),(1776,7,23,1),(1776,12,4,0),1776,12,4,1),(1776,12,14,0),(1776,12,14,1),(1776,12,23,0),(1776,12,23,1),(1789,7,4,0),(1789,7,4,1),(1789,7,14,0),(1789,7,14,1),(1789,7,23,0),(1789,7,23,1),(1789,12,4,0),(1789,12,4,1),(1789,12,14,0),(1789,12,14,1),(1789,12,23,0),(1789,12,23,1)] [1,2,3]x[30]x[500,100]=[(1,30,500),(1,30,100),(2,30,500),(2,30,100),(3,30,500),(3,30,100)] [1,2,3]x[]x[500,100]=[]
Ada
with Ada.Text_IO; use Ada.Text_Io;
with Ada.Containers.Doubly_Linked_Lists;
with Ada.Strings.Fixed;
procedure Cartesian is
type Element_Type is new Long_Integer;
package Lists is
new Ada.Containers.Doubly_Linked_Lists (Element_Type);
package List_Lists is
new Ada.Containers.Doubly_Linked_Lists (Lists.List, Lists."=");
subtype List is Lists.List;
subtype List_List is List_Lists.List;
function "*" (Left, Right : List) return List_List is
Result : List_List;
Sub : List;
begin
for Outer of Left loop
for Inner of Right loop
Sub.Clear;
Sub.Append (Outer);
Sub.Append (Inner);
Result.Append (Sub);
end loop;
end loop;
return Result;
end "*";
function "*" (Left : List_List;
Right : List) return List_List
is
Result : List_List;
Sub : List;
begin
for Outer of Left loop
for Inner of Right loop
Sub := Outer;
Sub.Append (Inner);
Result.Append (Sub);
end loop;
end loop;
return Result;
end "*";
procedure Put (L : List) is
use Ada.Strings;
First : Boolean := True;
begin
Put ("(");
for E of L loop
if not First then
Put (",");
end if;
Put (Fixed.Trim (E'Image, Left));
First := False;
end loop;
Put (")");
end Put;
procedure Put (LL : List_List) is
First : Boolean := True;
begin
Put ("{");
for E of LL loop
if not First then
Put (",");
end if;
Put (E);
First := False;
end loop;
Put ("}");
end Put;
function "&" (Left : List; Right : Element_Type) return List is
Result : List := Left;
begin
Result.Append (Right);
return Result;
end "&";
Nil : List renames Lists.Empty_List;
List_1_2 : constant List := Nil & 1 & 2;
List_3_4 : constant List := Nil & 3 & 4;
List_Empty : constant List := Nil;
List_1_2_3 : constant List := Nil & 1 & 2 & 3;
begin
Put (List_1_2 * List_3_4); New_Line;
Put (List_3_4 * List_1_2); New_Line;
Put (List_Empty * List_1_2); New_Line;
Put (List_1_2 * List_Empty); New_Line;
Put (List'(Nil & 1776 & 1789) * List'(Nil & 7 & 12) *
List'(Nil & 4 & 14 & 23) * List'(Nil & 0 & 1)); New_Line;
Put (List_1_2_3 * List'(Nil & 30) * List'(Nil & 500 & 100)); New_Line;
Put (List_1_2_3 * List_Empty * List'(Nil & 500 & 100)); New_Line;
end Cartesian;
- Output:
{(1,3),(1,4),(2,3),(2,4)}
{(3,1),(3,2),(4,1),(4,2)}
{}
{}
{(1776,7,4,0),(1776,7,4,1),(1776,7,14,0),(1776,7,14,1),(1776,7,23,0),(1776,7,23,1),(1776,12,4,0),(1776,12,4,1),(1776,12,14,0),(1776,12,14,1),(1776,12,23,0),(1776,12,23,1),(1789,7,4,0),(1789,7,4,1),(1789,7,14,0),(1789,7,14,1),(1789,7,23,0),(1789,7,23,1),(1789,12,4,0),(1789,12,4,1),(1789,12,14,0),(1789,12,14,1),(1789,12,23,0),(1789,12,23,1)}
{(1,30,500),(1,30,100),(2,30,500),(2,30,100),(3,30,500),(3,30,100)}
{}
ALGOL 68
Using a 1-dimensional array of INT to represent a list and a 2-dimensional array ( [,]INT ) to represent a product of two (or more) lists.
A list of lists is represented by a 1-dimensional array of 1-dimensional arrays of INT ([][]INT).
BEGIN # Cartesian Product #
# Cartesian product operators #
PRIO X = 7; # give X he same priority as * #
# returns the Cartesian product of the lists a and b #
OP X = ( []INT a, b )[,]INT:
BEGIN
[]INT a1 = a[ AT 1 ];
[]INT b1 = b[ AT 1 ];
INT len = UPB a1 * UPB b1;
[ 1 : len, 1 : IF len > 0 THEN 2 ELSE 0 FI ]INT result;
INT pos := 0;
FOR i TO UPB a1 DO
FOR j TO UPB b1 DO
pos +:= 1;
result[ pos, 1 ] := a1[ i ];
result[ pos, 2 ] := b1[ j ]
OD
OD;
result
END # X # ;
# returns the Cartesian product of the Cartesian product a and list b #
OP X = ( [,]INT a, []INT b )[,]INT:
BEGIN
[,]INT a1 = a[ AT 1, AT 1 ];
[]INT b1 = b[ AT 1 ];
INT len = 1 UPB a1 * UPB b1;
INT width = IF len <= 0 THEN 0 ELSE 2 UPB a1 + 1 FI;
[ 1 : len, 1 : width ]INT result;
INT pos := 0;
FOR i TO 1 UPB a1 DO
FOR j TO UPB b1 DO
result[ pos +:= 1, 1 : width - 1 ] := a1[ i, : ];
result[ pos, width ] := b1[ j ]
OD
OD;
result
END # X # ;
# returns the Cartesian product of the lists in a #
OP X = ( [][]INT a )[,]INT:
IF UPB a <= LWB a
THEN # zero or 1 list #
[,]INT()
ELSE # 2 or more lists #
FLEX[ 1 : 0, 1 : 0 ]INT result := a[ LWB a ] X a[ LWB a + 1 ];
FOR i FROM LWB a + 2 TO UPB a DO
result := result X a[ i ]
OD;
result
FI # X # ;
# print a Cartesian product #
PROC print product = ( [,]INT p )VOID:
BEGIN
print( ( "[" ) );
STRING close text := "]";
STRING open text := "(";
FOR i FROM 1 LWB p TO 1 UPB p DO
STRING separator := open text;
FOR j FROM 2 LWB p TO 2 UPB p DO
print( ( separator, whole( p[ i, j ], 0 ) ) );
separator := ","
OD;
open text := "),(";
close text := ")]"
OD;
print( ( close text ) )
END # print product # ;
# print a list #
PROC print list = ( []INT t )VOID:
BEGIN
print( ( "[" ) );
STRING separator := "";
FOR i FROM LWB t TO UPB t DO
print( ( separator, whole( t[ i ], 0 ) ) );
separator := ","
OD;
print( ( "]" ) )
END # print list # ;
BEGIN # test the X operators #
# prints the product of two lists #
PROC print lxl = ( []INT a, b )VOID:
BEGIN
print list( a );print( ( "X" ) );print list( b );
print( ( "=" ) );print product( a X b );
print( ( newline ) )
END # print lxl # ;
# prints the product of a list of lists #
PROC print xll = ( [][]INT a )VOID:
IF LWB a < UPB a THEN
# non empty list of lists #
print list( a[ LWB a ] );
FOR i FROM LWB a + 1 TO UPB a DO
print( ( "X" ) );print list( a[ i ] )
OD;
print( ( "=" ) );print product( X a );
print( ( newline ) )
FI # print xll # ;
print lxl( ( 1, 2 ), ( 3, 4 ) );
print lxl( ( 3, 4 ), ( 1, 2 ) );
print lxl( ( 1, 2 ), () );
print lxl( (), ( 1, 2 ) );
print xll( ( ( 1776, 1789 ), ( 7, 12 ), ( 4, 14, 23 ), ( 0, 1 ) ) );
print xll( ( ( 1, 2, 3 ), ( 30 ), ( 500, 100 ) ) );
print xll( ( ( 1, 2, 3 ), (), ( 500, 100 ) ) )
END
END- Output:
[1,2]X[3,4]=[(1,3),(1,4),(2,3),(2,4)] [3,4]X[1,2]=[(3,1),(3,2),(4,1),(4,2)] [1,2]X[]=[] []X[1,2]=[] [1776,1789]X[7,12]X[4,14,23]X[0,1]=[(1776,7,4,0),(1776,7,4,1),(1776,7,14,0),(1776,7,14,1),(1776,7,23,0),(1776,7,23,1),(1776,12,4,0),(1776,12,4,1),(1776,12,14,0),(1776,12,14,1),(1776,12,23,0),(1776,12,23,1),(1789,7,4,0),(1789,7,4,1),(1789,7,14,0),(1789,7,14,1),(1789,7,23,0),(1789,7,23,1),(1789,12,4,0),(1789,12,4,1),(1789,12,14,0),(1789,12,14,1),(1789,12,23,0),(1789,12,23,1)] [1,2,3]X[30]X[500,100]=[(1,30,500),(1,30,100),(2,30,500),(2,30,100),(3,30,500),(3,30,100)] [1,2,3]X[]X[500,100]=[]
APL
APL has a built-in outer product operator: X ∘.F Y will get you an ⍴X-by-⍴Y
matrix containing every corresponding value of x F y for all x∊X, y∊Y.
The Cartesian product can therefore be expressed as ∘.,, but as that would return
a matrix, and the task is asking for a list, you also need to ravel the result.
cart ← ,∘.,
- Output:
1 2 cart 3 4
1 3 1 4 2 3 2 4
3 4 cart 1 2
3 1 3 2 4 1 4 2
1 2 cart ⍬ ⍝ empty output
⍬ cart 1 2 ⍝ empty output again
This can be reduced over a list of lists to generate the Cartesian product of an arbitrary list of lists.
nary_cart ← ⊃(,∘.,)/
- Output:
The items are listed on separate lines (using ↑) for clarity.
↑nary_cart (1776 1789)(7 12)(4 14 23)(0 1)
1776 7 4 0
1776 7 4 1
1776 7 14 0
1776 7 14 1
1776 7 23 0
1776 7 23 1
1776 12 4 0
1776 12 4 1
1776 12 14 0
1776 12 14 1
1776 12 23 0
1776 12 23 1
1789 7 4 0
1789 7 4 1
1789 7 14 0
1789 7 14 1
1789 7 23 0
1789 7 23 1
1789 12 4 0
1789 12 4 1
1789 12 14 0
1789 12 14 1
1789 12 23 0
1789 12 23 1
↑nary_cart(1 2 3)(,30)(50 100)
1 30 50
1 30 100
2 30 50
2 30 100
3 30 50
3 30 100
↑nary_cart(1 2 3)(⍬)(50 100) ⍝ empty output
AppleScript
-- CARTESIAN PRODUCTS ---------------------------------------------------------
-- Two lists:
-- cartProd :: [a] -> [b] -> [(a, b)]
on cartProd(xs, ys)
script
on |λ|(x)
script
on |λ|(y)
[[x, y]]
end |λ|
end script
concatMap(result, ys)
end |λ|
end script
concatMap(result, xs)
end cartProd
-- N-ary – a function over a list of lists:
-- cartProdNary :: [[a]] -> [[a]]
on cartProdNary(xss)
script
on |λ|(accs, xs)
script
on |λ|(x)
script
on |λ|(a)
{x & a}
end |λ|
end script
concatMap(result, accs)
end |λ|
end script
concatMap(result, xs)
end |λ|
end script
foldr(result, {{}}, xss)
end cartProdNary
-- TESTS ----------------------------------------------------------------------
on run
set baseExamples to unlines(map(show, ¬
[cartProd({1, 2}, {3, 4}), ¬
cartProd({3, 4}, {1, 2}), ¬
cartProd({1, 2}, {}), ¬
cartProd({}, {1, 2})]))
set naryA to unlines(map(show, ¬
cartProdNary([{1776, 1789}, {7, 12}, {4, 14, 23}, {0, 1}])))
set naryB to show(cartProdNary([{1, 2, 3}, {30}, {500, 100}]))
set naryC to show(cartProdNary([{1, 2, 3}, {}, {500, 100}]))
intercalate(linefeed & linefeed, {baseExamples, naryA, naryB, naryC})
end run
-- GENERIC FUNCTIONS ----------------------------------------------------------
-- concatMap :: (a -> [b]) -> [a] -> [b]
on concatMap(f, xs)
set lst to {}
set lng to length of xs
tell mReturn(f)
repeat with i from 1 to lng
set lst to (lst & |λ|(item i of xs, i, xs))
end repeat
end tell
return lst
end concatMap
-- 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
-- intercalate :: Text -> [Text] -> Text
on intercalate(strText, lstText)
set {dlm, my text item delimiters} to {my text item delimiters, strText}
set strJoined to lstText as text
set my text item delimiters to dlm
return strJoined
end intercalate
-- 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
-- show :: a -> String
on show(e)
set c to class of e
if c = list then
script serialized
on |λ|(v)
show(v)
end |λ|
end script
"[" & intercalate(", ", map(serialized, e)) & "]"
else if c = record then
script showField
on |λ|(kv)
set {k, ev} to kv
"\"" & k & "\":" & show(ev)
end |λ|
end script
"{" & intercalate(", ", ¬
map(showField, zip(allKeys(e), allValues(e)))) & "}"
else if c = date then
"\"" & iso8601Z(e) & "\""
else if c = text then
"\"" & e & "\""
else if (c = integer or c = real) then
e as text
else if c = class then
"null"
else
try
e as text
on error
("«" & c as text) & "»"
end try
end if
end show
-- unlines :: [String] -> String
on unlines(xs)
intercalate(linefeed, xs)
end unlines
- Output:
[[1, 3], [1, 4], [2, 3], [2, 4]] [[3, 1], [3, 2], [4, 1], [4, 2]] [] [] [1776, 7, 4, 0] [1776, 7, 4, 1] [1776, 7, 14, 0] [1776, 7, 14, 1] [1776, 7, 23, 0] [1776, 7, 23, 1] [1776, 12, 4, 0] [1776, 12, 4, 1] [1776, 12, 14, 0] [1776, 12, 14, 1] [1776, 12, 23, 0] [1776, 12, 23, 1] [1789, 7, 4, 0] [1789, 7, 4, 1] [1789, 7, 14, 0] [1789, 7, 14, 1] [1789, 7, 23, 0] [1789, 7, 23, 1] [1789, 12, 4, 0] [1789, 12, 4, 1] [1789, 12, 14, 0] [1789, 12, 14, 1] [1789, 12, 23, 0] [1789, 12, 23, 1] [[1, 30, 500], [1, 30, 100], [2, 30, 500], [2, 30, 100], [3, 30, 500], [3, 30, 100]] []
Arturo
loop [
[[1 2][3 4]]
[[3 4][1 2]]
[[1 2][]]
[[][1 2]]
[[1776 1789][7 12][4 14 23][0 1]]
[[1 2 3][30][500 100]]
[[1 2 3][][500 100]]
] 'lst [
print as.code product.cartesian lst
]
- Output:
[[1 3] [1 4] [2 3] [2 4]] [[3 1] [3 2] [4 1] [4 2]] [] [] [[1776 7 4 0] [1776 7 4 1] [1776 7 14 0] [1776 7 14 1] [1776 7 23 0] [1776 7 23 1] [1776 12 4 0] [1776 12 4 1] [1776 12 14 0] [1776 12 14 1] [1776 12 23 0] [1776 12 23 1] [1789 7 4 0] [1789 7 4 1] [1789 7 14 0] [1789 7 14 1] [1789 7 23 0] [1789 7 23 1] [1789 12 4 0] [1789 12 4 1] [1789 12 14 0] [1789 12 14 1] [1789 12 23 0] [1789 12 23 1]] [[1 30 500] [1 30 100] [2 30 500] [2 30 100] [3 30 500] [3 30 100]] []
AutoHotkey
example := [
(join,
[[1, 2], [3, 4]]
[[3, 4], [1, 2]]
[[1, 2], []]
[[], [1, 2]]
[[1776, 1789], [7, 12], [4, 14, 23], [0, 1]]
[[1, 2, 3], [30] , [500, 100]]
[[1, 2, 3], [] , [500, 100]]
)]
for i, obj in example
{
Product := CartesianProduct(obj)
out := dispRes(Product)
result .= out "`n`n"
}
MsgBox % result
return
dispRes(Product){
for i, o in Product
{
for j, v in o
output .= v ", "
output := Trim(output, ", ")
output .= "], ["
}
return "[[" trim(output, ", []") "]]"
}
CartesianProduct(obj){
CP(obj, Product:=[], [])
return Product
}
CP(obj, Product, stack, v:=""){
oClone := obj.clone()
oClone.RemoveAt(1)
stack.= v ","
for i, o in obj
{
for j, v in o
CP(oClone, Product, stack, v)
return
}
stack := trim(stack, ",")
oTemp := []
for i, v in StrSplit(stack, ",")
oTemp.Push(v)
Product.push(oTemp)
}
- Output:
[[1, 3], [1, 4], [2, 3], [2, 4]] [[3, 1], [3, 2], [4, 1], [4, 2]] [] [] [[1776, 7, 4, 0], [1776, 7, 4, 1], [1776, 7, 14, 0], [1776, 7, 14, 1], [1776, 7, 23, 0], [1776, 7, 23, 1], [1776, 12, 4, 0], [1776, 12, 4, 1], [1776, 12, 14, 0], [1776, 12, 14, 1], [1776, 12, 23, 0], [1776, 12, 23, 1], [1789, 7, 4, 0], [1789, 7, 4, 1], [1789, 7, 14, 0], [1789, 7, 14, 1], [1789, 7, 23, 0], [1789, 7, 23, 1], [1789, 12, 4, 0], [1789, 12, 4, 1], [1789, 12, 14, 0], [1789, 12, 14, 1], [1789, 12, 23, 0], [1789, 12, 23, 1]] [[1, 30, 500], [1, 30, 100], [2, 30, 500], [2, 30, 100], [3, 30, 500], [3, 30, 100]] []
Bracmat
( ( mul
= R a b A B
. :?R
& !arg:(.?A) (.?B)
& ( !A
: ?
( %@?a
& !B
: ?
( (%@?b|(.?b))
& !R (.!a !b):?R
& ~
)
?
)
?
| (.!R)
)
)
& ( cartprod
= a
. !arg:%?a %?arg&mul$(!a cartprod$!arg)
| !arg
)
& out
$ ( cartprod
$ ( (.1776 1789)
(.7 12)
(.4 14 23)
(.0 1)
)
)
& out$(cartprod$((.1 2 3) (.30) (.500 100)))
& out$(cartprod$((.1 2 3) (.) (.500 100)))
). (.1776 7 4 0)
(.1776 7 4 1)
(.1776 7 14 0)
(.1776 7 14 1)
(.1776 7 23 0)
(.1776 7 23 1)
(.1776 12 4 0)
(.1776 12 4 1)
(.1776 12 14 0)
(.1776 12 14 1)
(.1776 12 23 0)
(.1776 12 23 1)
(.1789 7 4 0)
(.1789 7 4 1)
(.1789 7 14 0)
(.1789 7 14 1)
(.1789 7 23 0)
(.1789 7 23 1)
(.1789 12 4 0)
(.1789 12 4 1)
(.1789 12 14 0)
(.1789 12 14 1)
(.1789 12 23 0)
(.1789 12 23 1)
. (.1 30 500)
(.1 30 100)
(.2 30 500)
(.2 30 100)
(.3 30 500)
(.3 30 100)
.
BASIC
Applesoft BASIC
100 HOME : rem 10 CLS FOR Chipmunk Basic & GW-BASIC
110 DIM array(2,2)
120 array(1,1) = 1 : array(1,2) = 2
130 array(2,1) = 3 : array(2,2) = 4
140 GOSUB 190
150 array(1,1) = 3 : array(1,2) = 4
160 array(2,1) = 1 : array(2,2) = 2
170 GOSUB 190
180 END
190 rem SUB cartesian(list)
200 u1 = 2 : u2 = 2
210 FOR i = 1 TO u1
220 PRINT "{ ";
230 FOR j = 1 TO u2
240 PRINT array(i,j);
250 IF j < u1 THEN PRINT ", ";
260 NEXT j
270 PRINT "}";
280 IF i < u2 THEN PRINT " x ";
290 NEXT i
300 PRINT " = { ";
310 FOR i = 1 TO u1
320 FOR j = 1 TO u2
330 PRINT "{ "; array(1,i); ", "; array(2,j); "} ";
340 IF i < u2 THEN PRINT ", ";
350 IF i => u2 THEN IF j < u1 THEN PRINT ", ";
360 NEXT j
370 NEXT i
380 PRINT "}"
390 RETURN
BASIC256
arraybase 1
subroutine cartesian(list)
u1 = list[?][]
u2 = list[][?]
for i = 1 to u1
print "{";
for j = 1 to u2
print list[i,j];
if j < u1 then print ", ";
next
print "}";
if i < u2 then print " x ";
next i
print " = { ";
for i = 1 to u1
for j = 1 to u2
print "{"; list[1, i]; ", "; list[2, j]; "} ";
if i < u2 then
print ", ";
else
if j < u1 then print ", ";
end if
next j
next i
print "}"
end subroutine
dim list1 = {{1,2},{3,4}}
dim list2 = {{3,4},{1,2}}
call cartesian(list1)
call cartesian(list2)
end
- Output:
{1, 2} x {3, 4} = { {1, 3} , {1, 4} , {2, 3} , {2, 4} }
{3, 4} x {1, 2} = { {3, 1} , {3, 2} , {4, 1} , {4, 2} }
Chipmunk Basic
100 cls
110 dim array(2,2)
120 array(1,1) = 1 : array(1,2) = 2
130 array(2,1) = 3 : array(2,2) = 4
140 gosub 190
150 array(1,1) = 3 : array(1,2) = 4
160 array(2,1) = 1 : array(2,2) = 2
170 gosub 190
180 end
190 rem sub cartesian(list)
200 u1 = 2 : u2 = 2
210 for i = 1 to u1
220 print "{ ";
230 for j = 1 to u2
240 print array(i,j);
250 if j < u1 then print ", ";
260 next j
270 print "}";
280 if i < u2 then print " x ";
290 next i
300 print " = { ";
310 for i = 1 to u1
320 for j = 1 to u2
330 print "{ ";array(1,i);", ";array(2,j);"} ";
340 if i < u2 then
350 print ", ";
360 else
370 if j < u1 then print ", ";
380 endif
390 next j
400 next i
410 print "}"
420 return
Gambas
Public array[2, 2] As Integer
Public Sub Main()
array[0, 0] = 1
array[0, 1] = 2
array[1, 0] = 3
array[1, 1] = 4
cartesian(array)
array[0, 0] = 3
array[0, 1] = 4
array[1, 0] = 1
array[1, 1] = 2
cartesian(array)
End
Sub cartesian(arr As Integer[])
Dim u1 As Integer = arr.Max - 2
Dim u2 As Integer = arr.Max - 2
Dim i As Integer, j As Integer
For i = 0 To u1
Print "{";
For j = 0 To u2
Print arr[i, j];
If j < u1 Then Print ",";
Next
Print "}";
If i < u2 Then Print " x ";
Next
Print " = {";
For i = 0 To u1
For j = 0 To u2
Print "{"; arr[0, i]; ","; arr[1, j]; "}";
If i < u2 Then
Print ", ";
Else
If j < u1 Then Print ", ";
End If
Next
Next
Print "}"
End Sub
GW-BASIC
100 CLS
110 DIM ARR(2,2)
120 ARR(1,1) = (1) : ARR(1,2) = (2)
130 ARR(2,1) = (3) : ARR(2,2) = (4)
140 GOSUB 190
150 ARR(1,1) = 3 : ARR(1,2) = 4
160 ARR(2,1) = 1 : ARR(2,2) = 2
170 GOSUB 190
180 END
190 REM SUB cartesian(list)
200 U1 = 2 : U2 = 2
210 FOR I = 1 TO U1
220 PRINT "{";
230 FOR J = 1 TO U2
240 PRINT ARR(I,J);
250 IF J < U1 THEN PRINT ",";
260 NEXT J
270 PRINT "}";
280 IF I < U2 THEN PRINT " x ";
290 NEXT I
300 PRINT " = {";
310 FOR I = 1 TO U1
320 FOR J = 1 TO U2
330 PRINT "{"; ARR(1,I); ","; ARR(2,J); "}";
340 IF I < U2 THEN PRINT ", ";
350 IF I => U2 THEN IF J < U1 THEN PRINT ",";
360 NEXT J
370 NEXT I
380 PRINT "}"
390 RETURN
MSX Basic
The GW-BASIC solution works without any changes.
QBasic
DECLARE SUB cartesian (arr!())
CLS
DIM array(2, 2)
array(1, 1) = 1: array(1, 2) = 2
array(2, 1) = 3: array(2, 2) = 4
CALL cartesian(array())
array(1, 1) = 3: array(1, 2) = 4
array(2, 1) = 1: array(2, 2) = 2
CALL cartesian(array())
END
SUB cartesian (arr())
u1 = 2: u2 = 2
FOR i = 1 TO u1
PRINT "{";
FOR j = 1 TO u2
PRINT arr(i, j);
IF j < u1 THEN PRINT ",";
NEXT j
PRINT "}";
IF i < u2 THEN PRINT " x ";
NEXT i
PRINT " = {";
FOR i = 1 TO u1
FOR j = 1 TO u2
PRINT "{"; arr(1, i); ","; arr(2, j); "}";
IF i < u2 THEN
PRINT ", ";
ELSE
IF j < u1 THEN PRINT ", ";
END IF
NEXT j
NEXT i
PRINT "}"
END SUB
Run BASIC
cls
dim array(2,2)
array(1,1) = 1 : array(1,2) = 2
array(2,1) = 3 : array(2,2) = 4
gosub [cartesian]
array(1,1) = 3 : array(1,2) = 4
array(2,1) = 1 : array(2,2) = 2
gosub [cartesian]
end
[cartesian]
u1 = 2 : u2 = 2
for i = 1 to u1
print "{";
for j = 1 to u2
print array(i,j);
if j < u1 then print ",";
next j
print "}";
if i < u2 then print " x ";
next i
print " = {";
for i = 1 to u1
for j = 1 to u2
print "{"; array(1,i); ","; array(2,j); "}";
if i < u2 then
print ",";
else
if j < u1 then print ",";
end if
next j
next i
print "}"
return
BQN
Cp ← ⥊∾⌜
⍉≍⟨
⟨1, 2⟩ Cp ⟨3, 4⟩
⟨3, 4⟩ Cp ⟨1, 2⟩
⟨1, 2⟩ Cp ⟨⟩
⟨⟩ Cp ⟨1, 2⟩
∘‿3⥊ Cp´ ⟨1776, 1789⟩‿⟨7, 12⟩‿⟨4, 14, 23⟩‿⟨0, 1⟩
∘‿3⥊ Cp´ ⟨1, 2, 3⟩‿⟨30⟩‿⟨500, 100⟩
Cp´ ⟨1, 2, 3⟩‿⟨⟩‿⟨500, 100⟩
⟩
- Output:
┌─
╵ ⟨ ⟨ 1 3 ⟩ ⟨ 1 4 ⟩ ⟨ 2 3 ⟩ ⟨ 2 4 ⟩ ⟩
⟨ ⟨ 3 1 ⟩ ⟨ 3 2 ⟩ ⟨ 4 1 ⟩ ⟨ 4 2 ⟩ ⟩
⟨⟩
⟨⟩
┌─
╵ ⟨ 1776 7 4 0 ⟩ ⟨ 1776 7 4 1 ⟩ ⟨ 1776 7 14 0 ⟩
⟨ 1776 7 14 1 ⟩ ⟨ 1776 7 23 0 ⟩ ⟨ 1776 7 23 1 ⟩
⟨ 1776 12 4 0 ⟩ ⟨ 1776 12 4 1 ⟩ ⟨ 1776 12 14 0 ⟩
⟨ 1776 12 14 1 ⟩ ⟨ 1776 12 23 0 ⟩ ⟨ 1776 12 23 1 ⟩
⟨ 1789 7 4 0 ⟩ ⟨ 1789 7 4 1 ⟩ ⟨ 1789 7 14 0 ⟩
⟨ 1789 7 14 1 ⟩ ⟨ 1789 7 23 0 ⟩ ⟨ 1789 7 23 1 ⟩
⟨ 1789 12 4 0 ⟩ ⟨ 1789 12 4 1 ⟩ ⟨ 1789 12 14 0 ⟩
⟨ 1789 12 14 1 ⟩ ⟨ 1789 12 23 0 ⟩ ⟨ 1789 12 23 1 ⟩
┘
┌─
╵ ⟨ 1 30 500 ⟩ ⟨ 1 30 100 ⟩ ⟨ 2 30 500 ⟩
⟨ 2 30 100 ⟩ ⟨ 3 30 500 ⟩ ⟨ 3 30 100 ⟩
┘
⟨⟩
┘
C
Recursive implementation for computing the Cartesian product of lists. In the pursuit of making it as interactive as possible, the parsing function ended up taking the most space. The product set expression must be supplied enclosed by double quotes. Prints out usage on incorrect invocation.
#include<string.h>
#include<stdlib.h>
#include<stdio.h>
void cartesianProduct(int** sets, int* setLengths, int* currentSet, int numSets, int times){
int i,j;
if(times==numSets){
printf("(");
for(i=0;i<times;i++){
printf("%d,",currentSet[i]);
}
printf("\b),");
}
else{
for(j=0;j<setLengths[times];j++){
currentSet[times] = sets[times][j];
cartesianProduct(sets,setLengths,currentSet,numSets,times+1);
}
}
}
void printSets(int** sets, int* setLengths, int numSets){
int i,j;
printf("\nNumber of sets : %d",numSets);
for(i=0;i<numSets+1;i++){
printf("\nSet %d : ",i+1);
for(j=0;j<setLengths[i];j++){
printf(" %d ",sets[i][j]);
}
}
}
void processInputString(char* str){
int **sets, *currentSet, *setLengths, setLength, numSets = 0, i,j,k,l,start,counter=0;
char *token,*holder,*holderToken;
for(i=0;str[i]!=00;i++)
if(str[i]=='x')
numSets++;
if(numSets==0){
printf("\n%s",str);
return;
}
currentSet = (int*)calloc(sizeof(int),numSets + 1);
setLengths = (int*)calloc(sizeof(int),numSets + 1);
sets = (int**)malloc((numSets + 1)*sizeof(int*));
token = strtok(str,"x");
while(token!=NULL){
holder = (char*)malloc(strlen(token)*sizeof(char));
j = 0;
for(i=0;token[i]!=00;i++){
if(token[i]>='0' && token[i]<='9')
holder[j++] = token[i];
else if(token[i]==',')
holder[j++] = ' ';
}
holder[j] = 00;
setLength = 0;
for(i=0;holder[i]!=00;i++)
if(holder[i]==' ')
setLength++;
if(setLength==0 && strlen(holder)==0){
printf("\n{}");
return;
}
setLengths[counter] = setLength+1;
sets[counter] = (int*)malloc((1+setLength)*sizeof(int));
k = 0;
start = 0;
for(l=0;holder[l]!=00;l++){
if(holder[l+1]==' '||holder[l+1]==00){
holderToken = (char*)malloc((l+1-start)*sizeof(char));
strncpy(holderToken,holder + start,l+1-start);
sets[counter][k++] = atoi(holderToken);
start = l+2;
}
}
counter++;
token = strtok(NULL,"x");
}
printf("\n{");
cartesianProduct(sets,setLengths,currentSet,numSets + 1,0);
printf("\b}");
}
int main(int argC,char* argV[])
{
if(argC!=2)
printf("Usage : %s <Set product expression enclosed in double quotes>",argV[0]);
else
processInputString(argV[1]);
return 0;
}
Invocation and output :
C:\My Projects\threeJS>cartesianProduct.exe "{1,2} x {3,4}"
{(1,3),(1,4),(2,3),(2,4)}
C:\My Projects\threeJS>cartesianProduct.exe "{3,4} x {1,2}"
{(3,1),(3,2),(4,1),(4,2)}
C:\My Projects\threeJS>cartesianProduct.exe "{1,2} x {}"
{}
C:\My Projects\threeJS>cartesianProduct.exe "{} x {1,2}"
{}
C:\My Projects\threeJS>cartesianProduct.exe "{1776, 1789} x {7, 12} x {4, 14, 23} x {0, 1}"
{(1776,7,4,0),(1776,7,4,1),(1776,7,14,0),(1776,7,14,1),(1776,7,23,0),(1776,7,23,1),(1776,12,4,0),(1776,12,4,1),(1776,12,14,0),(1776,12,14,1),(1776,12,23,0),(1776,12,23,1),(1789,7,4,0),(1789,9,12,14,1),(1789,12,23,0),(1789,12,23,1)}
C:\My Projects\threeJS>cartesianProduct.exe "{1, 2, 3} x {30} x {500, 100}"
{(1,30,500),(1,30,100),(2,30,500),(2,30,100),(3,30,500),(3,30,100)}
C:\My Projects\threeJS>cartesianProduct.exe "{1, 2, 3} x {} x {500, 100}"
{}
C#
using System;
public class Program
{
public static void Main()
{
int[] empty = new int[0];
int[] list1 = { 1, 2 };
int[] list2 = { 3, 4 };
int[] list3 = { 1776, 1789 };
int[] list4 = { 7, 12 };
int[] list5 = { 4, 14, 23 };
int[] list6 = { 0, 1 };
int[] list7 = { 1, 2, 3 };
int[] list8 = { 30 };
int[] list9 = { 500, 100 };
foreach (var sequenceList in new [] {
new [] { list1, list2 },
new [] { list2, list1 },
new [] { list1, empty },
new [] { empty, list1 },
new [] { list3, list4, list5, list6 },
new [] { list7, list8, list9 },
new [] { list7, empty, list9 }
}) {
var cart = sequenceList.CartesianProduct()
.Select(tuple => $"({string.Join(", ", tuple)})");
Console.WriteLine($"{{{string.Join(", ", cart)}}}");
}
}
}
public static class Extensions
{
public static IEnumerable<IEnumerable<T>> CartesianProduct<T>(this IEnumerable<IEnumerable<T>> sequences) {
IEnumerable<IEnumerable<T>> emptyProduct = new[] { Enumerable.Empty<T>() };
return sequences.Aggregate(
emptyProduct,
(accumulator, sequence) =>
from acc in accumulator
from item in sequence
select acc.Concat(new [] { item }));
}
}
- Output:
{(1, 3), (1, 4), (2, 3), (2, 4)}
{(3, 1), (3, 2), (4, 1), (4, 2)}
{}
{}
{(1776, 7, 4, 0), (1776, 7, 4, 1), (1776, 7, 14, 0), (1776, 7, 14, 1), (1776, 7, 23, 0), (1776, 7, 23, 1), (1776, 12, 4, 0), (1776, 12, 4, 1), (1776, 12, 14, 0), (1776, 12, 14, 1), (1776, 12, 23, 0), (1776, 12, 23, 1), (1789, 7, 4, 0), (1789, 7, 4, 1), (1789, 7, 14, 0), (1789, 7, 14, 1), (1789, 7, 23, 0), (1789, 7, 23, 1), (1789, 12, 4, 0), (1789, 12, 4, 1), (1789, 12, 14, 0), (1789, 12, 14, 1), (1789, 12, 23, 0), (1789, 12, 23, 1)}
{(1, 30, 500), (1, 30, 100), (2, 30, 500), (2, 30, 100), (3, 30, 500), (3, 30, 100)}
{}
If the number of lists is known, LINQ provides an easier solution:
public static void Main()
{
///...
var cart1 =
from a in list1
from b in list2
select (a, b); // C# 7.0 tuple
Console.WriteLine($"{{{string.Join(", ", cart1)}}}");
var cart2 =
from a in list7
from b in list8
from c in list9
select (a, b, c);
Console.WriteLine($"{{{string.Join(", ", cart2)}}}");
}
- Output:
{(1, 3), (1, 4), (2, 3), (2, 4)}
{(1, 30, 500), (1, 30, 100), (2, 30, 500), (2, 30, 100), (3, 30, 500), (3, 30, 100)}
C++
#include <iostream>
#include <vector>
#include <algorithm>
void print(const std::vector<std::vector<int>>& v) {
std::cout << "{ ";
for (const auto& p : v) {
std::cout << "(";
for (const auto& e : p) {
std::cout << e << " ";
}
std::cout << ") ";
}
std::cout << "}" << std::endl;
}
auto product(const std::vector<std::vector<int>>& lists) {
std::vector<std::vector<int>> result;
if (std::find_if(std::begin(lists), std::end(lists),
[](auto e) -> bool { return e.size() == 0; }) != std::end(lists)) {
return result;
}
for (auto& e : lists[0]) {
result.push_back({ e });
}
for (size_t i = 1; i < lists.size(); ++i) {
std::vector<std::vector<int>> temp;
for (auto& e : result) {
for (auto f : lists[i]) {
auto e_tmp = e;
e_tmp.push_back(f);
temp.push_back(e_tmp);
}
}
result = temp;
}
return result;
}
int main() {
std::vector<std::vector<int>> prods[] = {
{ { 1, 2 }, { 3, 4 } },
{ { 3, 4 }, { 1, 2} },
{ { 1, 2 }, { } },
{ { }, { 1, 2 } },
{ { 1776, 1789 }, { 7, 12 }, { 4, 14, 23 }, { 0, 1 } },
{ { 1, 2, 3 }, { 30 }, { 500, 100 } },
{ { 1, 2, 3 }, { }, { 500, 100 } }
};
for (const auto& p : prods) {
print(product(p));
}
std::cin.ignore();
std::cin.get();
return 0;
}
- Output:
{ (1 3) (1 4) (2 3) (2 4) }
{ (3 1) (3 2) (4 1) (4 2) }
{ }
{ }
{ (1776 7 4 0) (1776 7 4 1) (1776 7 14 0) (1776 7 14 1) (1776 7 23 0) (1776 7 23 1) (1776 12 4 0) (1776 12 4 1) (1776 12 14 0) (1776 12 14 1) (1776 12 23 0) (1776 12 23 1) (1789 7 4 0) (1789 7 4 1) (1789 7 14 0) (1789 7 14 1) (1789 7 23 0) (1789 7 23 1) (1789 12 4 0) (1789 12 4 1) (1789 12 14 0) (1789 12 14 1) (1789 12 23 0) (1789 12 23 1) }
{ (1 30 500) (1 30 100) (2 30 500) (2 30 100) (3 30 500) (3 30 100) }
{ }
Clojure
(ns clojure.examples.product
(:gen-class)
(:require [clojure.pprint :as pp]))
(defn cart [colls]
"Compute the cartesian product of list of lists"
(if (empty? colls)
'(())
(for [more (cart (rest colls))
x (first colls)]
(cons x more))))
Output
(doseq [lst [ [[1,2],[3,4]],
[[3,4],[1,2]], [[], [1, 2]],
[[1, 2], []],
[[1776, 1789], [7, 12], [4, 14, 23], [0, 1]],
[[1, 2, 3], [30,], [500, 100]],
[[1, 2, 3], [], [500, 100]]
]
]
(println lst "=>")
(pp/pprint (cart lst)))
[[1 2] [3 4]] => ((1 3) (2 3) (1 4) (2 4)) [[3 4] [1 2]] => ((3 1) (4 1) (3 2) (4 2)) [[] [1 2]] => () [[1 2] []] => () [[1776 1789] [7 12] [4 14 23] [0 1]] => ((1776 7 4 0) (1789 7 4 0) (1776 12 4 0) (1789 12 4 0) (1776 7 14 0) (1789 7 14 0) (1776 12 14 0) (1789 12 14 0) (1776 7 23 0) (1789 7 23 0) (1776 12 23 0) (1789 12 23 0) (1776 7 4 1) (1789 7 4 1) (1776 12 4 1) (1789 12 4 1) (1776 7 14 1) (1789 7 14 1) (1776 12 14 1) (1789 12 14 1) (1776 7 23 1) (1789 7 23 1) (1776 12 23 1) (1789 12 23 1)) [[1 2 3] [30] [500 100]] => ((1 30 500) (2 30 500) (3 30 500) (1 30 100) (2 30 100) (3 30 100)) [[1 2 3] [] [500 100]] => ()
Common Lisp
(defun cartesian-product (s1 s2)
"Compute the cartesian product of two sets represented as lists"
(loop for x in s1
nconc (loop for y in s2 collect (list x y))))
Output
CL-USER> (cartesian-product '(1 2) '(3 4))
((1 3) (1 4) (2 3) (2 4))
CL-USER> (cartesian-product '(3 4) '(1 2))
((3 1) (3 2) (4 1) (4 2))
CL-USER> (cartesian-product '(1 2) '())
NIL
CL-USER> (cartesian-product '() '(1 2))
NIL
Extra credit:
(defun n-cartesian-product (l)
"Compute the n-cartesian product of a list of sets (each of them represented as list).
Algorithm:
If there are no sets, then produce an empty set of tuples;
otherwise, for all the elements x of the first set, concatenate the sets obtained by
inserting x at the beginning of each tuple of the n-cartesian product of the remaining sets."
(if (null l)
(list nil)
(loop for x in (car l)
nconc (loop for y in (n-cartesian-product (cdr l))
collect (cons x y)))))
Output:
CL-USER> (n-cartesian-product '((1776 1789) (7 12) (4 14 23) (0 1)))
((1776 7 4 0) (1776 7 4 1) (1776 7 14 0) (1776 7 14 1) (1776 7 23 0) (1776 7 23 1) (1776 12 4 0) (1776 12 4 1) (1776 12 14 0) (1776 12 14 1) (1776 12 23 0) (1776 12 23 1) (1789 7 4 0) (1789 7 4 1) (1789 7 14 0) (1789 7 14 1) (1789 7 23 0) (1789 7 23 1) (1789 12 4 0) (1789 12 4 1) (1789 12 14 0) (1789 12 14 1) (1789 12 23 0) (1789 12 23 1))
CL-USER> (n-cartesian-product '((1 2 3) (30) (500 100)))
((1 30 500) (1 30 100) (2 30 500) (2 30 100) (3 30 500) (3 30 100))
CL-USER> (n-cartesian-product '((1 2 3) () (500 100)))
NIL
Crystal
The first function is the basic task. The version overloaded for one argument is the extra credit task, implemented using recursion.
def cartesian_product(a, b)
return a.flat_map { |i| b.map { |j| [i, j] } }
end
def cartesian_product(l)
if l.size <= 1
return l
elsif l.size == 2
return cartesian_product(l[0], l[1])
end
return l[0].flat_map { |i|
cartesian_product(l[1..]).map { |j|
[i, j].flatten
}
}
end
tests = [ [[1, 2], [3, 4]],
[[3, 4], [1, 2]],
[[1, 2], [] of Int32],
[[] of Int32, [1, 2]],
[[1, 2, 3], [30], [500, 100]],
[[1, 2, 3], [] of Int32, [500, 100]],
[[1776, 1789], [7, 12], [4, 14, 23], [0, 1]] ]
tests.each { |test|
puts "#{test.join(" x ")} ->"
puts " #{cartesian_product(test)}"
puts ""
}
- Output:
[1, 2] x [3, 4] ->
[[1, 3], [1, 4], [2, 3], [2, 4]]
[3, 4] x [1, 2] ->
[[3, 1], [3, 2], [4, 1], [4, 2]]
[1, 2] x [] ->
[]
[] x [1, 2] ->
[]
[1, 2, 3] x [30] x [500, 100] ->
[[1, 30, 500], [1, 30, 100], [2, 30, 500], [2, 30, 100], [3, 30, 500], [3, 30, 100]]
[1, 2, 3] x [] x [500, 100] ->
[]
[1776, 1789] x [7, 12] x [4, 14, 23] x [0, 1] ->
[[1776, 7, 4, 0], [1776, 7, 4, 1], [1776, 7, 14, 0], [1776, 7, 14, 1], [1776, 7, 23, 0], [1776, 7, 23, 1], [1776, 12, 4, 0], [1776, 12, 4, 1], [1776, 12, 14, 0], [1776, 12, 14, 1], [1776, 12, 23, 0], [1776, 12, 23, 1], [1789, 7, 4, 0], [1789, 7, 4, 1], [1789, 7, 14, 0], [1789, 7, 14, 1], [1789, 7, 23, 0], [1789, 7, 23, 1], [1789, 12, 4, 0], [1789, 12, 4, 1], [1789, 12, 14, 0], [1789, 12, 14, 1], [1789, 12, 23, 0], [1789, 12, 23, 1]]
D
import std.stdio;
void main() {
auto a = listProduct([1,2], [3,4]);
writeln(a);
auto b = listProduct([3,4], [1,2]);
writeln(b);
auto c = listProduct([1,2], []);
writeln(c);
auto d = listProduct([], [1,2]);
writeln(d);
}
auto listProduct(T)(T[] ta, T[] tb) {
struct Result {
int i, j;
bool empty() {
return i>=ta.length
|| j>=tb.length;
}
T[] front() {
return [ta[i], tb[j]];
}
void popFront() {
if (++j>=tb.length) {
j=0;
i++;
}
}
}
return Result();
}
- Output:
[[1, 3], [1, 4], [2, 3], [2, 4]] [[3, 1], [3, 2], [4, 1], [4, 2]] [] []
Delphi
program Cartesian_product_of_two_or_more_lists;
{$APPTYPE CONSOLE}
uses
System.SysUtils;
type
TList = TArray<Integer>;
TLists = TArray<TList>;
TListHelper = record helper for TList
function ToString: string;
end;
TListsHelper = record helper for TLists
function ToString(BreakLines: boolean = false): string;
end;
function cartN(arg: TLists): TLists;
var
b, n: TList;
argc: Integer;
begin
argc := length(arg);
var c := 1;
for var a in arg do
c := c * length(a);
if c = 0 then
exit;
SetLength(result, c);
SetLength(b, c * argc);
SetLength(n, argc);
var s := 0;
for var i := 0 to c - 1 do
begin
var e := s + argc;
var Resi := copy(b, s, e - s);
Result[i] := Resi;
s := e;
for var j := 0 to high(n) do
begin
var nj := n[j];
Resi[j] := arg[j, nj];
end;
for var j := high(n) downto 0 do
begin
inc(n[j]);
if n[j] < Length(arg[j]) then
Break;
n[j] := 0;
end;
end;
end;
{ TListHelper }
function TListHelper.ToString: string;
begin
Result := '[';
for var i := 0 to High(self) do
begin
Result := Result + self[i].ToString;
if i < High(self) then
Result := Result + ' ';
end;
Result := Result + ']';
end;
{ TListsHelper }
function TListsHelper.ToString(BreakLines: boolean = false): string;
begin
Result := '[';
for var i := 0 to High(self) do
begin
Result := Result + self[i].ToString;
if i < High(self) then
begin
if BreakLines then
Result := Result + #10
else
Result := Result + ' ';
end;
end;
Result := Result + ']';
end;
begin
writeln(#10, cartN([[1, 2], [3, 4]]).ToString);
writeln(#10, cartN([[3, 4], [1, 2]]).ToString);
writeln(#10, cartN([[1, 2], []]).ToString);
writeln(#10, cartN([[], [1, 2]]).ToString);
writeln(#10, cartN([[1776, 1789], [17, 12], [4, 14, 23], [0, 1]]).ToString(True));
writeln(#10, cartN([[1, 2, 3], [30], [500, 100]]).ToString);
writeln(#10, cartN([[1, 2, 3], [], [500, 100]]).ToString);
{$IFNDEF UNIX} readln; {$ENDIF}
end.
- Output:
[[1 3] [1 4] [2 3] [2 4]] [[3 1] [3 2] [4 1] [4 2]] [] [] [[1776 17 4 0] [1776 17 4 1] [1776 17 14 0] [1776 17 14 1] [1776 17 23 0] [1776 17 23 1] [1776 12 4 0] [1776 12 4 1] [1776 12 14 0] [1776 12 14 1] [1776 12 23 0] [1776 12 23 1] [1789 17 4 0] [1789 17 4 1] [1789 17 14 0] [1789 17 14 1] [1789 17 23 0] [1789 17 23 1] [1789 12 4 0] [1789 12 4 1] [1789 12 14 0] [1789 12 14 1] [1789 12 23 0] [1789 12 23 1]] [[1 30 500] [1 30 100] [2 30 500] [2 30 100] [3 30 500] [3 30 100]] []
DuckDB
See #SQL on this page for one approach to computing cartesian products.
Since DuckDB also supports arrays as first-class citizens, the remainder of this entry will illustrate how to compute the cartesian product of two arrays using list notation for both input and output:
select array_agg([a,b]) as product
from (select unnest([1,2]) as a), (select unnest([3,4]) as b);
The output is shown below.
We can go one step further to define a scalar function that returns the cartesian product of its two list arguments:
create or replace macro cp(x,y) as (
select array_agg([a,b] order by ix, iy)
from (select unnest(x) as a, unnest(range(0, length(x))) as ix),
(select unnest(y) as b, unnest(range(0, length(y))) as iy)
);
# Example:
select cp([1,2], [3,4]) as product;
- Output:
┌──────────────────────────────────┐ │ product │ │ int32[][] │ ├──────────────────────────────────┤ │ [[1, 3], [1, 4], [2, 3], [2, 4]] │ └──────────────────────────────────┘
EasyLang
proc cart2 a[] b[] . p[][] .
p[][] = [ ]
for a in a[]
for b in b[]
p[][] &= [ a b ]
.
.
.
cart2 [ 1 2 ] [ 3 4 ] r[][]
print r[][]
cart2 [ 3 4 ] [ 1 2 ] r[][]
print r[][]
cart2 [ 1 2 ] [ ] r[][]
print r[][]
cart2 [ ] [ 1 2 ] r[][]
print r[][]
Erlang
Can do this with list comprehensions.
-module(cartesian).
-export([product/2]).
product(S1, S2) -> [{A,B} || A <- S1, B <- S2].
- Output:
2> cartesian:product([],[1,2,3]).
[]
3> cartesian:product([1,2,3],[]).
[]
4> cartesian:product([1,2],[3,4]).
[{1,3},{1,4},{2,3},{2,4}]
5> cartesian:product([3,4],[1,2]).
[{3,1},{3,2},{4,1},{4,2}]
F#
The Task
//Nigel Galloway February 12th., 2018
let cP2 n g = List.map (fun (n,g)->[n;g]) (List.allPairs n g)
- Output:
cP2 [1;2] [3;4] -> [[1; 3]; [1; 4]; [2; 3]; [2; 4]] cP2 [3;4] [1;2] -> [[3; 1]; [3; 2]; [4; 1]; [4; 2]] cP2 [1;2] [] -> [] cP2 [] [1;2] -> []
Extra Credit
//Nigel Galloway August 14th., 2018
let cP ng=Seq.foldBack(fun n g->[for n' in n do for g' in g do yield n'::g']) ng [[]]
- Output:
cP [[1;2];[3;4]] -> [[1; 3]; [1; 4]; [2; 3]; [2; 4]]
cP [[3;4];[1;2]] -> [[3; 1]; [3; 2]; [4; 1]; [4; 2]]
cP [[3;4];[]] ->[]
cP [[];[1;2]] ->[]
cP [[1776;1789];[7;12];[4;14;23];[0;1]] -> [[1776; 7; 4; 0]; [1776; 7; 4; 1]; [1776; 7; 14; 0]; [1776; 7; 14; 1];
[1776; 7; 23; 0]; [1776; 7; 23; 1]; [1776; 12; 4; 0]; [1776; 12; 4; 1];
[1776; 12; 14; 0]; [1776; 12; 14; 1]; [1776; 12; 23; 0]; [1776; 12; 23; 1];
[1789; 7; 4; 0]; [1789; 7; 4; 1]; [1789; 7; 14; 0]; [1789; 7; 14; 1];
[1789; 7; 23; 0]; [1789; 7; 23; 1]; [1789; 12; 4; 0]; [1789; 12; 4; 1];
[1789; 12; 14; 0]; [1789; 12; 14; 1]; [1789; 12; 23; 0]; [1789; 12; 23; 1]]
cP [[1;2;3];[30];[500;100]] -> [[1; 30; 500]; [1; 30; 100]; [2; 30; 500]; [2; 30; 100]; [3; 30; 500]; [3; 30; 100]]
cP [[1;2;3];[];[500;100]] -> []
Factor
IN: scratchpad { 1 2 } { 3 4 } cartesian-product .
{ { { 1 3 } { 1 4 } } { { 2 3 } { 2 4 } } }
IN: scratchpad { 3 4 } { 1 2 } cartesian-product .
{ { { 3 1 } { 3 2 } } { { 4 1 } { 4 2 } } }
IN: scratchpad { 1 2 } { } cartesian-product .
{ { } { } }
IN: scratchpad { } { 1 2 } cartesian-product .
{ }
Fortran
This implementation is hard to extend to n-ary products but it is simple and works well for binary products of lists of any length.
! Created by simon on 29/04/2021.
! ifort -o cartesian_product cartesian_product.f90 -check all
module tuple
implicit none
private
public :: tuple_t, operator(*), print
type tuple_t(n)
integer, len :: n
integer, private :: v(n)
contains
procedure, public :: print => print_tuple_t
generic, public :: assignment(=) => eq_tuple_t
procedure, public :: eq_tuple_t
end type tuple_t
interface print
module procedure print_tuple_a_t
end interface print
interface operator(*)
module procedure tup_times_tup
end interface
contains
subroutine eq_tuple_t(this, src)
class(tuple_t(*)), intent(inout) :: this
integer, intent(in) :: src(:)
this%v = src
end subroutine eq_tuple_t
pure function tup_times_tup(a, b) result(r)
type(tuple_t(*)), intent(in) :: a
type(tuple_t(*)), intent(in) :: b
type(tuple_t(2)), allocatable :: r(:)
integer :: i, j, k
allocate(r(a%n*b%n))
k = 0
do i=1,a%n
do j=1,b%n
k = k + 1
r(k)%v = [a%v(i),b%v(j)]
end do
end do
end function tup_times_tup
subroutine print_tuple_t(this)
class(tuple_t(*)), intent(in) :: this
integer :: i
write(*,fmt='(a)',advance='no') '{'
do i=1,size(this%v)
write(*,fmt='(i0)',advance='no') this%v(i)
if (i < size(this%v)) write(*,fmt='(a)',advance='no') ','
end do
write(*,fmt='(a)',advance='no') '}'
end subroutine print_tuple_t
subroutine print_tuple_a_t(r)
type(tuple_t(*)), intent(in) :: r(:)
integer :: i
write(*,fmt='(a)',advance='no') '{'
do i=1,size(r)
call r(i)%print
if (i < size(r)) write(*,fmt='(a)',advance='no') ','
end do
write(*,fmt='(a)') '}'
end subroutine print_tuple_a_t
end module tuple
program cartesian_product
use tuple
implicit none
type(tuple_t(2)) :: a, b
type(tuple_t(0)) :: z
a = [1,2]
b = [3,4]
call print_product(a, b)
call print_product(b, a)
call print_product(z, a)
call print_product(a, z)
stop
contains
subroutine print_product(s, t)
type(tuple_t(*)), intent(in) :: s
type(tuple_t(*)), intent(in) :: t
call s%print
write(*,fmt='(a)',advance='no') ' x '
call t%print
write(*,fmt='(a)',advance='no') ' = '
call print(s*t)
end subroutine print_product
end program cartesian_product
- Output:
{1,2} x {3,4} = {{1,3},{1,4},{2,3},{2,4}}
{3,4} x {1,2} = {{3,1},{3,2},{4,1},{4,2}}
{1,2} x {} = {}
{} x {1,2} = {}
FreeBASIC
I'll leave the extra credit part for someone else. It's just going to amount to repeatedly finding Cartesian products and flattening the result, so considerably less interesting than Cartesian products where the list items themselves can be lists.
#define MAXLEN 64
type listitem ' An item of a list may be a number
is_num as boolean ' or another list, so I have to account
union ' for both, implemented as a union.
list as any ptr ' FreeBASIC is twitchy about circularly
num as uinteger ' defined types, so one workaround is to
end union ' use a generic pointer that I will cast
end type ' later.
type list
length as uinteger 'simple, fixed storage length lists
item(1 to MAXLEN) as listitem 'are good enough for this example
end type
sub print_list( list as list )
print "{";
if list.length = 0 then print "}"; : return
for i as uinteger = 1 to list.length
if list.item(i).is_num then
print str(list.item(i).num);
else 'recursively print sublist
print_list( *cast(list ptr, list.item(i).list) )
end if
if i<list.length then print ", "; else print "}"; 'handle comma
next i 'gracefully
return
end sub
function cartprod( A as list, B as list ) as list
dim as uinteger i, j
dim as list C
dim as list ptr inner 'for brevity
C.length = 0
for i = 1 to A.length
for j = 1 to B.length
C.length += 1
C.item(C.length).is_num = false 'each item of the new list is a list itself
inner = allocate( sizeof(list) ) 'make space for it
C.item(C.length).list = inner
inner->length = 2 'each inner list contains two items
inner->item(1) = A.item(i) 'one from the first list
inner->item(2) = B.item(j) 'and one from the second
next j
next i
return C
end function
dim as list EMPTY, A, B, R
EMPTY.length = 0
A.length = 2
A.item(1).is_num = true : A.item(1).num = 1
A.item(2).is_num = true : A.item(2).num = 2
B.length = 2
B.item(1).is_num = true : B.item(1).num = 3
B.item(2).is_num = true : B.item(2).num = 4
R = cartprod(A, B)
print_list(R) : print 'print_list does not supply a final newline
R = cartprod(B, A) : print_list(R) : print
R = cartprod(A, EMPTY) : print_list(R) : print
R = cartprod(EMPTY, A) : print_list(R) : print
- Output:
{{1, 3}, {1, 4}, {2, 3}, {2, 4}}
{{3, 1}, {3, 2}, {4, 1}, {4, 2}}
{}
{}
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 is needed, the cartesian product is an intrinsic operation in Fōrmulæ
Test case 1. No commutativity
Test case 2. With an empty list
Test case 3. Extra credit. n-ary cartesian product
Go
Basic Task
package main
import "fmt"
type pair [2]int
func cart2(a, b []int) []pair {
p := make([]pair, len(a)*len(b))
i := 0
for _, a := range a {
for _, b := range b {
p[i] = pair{a, b}
i++
}
}
return p
}
func main() {
fmt.Println(cart2([]int{1, 2}, []int{3, 4}))
fmt.Println(cart2([]int{3, 4}, []int{1, 2}))
fmt.Println(cart2([]int{1, 2}, nil))
fmt.Println(cart2(nil, []int{1, 2}))
}
- Output:
[[1 3] [1 4] [2 3] [2 4]] [[3 1] [3 2] [4 1] [4 2]] [] []
Extra credit 1
This solution minimizes allocations and computes and fills the result sequentially.
package main
import "fmt"
func cartN(a ...[]int) [][]int {
c := 1
for _, a := range a {
c *= len(a)
}
if c == 0 {
return nil
}
p := make([][]int, c)
b := make([]int, c*len(a))
n := make([]int, len(a))
s := 0
for i := range p {
e := s + len(a)
pi := b[s:e]
p[i] = pi
s = e
for j, n := range n {
pi[j] = a[j][n]
}
for j := len(n) - 1; j >= 0; j-- {
n[j]++
if n[j] < len(a[j]) {
break
}
n[j] = 0
}
}
return p
}
func main() {
fmt.Println(cartN([]int{1, 2}, []int{3, 4}))
fmt.Println(cartN([]int{3, 4}, []int{1, 2}))
fmt.Println(cartN([]int{1, 2}, nil))
fmt.Println(cartN(nil, []int{1, 2}))
fmt.Println()
fmt.Println("[")
for _, p := range cartN(
[]int{1776, 1789},
[]int{7, 12},
[]int{4, 14, 23},
[]int{0, 1},
) {
fmt.Println(" ", p)
}
fmt.Println("]")
fmt.Println(cartN([]int{1, 2, 3}, []int{30}, []int{500, 100}))
fmt.Println(cartN([]int{1, 2, 3}, []int{}, []int{500, 100}))
fmt.Println()
fmt.Println(cartN(nil))
fmt.Println(cartN())
}
- Output:
[[1 3] [1 4] [2 3] [2 4]] [[3 1] [3 2] [4 1] [4 2]] [] [] [ [1776 7 4 0] [1776 7 4 1] [1776 7 14 0] [1776 7 14 1] [1776 7 23 0] [1776 7 23 1] [1776 12 4 0] [1776 12 4 1] [1776 12 14 0] [1776 12 14 1] [1776 12 23 0] [1776 12 23 1] [1789 7 4 0] [1789 7 4 1] [1789 7 14 0] [1789 7 14 1] [1789 7 23 0] [1789 7 23 1] [1789 12 4 0] [1789 12 4 1] [1789 12 14 0] [1789 12 14 1] [1789 12 23 0] [1789 12 23 1] ] [[1 30 500] [1 30 100] [2 30 500] [2 30 100] [3 30 500] [3 30 100]] [] [] [[]]
Extra credit 2
Code here is more compact, but with the cost of more garbage produced. It produces the same result as cartN above.
func cartN(a ...[]int) (c [][]int) {
if len(a) == 0 {
return [][]int{nil}
}
r := cartN(a[1:]...)
for _, e := range a[0] {
for _, p := range r {
c = append(c, append([]int{e}, p...))
}
}
return
}
Extra credit 3
This is a compact recursive version like Extra credit 2 but the result list is ordered differently. This is still a correct result if you consider a cartesian product to be a set, which is an unordered collection. Note that the set elements are still ordered lists. A cartesian product is an unordered collection of ordered collections. It draws attention though to the gloss of using list representations as sets. Any of the functions here will accept duplicate elements in the input lists, and then produce duplicate elements in the result.
func cartN(a ...[]int) (c [][]int) {
if len(a) == 0 {
return [][]int{nil}
}
last := len(a) - 1
l := cartN(a[:last]...)
for _, e := range a[last] {
for _, p := range l {
c = append(c, append(p, e))
}
}
return
}
Groovy
Solution:
The following CartesianCategory class allows for modification of regular Iterable interface behavior, overloading Iterable's multiply (*) operator to perform a Cartesian Product when the second operand is also an Iterable.
class CartesianCategory {
static Iterable multiply(Iterable a, Iterable b) {
assert [a,b].every { it != null }
def (m,n) = [a.size(),b.size()]
(0..<(m*n)).inject([]) { prod, i -> prod << [a[i.intdiv(n)], b[i%n]].flatten() }
}
}
Test:
The mixin method call is necessary to make the multiply (*) operator work.
Iterable.metaClass.mixin CartesianCategory
println "\nCore Solution:"
println "[1, 2] × [3, 4] = ${[1, 2] * [3, 4]}"
println "[3, 4] × [1, 2] = ${[3, 4] * [1, 2]}"
println "[1, 2] × [] = ${[1, 2] * []}"
println "[] × [1, 2] = ${[] * [1, 2]}"
println "\nExtra Credit:"
println "[1776, 1789] × [7, 12] × [4, 14, 23] × [0, 1] = ${[1776, 1789] * [7, 12] * [4, 14, 23] * [0, 1]}"
println "[1, 2, 3] × [30] × [500, 100] = ${[1, 2, 3] * [30] * [500, 100]}"
println "[1, 2, 3] × [] × [500, 100] = ${[1, 2, 3] * [] * [500, 100]}"
println "\nNon-Numeric Example:"
println "[John,Paul,George,Ringo] × [Emerson,Lake,Palmer] × [Simon,Garfunkle] = ["
( ["John","Paul","George","Ringo"] * ["Emerson","Lake","Palmer"] * ["Simon","Garfunkle"] ).each { println "\t${it}," }
println "]"
Output:
Core Solution: [1, 2] × [3, 4] = [[1, 3], [1, 4], [2, 3], [2, 4]] [3, 4] × [1, 2] = [[3, 1], [3, 2], [4, 1], [4, 2]] [1, 2] × [] = [] [] × [1, 2] = [] Extra Credit: [1776, 1789] × [7, 12] × [4, 14, 23] × [0, 1] = [[1776, 7, 4, 0], [1776, 7, 4, 1], [1776, 7, 14, 0], [1776, 7, 14, 1], [1776, 7, 23, 0], [1776, 7, 23, 1], [1776, 12, 4, 0], [1776, 12, 4, 1], [1776, 12, 14, 0], [1776, 12, 14, 1], [1776, 12, 23, 0], [1776, 12, 23, 1], [1789, 7, 4, 0], [1789, 7, 4, 1], [1789, 7, 14, 0], [1789, 7, 14, 1], [1789, 7, 23, 0], [1789, 7, 23, 1], [1789, 12, 4, 0], [1789, 12, 4, 1], [1789, 12, 14, 0], [1789, 12, 14, 1], [1789, 12, 23, 0], [1789, 12, 23, 1]] [1, 2, 3] × [30] × [500, 100] = [[1, 30, 500], [1, 30, 100], [2, 30, 500], [2, 30, 100], [3, 30, 500], [3, 30, 100]] [1, 2, 3] × [] × [500, 100] = [] Non-Numeric Example: [John,Paul,George,Ringo] × [Emerson,Lake,Palmer] × [Simon,Garfunkle] = [ [John, Emerson, Simon], [John, Emerson, Garfunkle], [John, Lake, Simon], [John, Lake, Garfunkle], [John, Palmer, Simon], [John, Palmer, Garfunkle], [Paul, Emerson, Simon], [Paul, Emerson, Garfunkle], [Paul, Lake, Simon], [Paul, Lake, Garfunkle], [Paul, Palmer, Simon], [Paul, Palmer, Garfunkle], [George, Emerson, Simon], [George, Emerson, Garfunkle], [George, Lake, Simon], [George, Lake, Garfunkle], [George, Palmer, Simon], [George, Palmer, Garfunkle], [Ringo, Emerson, Simon], [Ringo, Emerson, Garfunkle], [Ringo, Lake, Simon], [Ringo, Lake, Garfunkle], [Ringo, Palmer, Simon], [Ringo, Palmer, Garfunkle], ]
Haskell
Various routes can be taken to Cartesian products in Haskell. For the product of two lists we could write:
cartProd :: [a] -> [b] -> [(a, b)]
cartProd xs ys =
[ (x, y)
| x <- xs
, y <- ys ]
more directly:
cartProd :: [a] -> [b] -> [(a, b)]
cartProd xs ys = xs >>= \x -> ys >>= \y -> [(x, y)]
applicatively:
cartProd :: [a] -> [b] -> [(a, b)]
cartProd xs ys = (,) <$> xs <*> ys
parsimoniously:
cartProd :: [a] -> [b] -> [(a, b)]
cartProd = (<*>) . fmap (,)
We might test any of these with:
main :: IO ()
main =
mapM_ print $
uncurry cartProd <$>
[([1, 2], [3, 4]), ([3, 4], [1, 2]), ([1, 2], []), ([], [1, 2])]
- Output:
[(1,3),(1,4),(2,3),(2,4)] [(3,1),(3,2),(4,1),(4,2)] [] []
For the n-ary Cartesian product of an arbitrary number of lists, we could apply the Prelude's standard sequence function to a list of lists,
cartProdN :: [[a]] -> [[a]]
cartProdN = sequence
main :: IO ()
main = print $ cartProdN [[1, 2], [3, 4], [5, 6]]
- Output:
[[1,3,5],[1,3,6],[1,4,5],[1,4,6],[2,3,5],[2,3,6],[2,4,5],[2,4,6]]
or we could define ourselves an equivalent function over a list of lists in terms of a fold, for example as:
cartProdN :: [[a]] -> [[a]]
cartProdN = foldr (\xs as -> xs >>= (<$> as) . (:)) [[]]
or, equivalently, as:
cartProdN :: [[a]] -> [[a]]
cartProdN = foldr
(\xs as ->
[ x : a
| x <- xs
, a <- as ])
[[]]
testing any of these with something like:
main :: IO ()
main = do
mapM_ print $
cartProdN [[1776, 1789], [7,12], [4, 14, 23], [0,1]]
putStrLn ""
print $ cartProdN [[1,2,3], [30], [500, 100]]
putStrLn ""
print $ cartProdN [[1,2,3], [], [500, 100]]
- Output:
[1776,7,4,0] [1776,7,4,1] [1776,7,14,0] [1776,7,14,1] [1776,7,23,0] [1776,7,23,1] [1776,12,4,0] [1776,12,4,1] [1776,12,14,0] [1776,12,14,1] [1776,12,23,0] [1776,12,23,1] [1789,7,4,0] [1789,7,4,1] [1789,7,14,0] [1789,7,14,1] [1789,7,23,0] [1789,7,23,1] [1789,12,4,0] [1789,12,4,1] [1789,12,14,0] [1789,12,14,1] [1789,12,23,0] [1789,12,23,1] [[1,30,500],[1,30,100],[2,30,500],[2,30,100],[3,30,500],[3,30,100]] []
J
The J primitive catalogue { forms the Cartesian Product of two or more boxed lists. The result is a multi-dimensional array (which can be reshaped to a simple list of lists if desired).
{ 1776 1789 ; 7 12 ; 4 14 23 ; 0 1 NB. result is 4 dimensional array with shape 2 2 3 2
┌────────────┬────────────┐
│1776 7 4 0 │1776 7 4 1 │
├────────────┼────────────┤
│1776 7 14 0 │1776 7 14 1 │
├────────────┼────────────┤
│1776 7 23 0 │1776 7 23 1 │
└────────────┴────────────┘
┌────────────┬────────────┐
│1776 12 4 0 │1776 12 4 1 │
├────────────┼────────────┤
│1776 12 14 0│1776 12 14 1│
├────────────┼────────────┤
│1776 12 23 0│1776 12 23 1│
└────────────┴────────────┘
┌────────────┬────────────┐
│1789 7 4 0 │1789 7 4 1 │
├────────────┼────────────┤
│1789 7 14 0 │1789 7 14 1 │
├────────────┼────────────┤
│1789 7 23 0 │1789 7 23 1 │
└────────────┴────────────┘
┌────────────┬────────────┐
│1789 12 4 0 │1789 12 4 1 │
├────────────┼────────────┤
│1789 12 14 0│1789 12 14 1│
├────────────┼────────────┤
│1789 12 23 0│1789 12 23 1│
└────────────┴────────────┘
{ 1 2 3 ; 30 ; 50 100 NB. result is a 2-dimensional array with shape 2 3
┌───────┬────────┐
│1 30 50│1 30 100│
├───────┼────────┤
│2 30 50│2 30 100│
├───────┼────────┤
│3 30 50│3 30 100│
└───────┴────────┘
{ 1 2 3 ; '' ; 50 100 NB. result is an empty 3-dimensional array with shape 3 0 2
Java
import static java.util.Arrays.asList;
import static java.util.Collections.emptyList;
import static java.util.Optional.of;
import static java.util.stream.Collectors.toList;
import java.util.List;
public class CartesianProduct {
public List<?> product(List<?>... a) {
if (a.length >= 2) {
List<?> product = a[0];
for (int i = 1; i < a.length; i++) {
product = product(product, a[i]);
}
return product;
}
return emptyList();
}
private <A, B> List<?> product(List<A> a, List<B> b) {
return of(a.stream()
.map(e1 -> of(b.stream().map(e2 -> asList(e1, e2)).collect(toList())).orElse(emptyList()))
.flatMap(List::stream)
.collect(toList())).orElse(emptyList());
}
}
Using a generic class with a recursive function
import java.util.ArrayList;
import java.util.Arrays;
import java.util.List;
public class CartesianProduct<V> {
public List<List<V>> product(List<List<V>> lists) {
List<List<V>> product = new ArrayList<>();
// We first create a list for each value of the first list
product(product, new ArrayList<>(), lists);
return product;
}
private void product(List<List<V>> result, List<V> existingTupleToComplete, List<List<V>> valuesToUse) {
for (V value : valuesToUse.get(0)) {
List<V> newExisting = new ArrayList<>(existingTupleToComplete);
newExisting.add(value);
// If only one column is left
if (valuesToUse.size() == 1) {
// We create a new list with the exiting tuple for each value with the value
// added
result.add(newExisting);
} else {
// If there are still several columns, we go into recursion for each value
List<List<V>> newValues = new ArrayList<>();
// We build the next level of values
for (int i = 1; i < valuesToUse.size(); i++) {
newValues.add(valuesToUse.get(i));
}
product(result, newExisting, newValues);
}
}
}
public static void main(String[] args) {
List<Integer> list1 = new ArrayList<>(Arrays.asList(new Integer[] { 1776, 1789 }));
List<Integer> list2 = new ArrayList<>(Arrays.asList(new Integer[] { 7, 12 }));
List<Integer> list3 = new ArrayList<>(Arrays.asList(new Integer[] { 4, 14, 23 }));
List<Integer> list4 = new ArrayList<>(Arrays.asList(new Integer[] { 0, 1 }));
List<List<Integer>> input = new ArrayList<>();
input.add(list1);
input.add(list2);
input.add(list3);
input.add(list4);
CartesianProduct<Integer> cartesianProduct = new CartesianProduct<>();
List<List<Integer>> product = cartesianProduct.product(input);
System.out.println(product);
}
}
JavaScript
function cartesian(m){
if(!m.length)return[[]];
let tails=cartesian(m.slice(1));
return(m[0].flatMap(h=>tails.map(t=>[h].concat(t))));
}
ES6
Functional
Cartesian products fall quite naturally out of concatMap (Array.flatMap), and its argument-flipped twin bind.
For the Cartesian product of just two lists:
(() => {
// CARTESIAN PRODUCT OF TWO LISTS ---------------------
// cartProd :: [a] -> [b] -> [[a, b]]
const cartProd = xs => ys =>
xs.flatMap(x => ys.map(y => [x, y]))
// TEST -----------------------------------------------
return [
cartProd([1, 2])([3, 4]),
cartProd([3, 4])([1, 2]),
cartProd([1, 2])([]),
cartProd([])([1, 2]),
].map(JSON.stringify).join('\n');
})();
- Output:
[[1,3],[1,4],[2,3],[2,4]] [[3,1],[3,2],[4,1],[4,2]] [] []
Abstracting a little more, we can define the cartesian product quite economically in terms of a general applicative operator:
(() => {
// CARTESIAN PRODUCT OF TWO LISTS ---------------------
// cartesianProduct :: [a] -> [b] -> [(a, b)]
const cartesianProduct = xs =>
ap(xs.map(Tuple));
// GENERIC FUNCTIONS ----------------------------------
// e.g. [(*2),(/2), sqrt] <*> [1,2,3]
// --> ap([dbl, hlf, root], [1, 2, 3])
// --> [2,4,6,0.5,1,1.5,1,1.4142135623730951,1.7320508075688772]
// Each member of a list of functions applied to each
// of a list of arguments, deriving a list of new values.
// ap (<*>) :: [(a -> b)] -> [a] -> [b]
const ap = fs => xs =>
// The sequential application of each of a list
// of functions to each of a list of values.
fs.flatMap(
f => xs.map(f)
);
// Tuple (,) :: a -> b -> (a, b)
const Tuple = a => b => [a, b];
// TEST -----------------------------------------------
return [
cartesianProduct([1, 2])([3, 4]),
cartesianProduct([3, 4])([1, 2]),
cartesianProduct([1, 2])([]),
cartesianProduct([])([1, 2]),
]
.map(JSON.stringify)
.join('\n');
})();
- Output:
[[1,3],[1,4],[2,3],[2,4]] [[3,1],[3,2],[4,1],[4,2]] [] []
For the n-ary Cartesian product over a list of lists:
(() => {
const main = () => {
// n-ary Cartesian product of a list of lists.
// cartProdN :: [[a]] -> [[a]]
const cartProdN = foldr(
xs => as =>
bind(as)(
x => bind(xs)(
a => [
[a].concat(x)
]
)
)
)([
[]
]);
// TEST -------------------------------------------
return intercalate('\n\n')([
map(show)(
cartProdN([
[1776, 1789],
[7, 12],
[4, 14, 23],
[0, 1]
])
).join('\n'),
show(cartProdN([
[1, 2, 3],
[30],
[50, 100]
])),
show(cartProdN([
[1, 2, 3],
[],
[50, 100]
]))
])
};
// GENERIC FUNCTIONS ----------------------------------
// bind :: [a] -> (a -> [b]) -> [b]
const bind = xs => f => xs.flatMap(f);
// foldr :: (a -> b -> b) -> b -> [a] -> b
const foldr = f => a => xs =>
xs.reduceRight((a, x) => f(x)(a), a);
// intercalate :: String -> [a] -> String
const intercalate = s => xs => xs.join(s);
// map :: (a -> b) -> [a] -> [b]
const map = f => xs => xs.map(f);
// show :: a -> String
const show = x => JSON.stringify(x);
return main();
})();
- Output:
[1776,7,4,0] [1776,7,4,1] [1776,7,14,0] [1776,7,14,1] [1776,7,23,0] [1776,7,23,1] [1776,12,4,0] [1776,12,4,1] [1776,12,14,0] [1776,12,14,1] [1776,12,23,0] [1776,12,23,1] [1789,7,4,0] [1789,7,4,1] [1789,7,14,0] [1789,7,14,1] [1789,7,23,0] [1789,7,23,1] [1789,12,4,0] [1789,12,4,1] [1789,12,14,0] [1789,12,14,1] [1789,12,23,0] [1789,12,23,1] [[1,30,50],[1,30,100],[2,30,50],[2,30,100],[3,30,50],[3,30,100]] []
Imperative
Imperative implementations of Cartesian products are inevitably less compact and direct, but we can certainly write an iterative translation of a fold over nested applications of bind or concatMap:
(() => {
// n-ary Cartesian product of a list of lists
// ( Imperative implementation )
// cartProd :: [a] -> [b] -> [[a, b]]
const cartProd = lists => {
let ps = [],
acc = [
[]
],
i = lists.length;
while (i--) {
let subList = lists[i],
j = subList.length;
while (j--) {
let x = subList[j],
k = acc.length;
while (k--) ps.push([x].concat(acc[k]))
};
acc = ps;
ps = [];
};
return acc.reverse();
};
// GENERIC FUNCTIONS ------------------------------------------------------
// intercalate :: String -> [a] -> String
const intercalate = (s, xs) => xs.join(s);
// map :: (a -> b) -> [a] -> [b]
const map = (f, xs) => xs.map(f);
// show :: a -> String
const show = x => JSON.stringify(x);
// unlines :: [String] -> String
const unlines = xs => xs.join('\n');
// TEST -------------------------------------------------------------------
return intercalate('\n\n', [show(cartProd([
[1, 2],
[3, 4]
])),
show(cartProd([
[3, 4],
[1, 2]
])),
show(cartProd([
[1, 2],
[]
])),
show(cartProd([
[],
[1, 2]
])),
unlines(map(show, cartProd([
[1776, 1789],
[7, 12],
[4, 14, 23],
[0, 1]
]))),
show(cartProd([
[1, 2, 3],
[30],
[50, 100]
])),
show(cartProd([
[1, 2, 3],
[],
[50, 100]
]))
]);
})();
- Output:
[[1,4],[1,3],[2,4],[2,3]] [[3,2],[3,1],[4,2],[4,1]] [] [] [1776,12,4,1] [1776,12,4,0] [1776,12,14,1] [1776,12,14,0] [1776,12,23,1] [1776,12,23,0] [1776,7,4,1] [1776,7,4,0] [1776,7,14,1] [1776,7,14,0] [1776,7,23,1] [1776,7,23,0] [1789,12,4,1] [1789,12,4,0] [1789,12,14,1] [1789,12,14,0] [1789,12,23,1] [1789,12,23,0] [1789,7,4,1] [1789,7,4,0] [1789,7,14,1] [1789,7,14,0] [1789,7,23,1] [1789,7,23,0] [[1,30,50],[1,30,100],[2,30,50],[2,30,100],[3,30,50],[3,30,100]] []
jq
jq is stream-oriented and so we begin by defining a function that will emit a stream of the elements of the Cartesian product of two arrays:
def products: .[0][] as $x | .[1][] as $y | [$x,$y];To generate an array of these arrays, one would in practice most likely simply write `[products]`, but to comply with the requirements of this article, we can define `product` as:
def product: [products];For the sake of brevity, two illustrations should suffice:
[ [1,2], [3,4] ] | products
produces the stream:
[1,3] [1,4] [2,3] [2,4]
And
[[1,2], []] | productproduces:
[]
n-way Cartesian Product
Given an array of two or more arrays as input, `cartesians` as defined here produces a stream of the components of their Cartesian product:
def cartesians:
if length <= 2 then products
else .[0][] as $x
| (.[1:] | cartesians) as $y
| [$x] + $y
end;Again for brevity, in the following, we will just show the number of items in the Cartesian products:
[ [1776, 1789], [7, 12], [4, 14, 23], [0, 1]] | [cartesians] | length # 24
[[1, 2, 3], [30], [500, 100] ] | [cartesians] | length # 6
[[1, 2, 3], [], [500, 100] ] | [cartesians] | length # 0
Julia
Run in REPL.
# Product {1, 2} × {3, 4}
collect(Iterators.product([1, 2], [3, 4]))
# Product {3, 4} × {1, 2}
collect(Iterators.product([3, 4], [1, 2]))
# Product {1, 2} × {}
collect(Iterators.product([1, 2], []))
# Product {} × {1, 2}
collect(Iterators.product([], [1, 2]))
# Product {1776, 1789} × {7, 12} × {4, 14, 23} × {0, 1}
collect(Iterators.product([1776, 1789], [7, 12], [4, 14, 23], [0, 1]))
# Product {1, 2, 3} × {30} × {500, 100}
collect(Iterators.product([1, 2, 3], [30], [500, 100]))
# Product {1, 2, 3} × {} × {500, 100}
collect(Iterators.product([1, 2, 3], [], [500, 100]))
Kotlin
// version 1.1.2
fun flattenList(nestList: List<Any>): List<Any> {
val flatList = mutableListOf<Any>()
fun flatten(list: List<Any>) {
for (e in list) {
if (e !is List<*>)
flatList.add(e)
else
@Suppress("UNCHECKED_CAST")
flatten(e as List<Any>)
}
}
flatten(nestList)
return flatList
}
operator fun List<Any>.times(other: List<Any>): List<List<Any>> {
val prod = mutableListOf<List<Any>>()
for (e in this) {
for (f in other) {
prod.add(listOf(e, f))
}
}
return prod
}
fun nAryCartesianProduct(lists: List<List<Any>>): List<List<Any>> {
require(lists.size >= 2)
return lists.drop(2).fold(lists[0] * lists[1]) { cp, ls -> cp * ls }.map { flattenList(it) }
}
fun printNAryProduct(lists: List<List<Any>>) {
println("${lists.joinToString(" x ")} = ")
println("[")
println(nAryCartesianProduct(lists).joinToString("\n ", " "))
println("]\n")
}
fun main(args: Array<String>) {
println("[1, 2] x [3, 4] = ${listOf(1, 2) * listOf(3, 4)}")
println("[3, 4] x [1, 2] = ${listOf(3, 4) * listOf(1, 2)}")
println("[1, 2] x [] = ${listOf(1, 2) * listOf()}")
println("[] x [1, 2] = ${listOf<Any>() * listOf(1, 2)}")
println("[1, a] x [2, b] = ${listOf(1, 'a') * listOf(2, 'b')}")
println()
printNAryProduct(listOf(listOf(1776, 1789), listOf(7, 12), listOf(4, 14, 23), listOf(0, 1)))
printNAryProduct(listOf(listOf(1, 2, 3), listOf(30), listOf(500, 100)))
printNAryProduct(listOf(listOf(1, 2, 3), listOf<Int>(), listOf(500, 100)))
printNAryProduct(listOf(listOf(1, 2, 3), listOf(30), listOf('a', 'b')))
}
- Output:
[1, 2] x [3, 4] = [[1, 3], [1, 4], [2, 3], [2, 4]]
[3, 4] x [1, 2] = [[3, 1], [3, 2], [4, 1], [4, 2]]
[1, 2] x [] = []
[] x [1, 2] = []
[1, a] x [2, b] = [[1, 2], [1, b], [a, 2], [a, b]]
[1776, 1789] x [7, 12] x [4, 14, 23] x [0, 1] =
[
[1776, 7, 4, 0]
[1776, 7, 4, 1]
[1776, 7, 14, 0]
[1776, 7, 14, 1]
[1776, 7, 23, 0]
[1776, 7, 23, 1]
[1776, 12, 4, 0]
[1776, 12, 4, 1]
[1776, 12, 14, 0]
[1776, 12, 14, 1]
[1776, 12, 23, 0]
[1776, 12, 23, 1]
[1789, 7, 4, 0]
[1789, 7, 4, 1]
[1789, 7, 14, 0]
[1789, 7, 14, 1]
[1789, 7, 23, 0]
[1789, 7, 23, 1]
[1789, 12, 4, 0]
[1789, 12, 4, 1]
[1789, 12, 14, 0]
[1789, 12, 14, 1]
[1789, 12, 23, 0]
[1789, 12, 23, 1]
]
[1, 2, 3] x [30] x [500, 100] =
[
[1, 30, 500]
[1, 30, 100]
[2, 30, 500]
[2, 30, 100]
[3, 30, 500]
[3, 30, 100]
]
[1, 2, 3] x [] x [500, 100] =
[
]
[1, 2, 3] x [30] x [a, b] =
[
[1, 30, a]
[1, 30, b]
[2, 30, a]
[2, 30, b]
[3, 30, a]
[3, 30, b]
]
langur
val X = fn ...x:x
writeln mapX([1, 2], [3, 4], by=X) == [[1, 3], [1, 4], [2, 3], [2, 4]]
writeln mapX([3, 4], [1, 2], by=X) == [[3, 1], [3, 2], [4, 1], [4, 2]]
writeln mapX([1, 2], [], by=X) == []
writeln mapX([], [1, 2], by=X) == []
writeln()
writeln mapX([1776, 1789], [7, 12], [4, 14, 23], [0, 1], by=X)
writeln()
writeln mapX([1, 2, 3], [30], [500, 100], by=X)
writeln()
writeln mapX([1, 2, 3], [], [500, 100], by=X)
writeln()- Output:
true true true true [[1776, 7, 4, 0], [1776, 7, 4, 1], [1776, 7, 14, 0], [1776, 7, 14, 1], [1776, 7, 23, 0], [1776, 7, 23, 1], [1776, 12, 4, 0], [1776, 12, 4, 1], [1776, 12, 14, 0], [1776, 12, 14, 1], [1776, 12, 23, 0], [1776, 12, 23, 1], [1789, 7, 4, 0], [1789, 7, 4, 1], [1789, 7, 14, 0], [1789, 7, 14, 1], [1789, 7, 23, 0], [1789, 7, 23, 1], [1789, 12, 4, 0], [1789, 12, 4, 1], [1789, 12, 14, 0], [1789, 12, 14, 1], [1789, 12, 23, 0], [1789, 12, 23, 1]] [[1, 30, 500], [1, 30, 100], [2, 30, 500], [2, 30, 100], [3, 30, 500], [3, 30, 100]] []
Lua
Functional
An iterator is created to output the product items.
local pk,upk = table.pack, table.unpack
local getn = function(t)return #t end
local const = function(k)return function(e) return k end end
local function attachIdx(f)-- one-time-off function modifier
local idx = 0
return function(e)idx=idx+1 ; return f(e,idx)end
end
local function reduce(t,acc,f)
for i=1,t.n or #t do acc=f(acc,t[i])end
return acc
end
local function imap(t,f)
local r = {n=t.n or #t, r=reduce, u=upk, m=imap}
for i=1,r.n do r[i]=f(t[i])end
return r
end
local function prod(...)
local ts = pk(...)
local limit = imap(ts,getn)
local idx, cnt = imap(limit,const(1)), 0
local max = reduce(limit,1,function(a,b)return a*b end)
local function ret(t,i)return t[idx[i]] end
return function()
if cnt>=max then return end -- no more output
if cnt==0 then -- skip for 1st
cnt = cnt + 1
else
cnt, idx[#idx] = cnt + 1, idx[#idx] + 1
for i=#idx,2,-1 do -- update index list
if idx[i]<=limit[i] then
break -- no further update need
else -- propagate limit overflow
idx[i],idx[i-1] = 1, idx[i-1]+1
end
end
end
return cnt,imap(ts,attachIdx(ret)):u()
end
end
--- test
for i,a,b in prod({1,2},{3,4}) do
print(i,a,b)
end
print()
for i,a,b in prod({3,4},{1,2}) do
print(i,a,b)
end
- Output:
1 1 3 2 1 4 3 2 3 4 2 4 1 3 1 2 3 2 3 4 1 4 4 2
Using coroutines
I have not benchmarked this, but I believe that this should run faster than the functional implementation and also likely the imperative implementation, it has significantly fewer function calls per iteration, and only the stack changes during iteration (no garbage collection during iteration). On the other hand due to avoiding garbage collection, result is reused between returns, so mutating the returned result is unsafe.
It is possible that specialising descend by depth may yield a further improvement in performance, but it would only be able to eliminate the lookup of sets[depth] and the if test, because the reference to result[depth] is required; I doubt the increase in complexity would be worth the (potential) improvement in performance.
local function cartesian_product(sets)
local result = {}
local set_count = #sets
--[[ I believe that this should make the below go very slightly faster, because it doesn't need to lookup yield in coroutine each time it
yields, though perhaps the compiler optimises the lookup away? ]]
local yield = coroutine.yield
local function descend(depth)
if depth == set_count then
for k,v in pairs(sets[depth]) do
result[depth] = v
yield(result)
end
else
for k,v in pairs(sets[depth]) do
result[depth] = v
descend(depth + 1)
end
end
end
return coroutine.wrap(function() descend(1) end)
end
--- tests
local test_cases = {
{{1, 2}, {3, 4}},
{{3, 4}, {1, 2}},
{{1776, 1789}, {7, 12}, {4, 14, 23}, {0,1}},
{{1,2,3}, {30}, {500, 100}},
{{1,2,3}, {}, {500, 100}}
}
local function format_nested_list(list)
if list[1] and type(list[1]) == "table" then
local formatted_items = {}
for i, item in ipairs(list) do
formatted_items[i] = format_nested_list(item)
end
return format_nested_list(formatted_items)
else
return "{" .. table.concat(list, ",") .. "}"
end
end
for _,test_case in ipairs(test_cases) do
print(format_nested_list(test_case))
for product in cartesian_product(test_case) do
print(" " .. format_nested_list(product))
end
end
Imperative iterator
The functional implementation restated as an imperative iterator, also adjusted to not allocate a new result table on each iteration; this saves time, but makes mutating the returned table unsafe.
local function cartesian_product(sets)
local item_counts = {}
local indices = {}
local results = {}
local set_count = #sets
local combination_count = 1
for set_index=set_count, 1, -1 do
local set = sets[set_index]
local item_count = #set
item_counts[set_index] = item_count
indices[set_index] = 1
results[set_index] = set[1]
combination_count = combination_count * item_count
end
local combination_index = 0
return function()
if combination_index >= combination_count then return end -- no more output
if combination_index == 0 then goto skip_update end -- skip first index update
indices[set_count] = indices[set_count] + 1
for set_index=set_count, 1, -1 do -- update index list
local set = sets[set_index]
local index = indices[set_index]
if index <= item_counts[set_index] then
results[set_index] = set[index]
break -- no further update needed
else -- propagate item_counts overflow
results[set_index] = set[1]
indices[set_index] = 1
if set_index > 1 then
indices[set_index - 1] = indices[set_index - 1] + 1
end
end
end
::skip_update::
combination_index = combination_index + 1
return combination_index, results
end
end
--- tests
local test_cases = {
{{1, 2}, {3, 4}},
{{3, 4}, {1, 2}},
{{1776, 1789}, {7, 12}, {4, 14, 23}, {0,1}},
{{1,2,3}, {30}, {500, 100}},
{{1,2,3}, {}, {500, 100}}
}
local function format_nested_list(list)
if list[1] and type(list[1]) == "table" then
local formatted_items = {}
for i, item in ipairs(list) do
formatted_items[i] = format_nested_list(item)
end
return format_nested_list(formatted_items)
else
return "{" .. table.concat(list, ",") .. "}"
end
end
for _,test_case in ipairs(test_cases) do
print(format_nested_list(test_case))
for i, product in cartesian_product(test_case) do
print(i, format_nested_list(product))
end
end
Functional-esque (non-iterator)
Motivation: If a list-of-lists is passed into the cartesian product, then wouldn't a list-of-lists be the expected return type? Of course this is just personal opinion/preference, other implementations are fine as-is if you'd rather have an iterator.
-- support:
function T(t) return setmetatable(t, {__index=table}) end
table.clone = function(t) local s=T{} for k,v in ipairs(t) do s[k]=v end return s end
table.reduce = function(t,f,acc) for i=1,#t do acc=f(t[i],acc) end return acc end
-- implementation:
local function cartprod(sets)
local temp, prod = T{}, T{}
local function descend(depth)
for _,v in ipairs(sets[depth]) do
temp[depth] = v
if (depth==#sets) then prod[#prod+1]=temp:clone() else descend(depth+1) end
end
end
descend(1)
return prod
end
-- demonstration:
tests = {
{ {1, 2}, {3, 4} },
{ {3, 4}, {1, 2} },
{ {1, 2}, {} },
{ {}, {1, 2} },
{ {1776, 1789}, {7, 12}, {4, 14, 23}, {0, 1} },
{ {1, 2, 3}, {30}, {500, 100} },
{ {1, 2, 3}, {}, {500, 100} }
}
for _,test in ipairs(tests) do
local cp = cartprod(test)
print("{"..cp:reduce(function(t,a) return (a=="" and a or a..", ").."("..t:concat(", ")..")" end,"").."}")
end
- Output:
{(1, 3), (1, 4), (2, 3), (2, 4)}
{(3, 1), (3, 2), (4, 1), (4, 2)}
{}
{}
{(1776, 7, 4, 0), (1776, 7, 4, 1), (1776, 7, 14, 0), (1776, 7, 14, 1), (1776, 7, 23, 0), (1776, 7, 23, 1), (1776, 12, 4, 0), (1776, 12, 4, 1), (1776, 12, 14, 0), (1776, 12, 14, 1), (1776, 12, 23, 0), (1776, 12, 23, 1), (1789, 7, 4, 0), (1789, 7, 4, 1), (1789, 7, 14, 0), (1789, 7, 14, 1), (1789, 7, 23, 0), (1789, 7, 23, 1), (1789, 12, 4, 0), (1789, 12, 4, 1), (1789, 12, 14, 0), (1789, 12, 14, 1), (1789, 12, 23, 0), (1789, 12, 23, 1)}
{(1, 30, 500), (1, 30, 100), (2, 30, 500), (2, 30, 100), (3, 30, 500), (3, 30, 100)}
{}
Maple
cartmulti := proc ()
local m, v;
if [] in {args} then
return [];
else
m := Iterator:-CartesianProduct(args);
for v in m do
printf("%{}a\n", v);
end do;
end if;
end proc;Mathematica /Wolfram Language
cartesianProduct[args__] := Flatten[Outer[List, args], Length[{args}] - 1]
Maxima
Using built-in function cartesian_product
cartesian_product({1,2},{3,4});
/* {[1,3],[1,4],[2,3],[2,4]} */
cartesian_product({3,4},{1,2});
/* {[3,1],[3,2],[4,1],[4,2]} */
cartesian_product({1,2},{});
/* {} */
cartesian_product({},{1,2});
/* {} */
Using built-in function cartesian_product_list
cartesian_product_list([1,2],[3,4]);
/* [[1,3],[1,4],[2,3],[2,4]] */
cartesian_product_list([3,4],[1,2]);
/* [[3,1],[3,2],[4,1],[4,2]] */
cartesian_product_list([1,2],[]);
/* [] */
cartesian_product_list([],[1,2]);
/* [] */
Using built-in function create_list
create_list([i,j],i,[1,2],j,[3,4]);
/* [[1,3],[1,4],[2,3],[2,4]] */
create_list([i,j],i,[3,4],j,[1,2]);
/* [[3,1],[3,2],[4,1],[4,2]] */
create_list([i,j],i,[1,2],j,[]);
/* [] */
create_list([i,j],i,[],j,[1,2]);
/* [] */
Extra credit
my_cartesian(lst1,lst2):=create_list([i,j],i,lst1,j,lst2);
n_ary_cartesian(singleargument):=block(lreduce(my_cartesian,singleargument),map(flatten,%%));
[[1776,1789],[7,12],[4,14,23],[0,1]]$
n_ary_cartesian(%);
/* [[1776,7,4,0],[1776,7,4,1],[1776,7,14,0],[1776,7,14,1],[1776,7,23,0],[1776,7,23,1],[1776,12,4,0],[1776,12,4,1],[1776,12,14,0],[1776,12,14,1],[1776,12,23,0],[1776,12,23,1],[1789,7,4,0],[1789,7,4,1],[1789,7,14,0],[1789,7,14,1],[1789,7,23,0],[1789,7,23,1],[1789,12,4,0],[1789,12,4,1],[1789,12,14,0],[1789,12,14,1],[1789,12,23,0],[1789,12,23,1]] */
[[1,2,3],[30],[500,100]]$
n_ary_cartesian(%);
/* [[1,30,500],[1,30,100],[2,30,500],[2,30,100],[3,30,500],[3,30,100]] */
[[1,2,3],[],[500,100]]$
n_ary_cartesian(%);
/* [] */
Modula-2
MODULE CartesianProduct;
FROM FormatString IMPORT FormatString;
FROM Terminal IMPORT WriteString,WriteLn,ReadChar;
PROCEDURE WriteInt(a : INTEGER);
VAR buf : ARRAY[0..9] OF CHAR;
BEGIN
FormatString("%i", buf, a);
WriteString(buf)
END WriteInt;
PROCEDURE Cartesian(a,b : ARRAY OF INTEGER);
VAR i,j : CARDINAL;
BEGIN
WriteString("[");
FOR i:=0 TO HIGH(a) DO
FOR j:=0 TO HIGH(b) DO
IF (i>0) OR (j>0) THEN
WriteString(",");
END;
WriteString("[");
WriteInt(a[i]);
WriteString(",");
WriteInt(b[j]);
WriteString("]")
END
END;
WriteString("]");
WriteLn
END Cartesian;
TYPE
AP = ARRAY[0..1] OF INTEGER;
E = ARRAY[0..0] OF INTEGER;
VAR
a,b : AP;
BEGIN
a := AP{1,2};
b := AP{3,4};
Cartesian(a,b);
a := AP{3,4};
b := AP{1,2};
Cartesian(a,b);
(* If there is a way to create an empty array, I do not know of it *)
ReadChar
END CartesianProduct.
newLISP
(define (cartesian-product lists)
(if (null? lists)
'(())
(let (subproduct (cartesian-product (rest lists)))
(apply append
(map
(fn (x) (map (curry cons x) subproduct))
(first lists))))))
(cartesian-product '((1776 1789) (7 12) (4 14 23) (0 1)))
((1776 7 4 0) (1776 7 4 1) (1776 7 14 0) (1776 7 14 1) (1776 7 23 0)
(1776 7 231) (1776 12 4 0) (1776 12 4 1) (1776 12 14 0) (1776 12 14 1)
(1776 12 23 0) (1776 12 23 1) (1789 7 4 0) (1789 7 4 1) (1789 7 14 0)
(1789 7 14 1) (1789 7 23 0) (1789 7 23 1) (1789 12 4 0) (1789 12 4 1)
(1789 12 14 0) (1789 12 14 1) (1789 12 23 0) (1789 12 23 1))
(cartesian-product '((1 2 3) (30) (500 100)))
((1 30 500) (1 30 100) (2 30 500) (2 30 100) (3 30 500) (3 30 100))
(cartesian-product '((1 2 3) () (500 100)))
()
Nim
Task: product of two lists
To compute the product of two lists (Nim arrays or sequences), we use an iterator. Obtaining a sequence from an iterator is easily done using "toSeq" from the module “sequtils” of the standard library.
The procedure allows to mix sequences of different types, for instance integers and characters.
In order to display the result using mathematical formalism, we have created a special procedure “$$” for the sequences and have overloaded the procedure “$” for tuples.
iterator product[T1, T2](a: openArray[T1]; b: openArray[T2]): tuple[a: T1, b: T2] =
# Yield the element of the cartesian product of "a" and "b".
# Yield tuples rather than arrays as it allows T1 and T2 to be different.
for x in a:
for y in b:
yield (x, y)
#———————————————————————————————————————————————————————————————————————————————————————————————————
when isMainModule:
from seqUtils import toSeq
import strformat
from strutils import addSep
#-------------------------------------------------------------------------------------------------
proc `$`[T1, T2](t: tuple[a: T1, b: T2]): string =
## Overloading of `$` to display a tuple without the field names.
&"({t.a}, {t.b})"
proc `$$`[T](s: seq[T]): string =
## New operator to display a sequence using mathematical set notation.
result = "{"
for item in s:
result.addSep(", ", 1)
result.add($item)
result.add('}')
#-------------------------------------------------------------------------------------------------
const Empty = newSeq[int]() # Empty list of "int".
for (a, b) in [(@[1, 2], @[3, 4]),
(@[3, 4], @[1, 2]),
(@[1, 2], Empty ),
( Empty, @[1, 2])]:
echo &"{$$a} x {$$b} = {$$toSeq(product(a, b))}"
- Output:
1, 2} x {3, 4} = {(1, 3), (1, 4), (2, 3), (2, 4)}
{3, 4} x {1, 2} = {(3, 1), (3, 2), (4, 1), (4, 2)}
{1, 2} x {} = {}
{} x {1, 2} = {}
Extra credit: product of n list
Recursive procedure
As iterators cannot be recursive, we have to use a procedure which returns the whole sequence. And as we don’t know the number of sequences, we use a “varargs”. So all the sequences must contain the same type of values, values which are returned as sequences and not tuples.
Note that there exists in the standard module “algorithm” a procedure which computes the product of sequences of a same type. It is not recursive and, so, likely more efficient that the following version.
proc product[T](a: varargs[seq[T]]): seq[seq[T]] =
## Return the product of several sets (sequences).
if a.len == 1:
for x in a[0]:
result.add(@[x])
else:
for x in a[0]:
for s in product(a[1..^1]):
result.add(x & s)
#———————————————————————————————————————————————————————————————————————————————————————————————————
when isMainModule:
import strformat
let
a = @[1, 2]
b = @[3, 4]
c = @[5, 6]
echo &"{a} x {b} x {c} = {product(a, b, c)}"
- Output:
@[1, 2] x @[3, 4] x @[5, 6] = @[@[1, 3, 5], @[1, 3, 6], @[1, 4, 5], @[1, 4, 6], @[2, 3, 5], @[2, 3, 6], @[2, 4, 5], @[2, 4, 6]]
Using a macro
Another way to compute the product consists to use a macro. It would be possible to create an iterator but it’s somewhat easier to produce the code to build the whole sequence. No recursion here: we generate nested loops, so the algorithm is the simplest possible.
With a macro, we are able to mix several value types: the “varrags” is no longer a problem as being used at compile time it may contain sequences of different types. And we are able to return tuples of n values instead of sequences of n values.
import macros
macro product(args: varargs[typed]): untyped =
## Macro to generate the code to build the product of several sequences.
let t = args[0].getType()
if t.kind != nnkBracketExpr or t[0].kind != nnkSym or $t[0] != "seq":
error("Arguments must be sequences", args)
# Build the result type i.e. a tuple with "args.len" elements.
# Fields are named "f0", "f1", etc.
let tupleTyNode = newNimNode(nnkTupleTy)
for idx, arg in args:
let identDefsNode = newIdentDefs(ident('f' & $idx), arg.getType()[1])
tupleTyNode.add(identDefsNode)
# Build the nested for loops with counter "i0", "i1", etc.
var stmtListNode = newStmtList()
let loopsNode = nnkForStmt.newTree(ident("i0"), ident($args[0]), stmtListNode)
var idx = 0
for arg in args[1..^1]:
inc idx
let loopNode = nnkForStmt.newTree(ident('i' & $idx), ident($arg))
stmtListNode.add(loopNode)
stmtListNode = newStmtList()
loopNode.add(stmtListNode)
# Build the instruction "result.add(i1, i2,...)".
let parNode = newPar()
let addNode = newCall(newDotExpr(ident("result"), ident("add")), parNode)
for i, arg in args:
parNode.add(ident('i' & $i))
stmtListNode.add(addNode)
# Build the tree.
result = nnkStmtListExpr.newTree(
nnkVarSection.newTree(
newIdentDefs(
ident("result"),
nnkBracketExpr.newTree(ident("seq"), tupleTyNode))),
loopsNode,
ident("result"))
#———————————————————————————————————————————————————————————————————————————————————————————————————
when isMainModule:
import strformat
import strutils
#-------------------------------------------------------------------------------------------------
proc `$`[T: tuple](t: T): string =
## Overloading of `$` to display a tuple without the field names.
result = "("
for f in t.fields:
result.addSep(", ", 1)
result.add($f)
result.add(']')
proc `$$`[T](s: seq[T]): string =
## New operator to display a sequence using mathematical set notation.
result = "{"
for item in s:
result.addSep(", ", 1)
result.add($item)
result.add('}')
#-------------------------------------------------------------------------------------------------
var a = @[1, 2]
var b = @['a', 'b']
var c = @[false, true]
echo &"{$$a} x {$$b} x {$$c} = {$$product(a, b, c)}"
- Output:
{1, 2} x {a, b} x {false, true} = {(1, a, false], (1, a, true], (1, b, false], (1, b, true], (2, a, false], (2, a, true], (2, b, false], (2, b, true]}
OCaml
The double semicolons are necessary only for the toplevel
Naive but more readable version
let rec product l1 l2 =
match l1, l2 with
| [], _ | _, [] -> []
| h1::t1, h2::t2 -> (h1,h2)::(product [h1] t2)@(product t1 l2)
;;
product [1;2] [3;4];;
(*- : (int * int) list = [(1, 3); (1, 4); (2, 3); (2, 4)]*)
product [3;4] [1;2];;
(*- : (int * int) list = [(3, 1); (3, 2); (4, 1); (4, 2)]*)
product [1;2] [];;
(*- : (int * 'a) list = []*)
product [] [1;2];;
(*- : ('a * int) list = []*)
Implementation with a bit more tail-call optimization, introducing a helper function. The order of the result is changed but it should not be an issue for most uses.
let product' l1 l2 =
let rec aux ~acc l1' l2' =
match l1', l2' with
| [], _ | _, [] -> acc
| h1::t1, h2::t2 ->
let acc = (h1,h2)::acc in
let acc = aux ~acc t1 l2' in
aux ~acc [h1] t2
in aux [] l1 l2
;;
product' [1;2] [3;4];;
(*- : (int * int) list = [(1, 4); (2, 4); (2, 3); (1, 3)]*)
product' [3;4] [1;2];;
(*- : (int * int) list = [(3, 2); (4, 2); (4, 1); (3, 1)]*)
product' [1;2] [];;
(*- : (int * 'a) list = []*)
product' [] [1;2];;
(*- : ('a * int) list = []*)
Implemented using nested folds:
let cart_prod l1 l2 =
List.fold_left (fun acc1 ele1 ->
List.fold_left (fun acc2 ele2 -> (ele1,ele2)::acc2) acc1 l2) [] l1 ;;
cart_prod [1; 2; 3] ['a'; 'b'; 'c'] ;;
(*- : (int * char) list = [(3, 'c'); (3, 'b'); (3, 'a'); (2, 'c'); (2, 'b'); (2, 'a'); (1, 'c'); (1, 'b'); (1, 'a')]*)
cart_prod [1; 2; 3] [] ;;
(*- : ('a * int) list = [] *)
Extra credit function. Since in OCaml a function can return only one type, and because tuples of different arities are different types, this returns a list of lists rather than a list of tuples. Since lists are homogeneous this version is restricted to products over a single type, eg integers.
let rec product'' l =
(* We need to do the cross product of our current list and all the others
* so we define a helper function for that *)
let rec aux ~acc l1 l2 = match l1, l2 with
| [], _ | _, [] -> acc
| h1::t1, h2::t2 ->
let acc = (h1::h2)::acc in
let acc = (aux ~acc t1 l2) in
aux ~acc [h1] t2
(* now we can do the actual computation *)
in match l with
| [] -> []
| [l1] -> List.map (fun x -> [x]) l1
| l1::tl ->
let tail_product = product'' tl in
aux ~acc:[] l1 tail_product
product'' [[1;2];[3;4]];;
(*- : int list list = [[1; 4]; [2; 4]; [2; 3]; [1; 3]]*)
product'' [[3;4];[1;2]];;
(*- : int list list = [[3; 2]; [4; 2]; [4; 1]; [3; 1]]*)
product'' [[1;2];[]];;
(*- : int list list = []*)
product'' [[];[1;2]];;
(*- : int list list = []*)
product'' [[1776; 1789];[7; 12];[4; 14; 23];[0; 1]];;
(*
- : int list list =
[[1776; 7; 4; 1]; [1776; 12; 4; 1]; [1776; 12; 14; 1]; [1776; 12; 23; 1];
[1776; 12; 23; 0]; [1776; 12; 14; 0]; [1776; 12; 4; 0]; [1776; 7; 14; 1];
[1776; 7; 23; 1]; [1776; 7; 23; 0]; [1776; 7; 14; 0]; [1789; 7; 4; 1];
[1789; 12; 4; 1]; [1789; 12; 14; 1]; [1789; 12; 23; 1]; [1789; 12; 23; 0];
[1789; 12; 14; 0]; [1789; 12; 4; 0]; [1789; 7; 14; 1]; [1789; 7; 23; 1];
[1789; 7; 23; 0]; [1789; 7; 14; 0]; [1789; 7; 4; 0]; [1776; 7; 4; 0]]
*)
product'' [[1; 2; 3];[30];[500; 100]];;
(*
- : int list list =
[[1; 30; 500]; [2; 30; 500]; [3; 30; 500]; [3; 30; 100]; [2; 30; 100];
[1; 30; 100]]
*)
product'' [[1; 2; 3];[];[500; 100]];;
(*- : int list list = []*)
Better type
In the latter example, our function has this signature:
val product'' : 'a list list -> 'a list list = <fun>
This lacks clarity as those two lists are not equivalent since one replaces a tuple. We can get a better signature by creating a tuple type:
type 'a tuple = 'a list
let rec product'' (l:'a list tuple) =
(* We need to do the cross product of our current list and all the others
* so we define a helper function for that *)
let rec aux ~acc l1 l2 = match l1, l2 with
| [], _ | _, [] -> acc
| h1::t1, h2::t2 ->
let acc = (h1::h2)::acc in
let acc = (aux ~acc t1 l2) in
aux ~acc [h1] t2
(* now we can do the actual computation *)
in match l with
| [] -> []
| [l1] -> List.map ~f:(fun x -> ([x]:'a tuple)) l1
| l1::tl ->
let tail_product = product'' tl in
aux ~acc:[] l1 tail_product
;;
type 'a tuple = 'a list
val product'' : 'a list tuple -> 'a tuple list = <fun>
PascalABC.NET
##
var empty := new List<integer>;
println(cartesian(|1, 2|, |3, 4|));
println(cartesian(|3, 4|, |1, 2|));
println(cartesian(|1, 2|, empty));
println(cartesian(empty, |1, 2|));
println(cartesian(|1776, 1789|, |7, 12|, |4, 14, 23|, |0, 1|));
println(cartesian(|1, 2, 3|, |30|, |500, 100|));
println(cartesian(|1, 2, 3|, empty, |500, 100|));
- Output:
[(1,3),(1,4),(2,3),(2,4)] [(3,1),(3,2),(4,1),(4,2)] [] [] [(1776,7,4,0),(1776,7,4,1),(1776,7,14,0),(1776,7,14,1),(1776,7,23,0),(1776,7,23,1),(1776,12,4,0),(1776,12,4,1),(1776,12,14,0),(1776,12,14,1),(1776,12,23,0),(1776,12,23,1),(1789,7,4,0),(1789,7,4,1),(1789,7,14,0),(1789,7,14,1),(1789,7,23,0),(1789,7,23,1),(1789,12,4,0),(1789,12,4,1),(1789,12,14,0),(1789,12,14,1),(1789,12,23,0),(1789,12,23,1)] [(1,30,500),(1,30,100),(2,30,500),(2,30,100),(3,30,500),(3,30,100)] []
Perl
Iterative
Nested loops, with a short-circuit to quit early if any term is an empty set.
sub cartesian {
my $sets = shift @_;
for (@$sets) { return [] unless @$_ }
my $products = [[]];
for my $set (reverse @$sets) {
my $partial = $products;
$products = [];
for my $item (@$set) {
for my $product (@$partial) {
push @$products, [$item, @$product];
}
}
}
$products;
}
sub product {
my($s,$fmt) = @_;
my $tuples;
for $a ( @{ cartesian( \@$s ) } ) { $tuples .= sprintf "($fmt) ", @$a; }
$tuples . "\n";
}
print
product([[1, 2], [3, 4] ], '%1d %1d' ).
product([[3, 4], [1, 2] ], '%1d %1d' ).
product([[1, 2], [] ], '%1d %1d' ).
product([[], [1, 2] ], '%1d %1d' ).
product([[1,2,3], [30], [500,100] ], '%1d %1d %3d' ).
product([[1,2,3], [], [500,100] ], '%1d %1d %3d' ).
product([[1776,1789], [7,12], [4,14,23], [0,1]], '%4d %2d %2d %1d')
- Output:
(1 3) (1 4) (2 3) (2 4) (3 1) (3 2) (4 1) (4 2) (1 30 500) (1 30 100) (2 30 500) (2 30 100) (3 30 500) (3 30 100) (1776 7 4 0) (1776 7 4 1) (1776 7 14 0) (1776 7 14 1) (1776 7 23 0) (1776 7 23 1) (1776 12 4 0) (1776 12 4 1) (1776 12 14 0) (1776 12 14 1) (1776 12 23 0) (1776 12 23 1) (1789 7 4 0) (1789 7 4 1) (1789 7 14 0) (1789 7 14 1) (1789 7 23 0) (1789 7 23 1) (1789 12 4 0) (1789 12 4 1) (1789 12 14 0) (1789 12 14 1) (1789 12 23 0) (1789 12 23 1)
Glob
This being Perl, there's more than one way to do it. A quick demonstration of how glob, more typically used for filename wildcard expansion, can solve the task.
$tuples = [ map { [split /:/] } glob '{1,2,3}:{30}:{500,100}' ];
for $a (@$tuples) { printf "(%1d %2d %3d) ", @$a; }
- Output:
(1 30 500) (1 30 100) (2 30 500) (2 30 100) (3 30 500) (3 30 100)
Modules
A variety of modules can do this correctly for an arbitrary number of lists (each of independent length). Arguably using modules is very idiomatic Perl.
use ntheory qw/forsetproduct/;
forsetproduct { say "@_" } [1,2,3],[qw/a b c/],[qw/@ $ !/];
use Set::Product qw/product/;
product { say "@_" } [1,2,3],[qw/a b c/],[qw/@ $ !/];
use Math::Cartesian::Product;
cartesian { say "@_" } [1,2,3],[qw/a b c/],[qw/@ $ !/];
use Algorithm::Loops qw/NestedLoops/;
NestedLoops([[1,2,3],[qw/a b c/],[qw/@ $ !/]], sub { say "@_"; });
Phix
with javascript_semantics function cart(sequence s) sequence res = {} for n=2 to length(s) do for i=1 to length(s[1]) do for j=1 to length(s[2]) do res = append(res,s[1][i]&s[2][j]) end for end for if length(s)=2 then exit end if s[1..2] = {res} res = {} end for return res end function ?cart({{1,2},{3,4}}) ?cart({{3,4},{1,2}}) ?cart({{1,2},{}}) ?cart({{},{1,2}}) ?cart({{1776, 1789},{7, 12},{4, 14, 23},{0, 1}}) ?cart({{1, 2, 3},{30},{500, 100}}) ?cart({{1, 2, 3},{},{500, 100}})
- Output:
{{1,3},{1,4},{2,3},{2,4}}
{{3,1},{3,2},{4,1},{4,2}}
{}
{}
{{1776,7,4,0},{1776,7,4,1},{1776,7,14,0},{1776,7,14,1},{1776,7,23,0},{1776,7,23,1},
{1776,12,4,0},{1776,12,4,1},{1776,12,14,0},{1776,12,14,1},{1776,12,23,0},{1776,12,23,1},
{1789,7,4,0},{1789,7,4,1},{1789,7,14,0},{1789,7,14,1},{1789,7,23,0},{1789,7,23,1},
{1789,12,4,0},{1789,12,4,1},{1789,12,14,0},{1789,12,14,1},{1789,12,23,0},{1789,12,23,1}}
{{1,30,500},{1,30,100},{2,30,500},{2,30,100},{3,30,500},{3,30,100}}
{}
Phixmonti
include ..\Utilitys.pmt
def cart
( ) var res
-1 get var ta -1 del
-1 get var he -1 del
ta "" != he "" != and if
he len nip for
he swap get var h drop
ta len nip for
ta swap get var t drop
( h t ) flatten res swap 0 put var res
endfor
endfor
len if res 0 put cart endif
endif
enddef
/# ---------- MAIN ---------- #/
( ( 1 2 ) ( 3 4 ) ) cart
drop res print nl nl
( ( 1776 1789 ) ( 7 12 ) ( 4 14 23 ) ( 0 1 ) ) cart
drop res print nl nl
( ( 1 2 3 ) ( 30 ) ( 500 100 ) ) cart
drop res print nl nl
( ( 1 2 ) ( ) ) cart
drop res print nl nlPicoLisp
(de 2lists (L1 L2)
(mapcan
'((I)
(mapcar
'((A) ((if (atom A) list cons) I A))
L2 ) )
L1 ) )
(de reduce (L . @)
(ifn (rest) L (2lists L (apply reduce (rest)))) )
(de cartesian (L . @)
(and L (rest) (pass reduce L)) )
(println
(cartesian (1 2)) )
(println
(cartesian NIL (1 2)) )
(println
(cartesian (1 2) (3 4)) )
(println
(cartesian (3 4) (1 2)) )
(println
(cartesian (1776 1789) (7 12) (4 14 23) (0 1)) )
(println
(cartesian (1 2 3) (30) (500 100)) )
(println
(cartesian (1 2 3) NIL (500 100)) )- Output:
NIL NIL ((1 3) (1 4) (2 3) (2 4)) ((3 1) (3 2) (4 1) (4 2)) ((1776 7 4 0) (1776 7 4 1) (1776 7 14 0) (1776 7 14 1) (1776 7 23 0) (1776 7 23 1) (1776 12 4 0) (1776 12 4 1) (1776 12 14 0) (1776 12 14 1) (1776 12 23 0) (1776 12 23 1) (1789 7 4 0) (1789 7 4 1) (1789 7 14 0) (1789 7 14 1) (1789 7 23 0) (1789 7 23 1) (1789 12 4 0) (1789 12 4 1) (1789 12 14 0) (1789 12 14 1) (1789 12 23 0) (1789 12 23 1)) ((1 30 500) (1 30 100) (2 30 500) (2 30 100) (3 30 500) (3 30 100)) NIL
Prolog
product([A|_], Bs, [A, B]) :- member(B, Bs).
product([_|As], Bs, X) :- product(As, Bs, X).
- Output:
?- findall(X, product([1,2],[3,4],X), S). S = [[1, 3], [1, 4], [2, 3], [2, 4]]. ?- findall(X, product([3,4],[1,2],X), S). S = [[3, 1], [3, 2], [4, 1], [4, 2]]. ?- findall(X, product([1,2,3],[],X), S). S = []. ?- findall(X, product([],[1,2,3],X), S). S = [].
Python
Using itertools
import itertools
def cp(lsts):
return list(itertools.product(*lsts))
if __name__ == '__main__':
from pprint import pprint as pp
for lists in [[[1,2],[3,4]], [[3,4],[1,2]], [[], [1, 2]], [[1, 2], []],
((1776, 1789), (7, 12), (4, 14, 23), (0, 1)),
((1, 2, 3), (30,), (500, 100)),
((1, 2, 3), (), (500, 100))]:
print(lists, '=>')
pp(cp(lists), indent=2)
- Output:
[[1, 2], [3, 4]] => [(1, 3), (1, 4), (2, 3), (2, 4)] [[3, 4], [1, 2]] => [(3, 1), (3, 2), (4, 1), (4, 2)] [[], [1, 2]] => [] [[1, 2], []] => [] ((1776, 1789), (7, 12), (4, 14, 23), (0, 1)) => [ (1776, 7, 4, 0), (1776, 7, 4, 1), (1776, 7, 14, 0), (1776, 7, 14, 1), (1776, 7, 23, 0), (1776, 7, 23, 1), (1776, 12, 4, 0), (1776, 12, 4, 1), (1776, 12, 14, 0), (1776, 12, 14, 1), (1776, 12, 23, 0), (1776, 12, 23, 1), (1789, 7, 4, 0), (1789, 7, 4, 1), (1789, 7, 14, 0), (1789, 7, 14, 1), (1789, 7, 23, 0), (1789, 7, 23, 1), (1789, 12, 4, 0), (1789, 12, 4, 1), (1789, 12, 14, 0), (1789, 12, 14, 1), (1789, 12, 23, 0), (1789, 12, 23, 1)] ((1, 2, 3), (30,), (500, 100)) => [ (1, 30, 500), (1, 30, 100), (2, 30, 500), (2, 30, 100), (3, 30, 500), (3, 30, 100)] ((1, 2, 3), (), (500, 100)) => []
Using the 'Applicative' abstraction
This task calls for alternative approaches to defining cartesian products, and one particularly compact alternative route to a native cartesian product (in a more mathematically reasoned idiom of programming) is through the Applicative abstraction (see Applicative Functor), which is slightly more general than the possibly better known monad structure. Applicative functions are provided off-the-shelf by languages like Agda, Idris, Haskell and Scala, and can usefully be implemented in any language, including Python, which supports higher-order functions.
If we write ourselves a re-usable Python ap function for the case of lists (applicative functions for other 'data containers' can also be written – this one applies a list of functions to a list of values):
# ap (<*>) :: [(a -> b)] -> [a] -> [b]
def ap(fs):
return lambda xs: foldl(
lambda a: lambda f: a + foldl(
lambda a: lambda x: a + [f(x)])([])(xs)
)([])(fs)
then one simple use of it will be to define the cartesian product of two lists (of possibly different type) as:
ap(map(Tuple, xs))
where Tuple is a constructor, and xs is bound to the first of two lists. The returned value is a function which can be applied to a second list.
For an nAry product, we can then use a fold (catamorphism) to lift the basic function over two lists cartesianProduct :: [a] -> [b] -> [(a, b)] to a function over a list of lists:
# nAryCartProd :: [[a], [b], [c] ...] -> [(a, b, c ...)]
def nAryCartProd(xxs):
return foldl1(cartesianProduct)(
xxs
)
For example:
# Two lists -> list of tuples
# cartesianProduct :: [a] -> [b] -> [(a, b)]
def cartesianProduct(xs):
return ap(map(Tuple, xs))
# List of lists -> list of tuples
# nAryCartProd :: [[a], [b], [c] ...] -> [(a, b, c ...)]
def nAryCartProd(xxs):
return foldl1(cartesianProduct)(
xxs
)
# main :: IO ()
def main():
# Product of lists of different types
print (
'Product of two lists of different types:'
)
print(
cartesianProduct(['a', 'b', 'c'])(
[1, 2]
)
)
# TESTS OF PRODUCTS OF TWO LISTS
print(
'\nSpecified tests of products of two lists:'
)
print(
cartesianProduct([1, 2])([3, 4]),
' <--> ',
cartesianProduct([3, 4])([1, 2])
)
print (
cartesianProduct([1, 2])([]),
' <--> ',
cartesianProduct([])([1, 2])
)
# TESTS OF N-ARY CARTESIAN PRODUCTS
print('\nSpecified tests of nAry products:')
for xs in nAryCartProd([[1776, 1789], [7, 12], [4, 14, 23], [0, 1]]):
print(xs)
for xs in (
map_(nAryCartProd)(
[
[[1, 2, 3], [30], [500, 100]],
[[1, 2, 3], [], [500, 100]]
]
)
):
print(
xs
)
# GENERIC -------------------------------------------------
# Applicative function for lists
# ap (<*>) :: [(a -> b)] -> [a] -> [b]
def ap(fs):
return lambda xs: foldl(
lambda a: lambda f: a + foldl(
lambda a: lambda x: a + [f(x)])([])(xs)
)([])(fs)
# foldl :: (a -> b -> a) -> a -> [b] -> a
def foldl(f):
def go(v, xs):
a = v
for x in xs:
a = f(a)(x)
return a
return lambda acc: lambda xs: go(acc, xs)
# foldl1 :: (a -> a -> a) -> [a] -> a
def foldl1(f):
return lambda xs: foldl(f)(xs[0])(
xs[1:]
) if xs else None
# map :: (a -> b) -> [a] -> [b]
def map_(f):
return lambda xs: list(map(f, xs))
# Tuple :: a -> b -> (a, b)
def Tuple(x):
return lambda y: (
x + (y,)
) if tuple is type(x) else (x, y)
# TEST ----------------------------------------------------
if __name__ == '__main__':
main()
- Output:
Product of two lists of different types:
[('a', 1), ('a', 2), ('b', 1), ('b', 2), ('c', 1), ('c', 2)]
Specified tests of products of two lists:
[(1, 3), (1, 4), (2, 3), (2, 4)] <--> [(3, 1), (3, 2), (4, 1), (4, 2)]
[] <--> []
Specified tests of nAry products:
(1776, 7, 4, 0)
(1776, 7, 4, 1)
(1776, 7, 14, 0)
(1776, 7, 14, 1)
(1776, 7, 23, 0)
(1776, 7, 23, 1)
(1776, 12, 4, 0)
(1776, 12, 4, 1)
(1776, 12, 14, 0)
(1776, 12, 14, 1)
(1776, 12, 23, 0)
(1776, 12, 23, 1)
(1789, 7, 4, 0)
(1789, 7, 4, 1)
(1789, 7, 14, 0)
(1789, 7, 14, 1)
(1789, 7, 23, 0)
(1789, 7, 23, 1)
(1789, 12, 4, 0)
(1789, 12, 4, 1)
(1789, 12, 14, 0)
(1789, 12, 14, 1)
(1789, 12, 23, 0)
(1789, 12, 23, 1)
[(1, 30, 500), (1, 30, 100), (2, 30, 500), (2, 30, 100), (3, 30, 500), (3, 30, 100)]
[]
Quackery
[ [] unrot
swap witheach
[ over witheach
[ over nested
swap nested join
nested dip rot join
unrot ]
drop ] drop ] is cartprod ( [ [ --> [ )
' [ 1 2 ] ' [ 3 4 ] cartprod echo cr
' [ 3 4 ] ' [ 1 2 ] cartprod echo cr
' [ 1 2 ] ' [ ] cartprod echo cr
' [ ] ' [ 1 2 ] cartprod echo cr- Output:
[ [ 1 3 ] [ 1 4 ] [ 2 3 ] [ 2 4 ] ] [ [ 3 1 ] [ 3 2 ] [ 4 1 ] [ 4 2 ] ] [ ] [ ]
R
one_w_many <- function(one, many) lapply(many, function(x) c(one,x))
# Let's define an infix operator to perform a cartesian product.
"%p%" <- function( a, b ) {
p = c( sapply(a, function (x) one_w_many(x, b) ) )
if (is.null(unlist(p))) list() else p}
display_prod <-
function (xs) { for (x in xs) cat( paste(x, collapse=", "), "\n" ) }
fmt_vec <- function(v) sprintf("(%s)", paste(v, collapse=', '))
go <- function (...) {
cat("\n", paste( sapply(list(...),fmt_vec), collapse=" * "), "\n")
prod = Reduce( '%p%', list(...) )
display_prod( prod ) }
Simple tests:
> display_prod( c(1, 2) %p% c(3, 4) )
1, 3
1, 4
2, 3
2, 4
> display_prod( c(3, 4) %p% c(1, 2) )
3, 1
3, 2
4, 1
4, 2
> display_prod( c(3, 4) %p% c() )
>
Tougher tests:
go( c(1776, 1789), c(7, 12), c(4, 14, 23), c(0, 1) )
go( c(1, 2, 3), c(30), c(500, 100) )
go( c(1, 2, 3), c(), c(500, 100) )
- Output:
(1776, 1789) * (7, 12) * (4, 14, 23) * (0, 1) 1776, 7, 4, 0 1776, 7, 4, 1 1776, 7, 14, 0 1776, 7, 14, 1 1776, 7, 23, 0 1776, 7, 23, 1 1776, 12, 4, 0 1776, 12, 4, 1 1776, 12, 14, 0 1776, 12, 14, 1 1776, 12, 23, 0 1776, 12, 23, 1 1789, 7, 4, 0 1789, 7, 4, 1 1789, 7, 14, 0 1789, 7, 14, 1 1789, 7, 23, 0 1789, 7, 23, 1 1789, 12, 4, 0 1789, 12, 4, 1 1789, 12, 14, 0 1789, 12, 14, 1 1789, 12, 23, 0 1789, 12, 23, 1 (1, 2, 3) * (30) * (500, 100) 1, 30, 500 1, 30, 100 2, 30, 500 2, 30, 100 3, 30, 500 3, 30, 100 (1, 2, 3) * () * (500, 100)
Racket
Racket has a built-in "cartesian-product" function:
#lang racket/base
(require rackunit
;; usually, included in "racket", but we're using racket/base so we
;; show where this comes from
(only-in racket/list cartesian-product))
;; these tests will pass silently
(check-equal? (cartesian-product '(1 2) '(3 4))
'((1 3) (1 4) (2 3) (2 4)))
(check-equal? (cartesian-product '(3 4) '(1 2))
'((3 1) (3 2) (4 1) (4 2)))
(check-equal? (cartesian-product '(1 2) '()) '())
(check-equal? (cartesian-product '() '(1 2)) '())
;; these will print
(cartesian-product '(1776 1789) '(7 12) '(4 14 23) '(0 1))
(cartesian-product '(1 2 3) '(30) '(500 100))
(cartesian-product '(1 2 3) '() '(500 100))
- Output:
'((1776 7 4 0) (1776 7 4 1) (1776 7 14 0) (1776 7 14 1) (1776 7 23 0) (1776 7 23 1) (1776 12 4 0) (1776 12 4 1) (1776 12 14 0) (1776 12 14 1) (1776 12 23 0) (1776 12 23 1) (1789 7 4 0) (1789 7 4 1) (1789 7 14 0) (1789 7 14 1) (1789 7 23 0) (1789 7 23 1) (1789 12 4 0) (1789 12 4 1) (1789 12 14 0) (1789 12 14 1) (1789 12 23 0) (1789 12 23 1)) '((1 30 500) (1 30 100) (2 30 500) (2 30 100) (3 30 500) (3 30 100)) '()
Raku
(formerly Perl 6)
The cross meta operator X will return the cartesian product of two lists. To apply the cross meta-operator to a variable number of lists, use the reduce cross meta operator [X].
# cartesian product of two lists using the X cross meta-operator
say (1, 2) X (3, 4);
say (3, 4) X (1, 2);
say (1, 2) X ( );
say ( ) X ( 1, 2 );
# cartesian product of variable number of lists using
# the [X] reduce cross meta-operator
say [X] (1776, 1789), (7, 12), (4, 14, 23), (0, 1);
say [X] (1, 2, 3), (30), (500, 100);
say [X] (1, 2, 3), (), (500, 100);
- Output:
((1 3) (1 4) (2 3) (2 4)) ((3 1) (3 2) (4 1) (4 2)) () () ((1776 7 4 0) (1776 7 4 1) (1776 7 14 0) (1776 7 14 1) (1776 7 23 0) (1776 7 23 1) (1776 12 4 0) (1776 12 4 1) (1776 12 14 0) (1776 12 14 1) (1776 12 23 0) (1776 12 23 1) (1789 7 4 0) (1789 7 4 1) (1789 7 14 0) (1789 7 14 1) (1789 7 23 0) (1789 7 23 1) (1789 12 4 0) (1789 12 4 1) (1789 12 14 0) (1789 12 14 1) (1789 12 23 0) (1789 12 23 1)) ((1 30 500) (1 30 100) (2 30 500) (2 30 100) (3 30 500) (3 30 100)) ()
REXX
version 1
This REXX version isn't limited by the number of lists or the number of sets within a list.
/*REXX program calculates the Cartesian product of two arbitrary-sized lists. */
@.= /*assign the default value to @. array*/
parse arg @.1 /*obtain the optional value of @.1 */
if @.1='' then do; @.1= "{1,2} {3,4}" /*Not specified? Then use the defaults*/
@.2= "{3,4} {1,2}" /* " " " " " " */
@.3= "{1,2} {}" /* " " " " " " */
@.4= "{} {3,4}" /* " " " " " " */
@.5= "{1,2} {3,4,5}" /* " " " " " " */
end
/* [↓] process each of the @.n values*/
do n=1 while @.n \= '' /*keep processing while there's a value*/
z= translate( space( @.n, 0), , ',') /*translate the commas to blanks. */
do #=1 until z=='' /*process each elements in first list. */
parse var z '{' x.# '}' z /*parse the list (contains elements). */
end /*#*/
$=
do i=1 for #-1 /*process the subsequent lists. */
do a=1 for words(x.i) /*obtain the elements of the first list*/
do j=i+1 for #-1 /* " " subsequent lists. */
do b=1 for words(x.j) /* " " elements of subsequent list*/
$=$',('word(x.i, a)","word(x.j, b)')' /*append partial Cartesian product ──►$*/
end /*b*/
end /*j*/
end /*a*/
end /*i*/
say 'Cartesian product of ' space(@.n) " is ───► {"substr($, 2)'}'
end /*n*/ /*stick a fork in it, we're all done. */
- output when using the default lists:
Cartesian product of {1,2} {3,4} is ───► {(1,3),(1,4),(2,3),(2,4)}
Cartesian product of {3,4} {1,2} is ───► {(3,1),(3,2),(4,1),(4,2)}
Cartesian product of {1,2} {} is ───► {}
Cartesian product of {} {3,4} is ───► {}
Cartesian product of {1,2} {3,4,5} is ───► {(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)}
version 2
/* REXX computes the Cartesian Product of up to 4 sets */
Call cart '{1, 2} x {3, 4}'
Call cart '{3, 4} x {1, 2}'
Call cart '{1, 2} x {}'
Call cart '{} x {1, 2}'
Call cart '{1776, 1789} x {7, 12} x {4, 14, 23} x {0, 1}'
Call cart '{1, 2, 3} x {30} x {500, 100}'
Call cart '{1, 2, 3} x {} x {500, 100}'
Exit
cart:
Parse Arg sl
Say sl
Do i=1 By 1 while pos('{',sl)>0
Parse Var sl '{' list '}' sl
Do j=1 By 1 While list<>''
Parse Var list e.i.j . ',' list
End
n.i=j-1
If n.i=0 Then Do /* an empty set */
Say '{}'
Say ''
Return
End
End
n=i-1
ct2.=0
Do i=1 To n.1
Do j=1 To n.2
z=ct2.0+1
ct2.z=e.1.i e.2.j
ct2.0=z
End
End
If n<3 Then
Return output(2)
ct3.=0
Do i=1 To ct2.0
Do k=1 To n.3
z=ct3.0+1
ct3.z=ct2.i e.3.k
ct3.0=z
End
End
If n<4 Then
Return output(3)
ct4.=0
Do i=1 To ct3.0
Do l=1 To n.4
z=ct4.0+1
ct4.z=ct3.i e.4.l
ct4.0=z
End
End
Return output(4)
output:
Parse Arg u
Do v=1 To value('ct'u'.0')
res='{'translate(value('ct'u'.'v),',',' ')'}'
Say res
End
Say ' '
Return 0
- Output:
{1, 2} x {3, 4}
{1,3}
{1,4}
{2,3}
{2,4}
{3, 4} x {1, 2}
{3,1}
{3,2}
{4,1}
{4,2}
{1, 2} x {}
{}
{} x {1, 2}
{}
{1776, 1789} x {7, 12} x {4, 14, 23} x {0, 1}
{1776,7,4,0}
{1776,7,4,1}
{1776,7,14,0}
{1776,7,14,1}
{1776,7,23,0}
{1776,7,23,1}
{1776,12,4,0}
{1776,12,4,1}
{1776,12,14,0}
{1776,12,14,1}
{1776,12,23,0}
{1776,12,23,1}
{1789,7,4,0}
{1789,7,4,1}
{1789,7,14,0}
{1789,7,14,1}
{1789,7,23,0}
{1789,7,23,1}
{1789,12,4,0}
{1789,12,4,1}
{1789,12,14,0}
{1789,12,14,1}
{1789,12,23,0}
{1789,12,23,1}
{1, 2, 3} x {30} x {500, 100}
{1,30,500}
{1,30,100}
{2,30,500}
{2,30,100}
{3,30,500}
{3,30,100}
{1, 2, 3} x {} x {500, 100}
{}
Ring
# Project : Cartesian product of two or more lists
list1 = [[1,2],[3,4]]
list2 = [[3,4],[1,2]]
cartesian(list1)
cartesian(list2)
func cartesian(list1)
for n = 1 to len(list1[1])
for m = 1 to len(list1[2])
see "(" + list1[1][n] + ", " + list1[2][m] + ")" + nl
next
next
see nlOutput:
(1, 3) (1, 4) (2, 3) (2, 4) (3, 1) (3, 2) (4, 1) (4, 2)
RPL
≪ → a b
≪ { }
IF a SIZE b SIZE AND THEN
1 a SIZE FOR j
1 b SIZE FOR k
a j GET b k GET 2 →LIST 1 →LIST +
NEXT
NEXT
END
≫ ≫ 'CROIX' STO
{1 2} {3 4} CROIX
{3 4} {1 2} CROIX
{1 2} {} CROIX
{} {1 2} CROIX
- Output:
4: {(1 3) (1 4) (2 3) (2 4)}
3: {(3 1) (3 2) (4 1) (4 2)}
2: {}
1: {}
Ruby
"product" is a method of arrays. It takes one or more arrays as argument and results in the Cartesian product:
p [1, 2].product([3, 4])
p [3, 4].product([1, 2])
p [1, 2].product([])
p [].product([1, 2])
p [1776, 1789].product([7, 12], [4, 14, 23], [0, 1])
p [1, 2, 3].product([30], [500, 100])
p [1, 2, 3].product([], [500, 100])
- Output:
[[1, 3], [1, 4], [2, 3], [2, 4]][[3, 1], [3, 2], [4, 1], [4, 2]] [] [] [[1776, 7, 4, 0], [1776, 7, 4, 1], [1776, 7, 14, 0], [1776, 7, 14, 1], [1776, 7, 23, 0], [1776, 7, 23, 1], [1776, 12, 4, 0], [1776, 12, 4, 1], [1776, 12, 14, 0], [1776, 12, 14, 1], [1776, 12, 23, 0], [1776, 12, 23, 1], [1789, 7, 4, 0], [1789, 7, 4, 1], [1789, 7, 14, 0], [1789, 7, 14, 1], [1789, 7, 23, 0], [1789, 7, 23, 1], [1789, 12, 4, 0], [1789, 12, 4, 1], [1789, 12, 14, 0], [1789, 12, 14, 1], [1789, 12, 23, 0], [1789, 12, 23, 1]] [[1, 30, 500], [1, 30, 100], [2, 30, 500], [2, 30, 100], [3, 30, 500], [3, 30, 100]] []
Rust
Specific Version
Type-Specific/No Generics/Fixed Input.
fn cartesian_product(lists: &Vec<Vec<u32>>) -> Vec<Vec<u32>> {
let mut res = vec![];
let mut list_iter = lists.iter();
if let Some(first_list) = list_iter.next() {
for &i in first_list {
res.push(vec![i]);
}
}
for l in list_iter {
let mut tmp = vec![];
for r in res {
for &el in l {
let mut tmp_el = r.clone();
tmp_el.push(el);
tmp.push(tmp_el);
}
}
res = tmp;
}
res
}
fn main() {
let cases = vec![
vec![vec![1, 2], vec![3, 4]],
vec![vec![3, 4], vec![1, 2]],
vec![vec![1, 2], vec![]],
vec![vec![], vec![1, 2]],
vec![vec![1776, 1789], vec![7, 12], vec![4, 14, 23], vec![0, 1]],
vec![vec![1, 2, 3], vec![30], vec![500, 100]],
vec![vec![1, 2, 3], vec![], vec![500, 100]],
];
for case in cases {
println!(
"{}\n{:?}\n",
case.iter().map(|c| format!("{:?}", c)).collect::<Vec<_>>().join(" × "),
cartesian_product(&case)
)
}
}
- Output:
[1, 2] × [3, 4] [[1, 3], [1, 4], [2, 3], [2, 4]] [3, 4] × [1, 2] [[3, 1], [3, 2], [4, 1], [4, 2]] [1, 2] × [] [] [] × [1, 2] [] [1776, 1789] × [7, 12] × [4, 14, 23] × [0, 1] [[1776, 7, 4, 0], [1776, 7, 4, 1], [1776, 7, 14, 0], [1776, 7, 14, 1], [1776, 7, 23, 0], [1776, 7, 23, 1], [1776, 12, 4, 0], [1776, 12, 4, 1], [1776, 12, 14, 0], [1776, 12, 14, 1], [1776, 12, 23, 0], [1776, 12, 23, 1], [1789, 7, 4, 0], [1789, 7, 4, 1], [1789, 7, 14, 0], [1789, 7, 14, 1], [1789, 7, 23, 0], [1789, 7, 23, 1], [1789, 12, 4, 0], [1789, 12, 4, 1], [1789, 12, 14, 0], [1789, 12, 14, 1], [1789, 12, 23, 0], [1789, 12, 23, 1]] [1, 2, 3] × [30] × [500, 100] [[1, 30, 500], [1, 30, 100], [2, 30, 500], [2, 30, 100], [3, 30, 500], [3, 30, 100]] [1, 2, 3] × [] × [500, 100] []
Generic Version
Generic/Reusable/Flexible Input.
fn cartesian_product<T: Clone>(sets: &[Vec<T>]) -> Vec<Vec<T>> {
if sets.is_empty() {
return vec![vec![]];
}
let mut result = vec![vec![]];
for set in sets {
let mut temp = Vec::new();
for res in &result {
for item in set {
let mut new_res = res.clone();
new_res.push(item.clone());
temp.push(new_res);
}
}
result = temp;
}
result
}Main function body and Output same as the above version.
Scala
Function returning the n-ary product of an arbitrary number of lists, each of arbitrary length:
def cartesianProduct[T](lst: List[T]*): List[List[T]] = {
/**
* Prepend single element to all lists of list
* @param e single elemetn
* @param ll list of list
* @param a accumulator for tail recursive implementation
* @return list of lists with prepended element e
*/
def pel(e: T,
ll: List[List[T]],
a: List[List[T]] = Nil): List[List[T]] =
ll match {
case Nil => a.reverse
case x :: xs => pel(e, xs, (e :: x) :: a )
}
lst.toList match {
case Nil => Nil
case x :: Nil => List(x)
case x :: _ =>
x match {
case Nil => Nil
case _ =>
lst.par.foldRight(List(x))( (l, a) =>
l.flatMap(pel(_, a))
).map(_.dropRight(x.size))
}
}
}and usage:
cartesianProduct(List(1, 2), List(3, 4))
.map(_.mkString("(", ", ", ")")).mkString("{",", ","}")- Output:
{(1, 3), (1, 4), (2, 3), (2, 4)}
cartesianProduct(List(3, 4), List(1, 2))
.map(_.mkString("(", ", ", ")")).mkString("{",", ","}")- Output:
{(3, 1), (3, 2), (4, 1), (4, 2)}
cartesianProduct(List(1, 2), List.empty)
.map(_.mkString("(", ", ", ")")).mkString("{",", ","}")- Output:
{}
cartesianProduct(List.empty, List(1, 2))
.map(_.mkString("(", ", ", ")")).mkString("{",", ","}")- Output:
{}
cartesianProduct(List(1776, 1789), List(7, 12), List(4, 14, 23), List(0, 1))
.map(_.mkString("(", ", ", ")")).mkString("{",", ","}")- Output:
{(1776, 7, 4, 0), (1776, 7, 4, 1), (1776, 7, 14, 0), (1776, 7, 14, 1), (1776, 7, 23, 0), (1776, 7, 23, 1), (1776, 12, 4, 0), (1776, 12, 4, 1), (1776, 12, 14, 0), (1776, 12, 14, 1), (1776, 12, 23, 0), (1776, 12, 23, 1), (1789, 7, 4, 0), (1789, 7, 4, 1), (1789, 7, 14, 0), (1789, 7, 14, 1), (1789, 7, 23, 0), (1789, 7, 23, 1), (1789, 12, 4, 0), (1789, 12, 4, 1), (1789, 12, 14, 0), (1789, 12, 14, 1), (1789, 12, 23, 0), (1789, 12, 23, 1)}
cartesianProduct(List(1, 2, 3), List(30), List(500, 100))
.map(_.mkString("(", ", ", ")")).mkString("{",", ","}")- Output:
{(1, 30, 500), (1, 30, 100), (2, 30, 500), (2, 30, 100), (3, 30, 500), (3, 30, 100)}
cartesianProduct(List(1, 2, 3), List.empty, List(500, 100))
.map(_.mkString("[", ", ", "]")).mkString("\n")- Output:
{}
Scheme
(define cartesian-product (lambda (xs ys)
(if (or (zero? (length xs)) (zero? (length ys)))
'()
(fold append (map (lambda (x) (map (lambda (y) (list x y)) ys)) xs)))))
(define nary-cartesian-product (lambda (ls)
(if (fold (lambda (a b) (or a b)) (map (compose zero? length) ls))
'()
(map flatten (fold cartesian-product ls)))))
> (cartesian-product '(1 2) '(3 4))
((1 3) (1 4) (2 3) (2 4))
> (cartesian-product '(3 4) '(1 2))
((3 1) (3 2) (4 1) (4 2))
> (cartesian-product '(1 2) '())
()
> (cartesian-product '() '(1 2))
()
> (nary-cartesian-product '((1 2)(a b)(x y)))
((1 a x) (1 a y) (1 b x) (1 b y) (2 a x) (2 a y) (2 b x) (2 b y))Sidef
In Sidef, the Cartesian product of an arbitrary number of arrays is built-in as Array.cartesian():
cartesian([[1,2], [3,4], [5,6]]).say
cartesian([[1,2], [3,4], [5,6]], {|*arr| say arr })Alternatively, a simple recursive implementation:
func cartesian_product(*arr) {
var c = []
var r = []
func {
if (c.len < arr.len) {
for item in (arr[c.len]) {
c.push(item)
__FUNC__()
c.pop
}
}
else {
r.push([c...])
}
}()
return r
}Completing the task:
say cartesian_product([1,2], [3,4])
say cartesian_product([3,4], [1,2])- Output:
[[1, 3], [1, 4], [2, 3], [2, 4]] [[3, 1], [3, 2], [4, 1], [4, 2]]
The product of an empty list with any other list is empty:
say cartesian_product([1,2], [])
say cartesian_product([], [1,2])- Output:
[] []
Extra credit:
cartesian_product([1776, 1789], [7, 12], [4, 14, 23], [0, 1]).each{ .say }- Output:
[1776, 7, 4, 0] [1776, 7, 4, 1] [1776, 7, 14, 0] [1776, 7, 14, 1] [1776, 7, 23, 0] [1776, 7, 23, 1] [1776, 12, 4, 0] [1776, 12, 4, 1] [1776, 12, 14, 0] [1776, 12, 14, 1] [1776, 12, 23, 0] [1776, 12, 23, 1] [1789, 7, 4, 0] [1789, 7, 4, 1] [1789, 7, 14, 0] [1789, 7, 14, 1] [1789, 7, 23, 0] [1789, 7, 23, 1] [1789, 12, 4, 0] [1789, 12, 4, 1] [1789, 12, 14, 0] [1789, 12, 14, 1] [1789, 12, 23, 0] [1789, 12, 23, 1]
say cartesian_product([1, 2, 3], [30], [500, 100])
say cartesian_product([1, 2, 3], [], [500, 100])- Output:
[[1, 30, 500], [1, 30, 100], [2, 30, 500], [2, 30, 100], [3, 30, 500], [3, 30, 100]] []
SQL
If we create lists as tables with one column, cartesian product is easy.
-- set up list 1
create table L1 (value integer);
insert into L1 values (1);
insert into L1 values (2);
-- set up list 2
create table L2 (value integer);
insert into L2 values (3);
insert into L2 values (4);
-- get the product
select * from L1, L2;- Output:
VALUE VALUE
---------- ----------
1 3
1 4
2 3
2 4You should be able to be more explicit should get the same result:
select * from L1 cross join L2;Product with an empty list works as expected (using the tables created above):
delete from L2;
select * from L1, L2;- Output:
no rows selected
I don't think "extra credit" is meaningful here because cartesian product is so hard-baked into SQL, so here's just one of the extra credit examples (again using the tables created above):
insert into L1 values (3);
insert into L2 values (30);
create table L3 (value integer);
insert into L3 values (500);
insert into L3 values (100);
-- product works the same for as many "lists" as you'd like
select * from L1, L2, L3;- Output:
VALUE VALUE VALUE
---------- ---------- ----------
1 30 500
2 30 500
3 30 500
1 30 100
2 30 100
3 30 100
Standard ML
fun prodList (nil, _) = nil
| prodList ((x::xs), ys) = map (fn y => (x,y)) ys @ prodList (xs, ys)
fun naryProdList zs = foldl (fn (xs, ys) => map op:: (prodList (xs, ys))) [[]] (rev zs)- Output:
- prodList ([1, 2], [3, 4]); val it = [(1,3),(1,4),(2,3),(2,4)] : (int * int) list - prodList ([3, 4], [1, 2]); val it = [(3,1),(3,2),(4,1),(4,2)] : (int * int) list - prodList ([1, 2], []); stdIn:8.1-8.22 Warning: type vars not generalized because of value restriction are instantiated to dummy types (X1,X2,...) val it = [] : (int * ?.X1) list - naryProdList [[1776, 1789], [7, 12], [4, 14, 23], [0, 1]]; val it = [[1776,7,4,0],[1776,7,4,1],[1776,7,14,0],[1776,7,14,1],[1776,7,23,0], [1776,7,23,1],[1776,12,4,0],[1776,12,4,1],[1776,12,14,0],[1776,12,14,1], [1776,12,23,0],[1776,12,23,1],[1789,7,4,0],[1789,7,4,1],[1789,7,14,0], [1789,7,14,1],[1789,7,23,0],[1789,7,23,1],[1789,12,4,0],[1789,12,4,1], [1789,12,14,0],[1789,12,14,1],[1789,12,23,0],[1789,12,23,1]] : int list list - naryProdList [[1, 2, 3], [30], [500, 100]]; val it = [[1,30,500],[1,30,100],[2,30,500],[2,30,100],[3,30,500],[3,30,100]] : int list list - naryProdList [[1, 2, 3], [], [500, 100]]; val it = [] : int list list
Stata
In Stata, the command fillin may be used to expand a dataset with all combinations of a number of variables. Thus it's easy to compute a cartesian product.
. list
+-------+
| a b |
|-------|
1. | 1 3 |
2. | 2 4 |
+-------+
. fillin a b
. list
+-----------------+
| a b _fillin |
|-----------------|
1. | 1 3 0 |
2. | 1 4 1 |
3. | 2 3 1 |
4. | 2 4 0 |
+-----------------+The other way around:
. list
+-------+
| a b |
|-------|
1. | 3 1 |
2. | 4 2 |
+-------+
. fillin a b
. list
+-----------------+
| a b _fillin |
|-----------------|
1. | 3 1 0 |
2. | 3 2 1 |
3. | 4 1 1 |
4. | 4 2 0 |
+-----------------+Note, however, that this is not equivalent to a cartesian product when one of the variables is "empty" (that is, only contains missing values).
. list
+-------+
| a b |
|-------|
1. | 1 . |
2. | 2 . |
+-------+
. fillin a b
. list
+-----------------+
| a b _fillin |
|-----------------|
1. | 1 . 0 |
2. | 2 . 0 |
+-----------------+This command works also if the varaibles have different numbers of nonmissing elements. However, this requires additional code to remove the observations with missing values.
. list
+-----------+
| a b c |
|-----------|
1. | 1 4 6 |
2. | 2 5 . |
3. | 3 . . |
+-----------+
. fillin a b c
. list
+---------------------+
| a b c _fillin |
|---------------------|
1. | 1 4 6 0 |
2. | 1 4 . 1 |
3. | 1 5 6 1 |
4. | 1 5 . 1 |
5. | 1 . 6 1 |
|---------------------|
6. | 1 . . 1 |
7. | 2 4 6 1 |
8. | 2 4 . 1 |
9. | 2 5 6 1 |
10. | 2 5 . 0 |
|---------------------|
11. | 2 . 6 1 |
12. | 2 . . 1 |
13. | 3 4 6 1 |
14. | 3 4 . 1 |
15. | 3 5 6 1 |
|---------------------|
16. | 3 5 . 1 |
17. | 3 . 6 1 |
18. | 3 . . 0 |
+---------------------+
. foreach var of varlist _all {
quietly drop if missing(`var')
}
. list
+---------------------+
| a b c _fillin |
|---------------------|
1. | 1 4 6 0 |
2. | 1 5 6 1 |
3. | 2 4 6 1 |
4. | 2 5 6 1 |
5. | 3 4 6 1 |
|---------------------|
6. | 3 5 6 1 |
+---------------------+Swift
func + <T>(el: T, arr: [T]) -> [T] {
var ret = arr
ret.insert(el, at: 0)
return ret
}
func cartesianProduct<T>(_ arrays: [T]...) -> [[T]] {
guard let head = arrays.first else {
return []
}
let first = Array(head)
func pel(
_ el: T,
_ ll: [[T]],
_ a: [[T]] = []
) -> [[T]] {
switch ll.count {
case 0:
return a.reversed()
case _:
let tail = Array(ll.dropFirst())
let head = ll.first!
return pel(el, tail, el + head + a)
}
}
return arrays.reversed()
.reduce([first], {res, el in el.flatMap({ pel($0, res) }) })
.map({ $0.dropLast(first.count) })
}
print(cartesianProduct([1, 2], [3, 4]))
print(cartesianProduct([3, 4], [1, 2]))
print(cartesianProduct([1, 2], []))
print(cartesianProduct([1776, 1789], [7, 12], [4, 14, 23], [0, 1]))
print(cartesianProduct([1, 2, 3], [30], [500, 100]))
print(cartesianProduct([1, 2, 3], [], [500, 100])- Output:
[[1, 3], [1, 4], [2, 3], [2, 4]] [[3, 1], [3, 2], [4, 1], [4, 2]] [] [[1776, 7, 4, 0], [1776, 7, 4, 1], [1776, 7, 14, 0], [1776, 7, 14, 1], [1776, 7, 23, 0], [1776, 7, 23, 1], [1776, 12, 4, 0], [1776, 12, 4, 1], [1776, 12, 14, 0], [1776, 12, 14, 1], [1776, 12, 23, 0], [1776, 12, 23, 1], [1789, 7, 4, 0], [1789, 7, 4, 1], [1789, 7, 14, 0], [1789, 7, 14, 1], [1789, 7, 23, 0], [1789, 7, 23, 1], [1789, 12, 4, 0], [1789, 12, 4, 1], [1789, 12, 14, 0], [1789, 12, 14, 1], [1789, 12, 23, 0], [1789, 12, 23, 1]] [[1, 30, 500], [1, 30, 100], [2, 30, 500], [2, 30, 100], [3, 30, 500], [3, 30, 100]] []
Tailspin
'{1,2}x{3,4} = $:[by [1,2]..., by [3,4]...];
' -> !OUT::write
'{3,4}x{1,2} = $:[by [3,4]..., by [1,2]...];
' -> !OUT::write
'{1,2}x{} = $:[by [1,2]..., by []...];
' -> !OUT::write
'{}x{1,2} = $:[by []..., by [1,2]...];
' -> !OUT::write
'{1776, 1789} × {7, 12} × {4, 14, 23} × {0, 1} = $:[by [1776, 1789]..., by [7, 12]..., by [4, 14, 23]..., by [0, 1]...];
' -> !OUT::write
'{1, 2, 3} × {30} × {500, 100} = $:[by [1, 2, 3] ..., by [30]..., by [500, 100]...];
' -> !OUT::write
'{1, 2, 3} × {} × {500, 100} = $:[by [1, 2, 3]..., by []..., by [500, 100]...];
' -> !OUT::write
// You can also generate structures with named fields
'year {1776, 1789} × month {7, 12} × day {4, 14, 23} = $:{by [1776, 1789]... -> (year:$), by [7, 12]... -> (month:$), by [4, 14, 23]... -> (day:$)};
' -> !OUT::write- Output:
{1,2}x{3,4} = [1, 3][2, 3][1, 4][2, 4]
{3,4}x{1,2} = [3, 1][4, 1][3, 2][4, 2]
{1,2}x{} =
{}x{1,2} =
{1776, 1789} × {7, 12} × {4, 14, 23} × {0, 1} = [1776, 7, 4, 0][1789, 7, 4, 0][1776, 12, 4, 0][1789, 12, 4, 0][1776, 7, 14, 0][1789, 7, 14, 0][1776, 12, 14, 0][1789, 12, 14, 0][1776, 7, 23, 0][1789, 7, 23, 0][1776, 12, 23, 0][1789, 12, 23, 0][1776, 7, 4, 1][1789, 7, 4, 1][1776, 12, 4, 1][1789, 12, 4, 1][1776, 7, 14, 1][1789, 7, 14, 1][1776, 12, 14, 1][1789, 12, 14, 1][1776, 7, 23, 1][1789, 7, 23, 1][1776, 12, 23, 1][1789, 12, 23, 1]
{1, 2, 3} × {30} × {500, 100} = [1, 30, 500][2, 30, 500][3, 30, 500][1, 30, 100][2, 30, 100][3, 30, 100]
{1, 2, 3} × {} × {500, 100} =
year {1776, 1789} × month {7, 12} × day {4, 14, 23} = {day=4, month=7, year=1776}{day=4, month=7, year=1789}{day=4, month=12, year=1776}{day=4, month=12, year=1789}{day=14, month=7, year=1776}{day=14, month=7, year=1789}{day=14, month=12, year=1776}{day=14, month=12, year=1789}{day=23, month=7, year=1776}{day=23, month=7, year=1789}{day=23, month=12, year=1776}{day=23, month=12, year=1789}
Tcl
proc cartesianProduct {l1 l2} {
set result {}
foreach el1 $l1 {
foreach el2 $l2 {
lappend result [list $el1 $el2]
}
}
return $result
}
puts "simple"
puts "result: [cartesianProduct {1 2} {3 4}]"
puts "result: [cartesianProduct {3 4} {1 2}]"
puts "result: [cartesianProduct {1 2} {}]"
puts "result: [cartesianProduct {} {3 4}]"
proc cartesianNaryProduct {lists} {
set result {{}}
foreach l $lists {
set res {}
foreach comb $result {
foreach el $l {
lappend res [linsert $comb end $el]
}
}
set result $res
}
return $result
}
puts "n-ary"
puts "result: [cartesianNaryProduct {{1776 1789} {7 12} {4 14 23} {0 1}}]"
puts "result: [cartesianNaryProduct {{1 2 3} {30} {500 100}}]"
puts "result: [cartesianNaryProduct {{1 2 3} {} {500 100}}]"- Output:
simple
result: {1 3} {1 4} {2 3} {2 4}
result: {3 1} {3 2} {4 1} {4 2}
result:
result:
n-ary
result: {1776 7 4 0} {1776 7 4 1} {1776 7 14 0} {1776 7 14 1} {1776 7 23 0} {1776 7 23 1} {1776 12 4 0} {1776 12 4 1} {1776 12 14 0} {1776 12 14 1} {1776 12 23 0} {1776 12 23 1} {1789 7 4 0} {1789 7 4 1} {1789 7 14 0} {1789 7 14 1} {1789 7 23 0} {1789 7 23 1} {1789 12 4 0} {1789 12 4 1} {1789 12 14 0} {1789 12 14 1} {1789 12 23 0} {1789 12 23 1}
result: {1 30 500} {1 30 100} {2 30 500} {2 30 100} {3 30 500} {3 30 100}
result:
Uiua
# Cartesian products
☇1⊞⊂[1 2] [3 4]
☇1⊞⊂[3 4] [1 2]
☇1⊞⊂[3 4] []
C ← /◇(☇1⊞⊂)
≡⍚C{{[1776 1789] [7 12] [4 14 23] [0 1]}
{[1 2 3] [30] [500 100]}
{[1 2 3] [] [500 100]}}- Output:
╭─
╷ 1 3
1 4
2 3
2 4
╯
╭─
╷ 3 1
3 2
4 1
4 2
╯
╭─
╷ 0×2 ℝ # i.e. empty
╯
╭─
╓─
╟ 1776 7 4 0
1776 7 4 1
1776 7 14 0
1776 7 14 1
1776 7 23 0
1776 7 23 1
1776 12 4 0
1776 12 4 1
1776 12 14 0 ╓─
1776 12 14 1 ╟ 1 30 500
1776 12 23 0 1 30 100
1776 12 23 1 2 30 500 ╓─
1789 7 4 0 2 30 100 ╟ 0×2 ℝ
1789 7 4 1 3 30 500 ╜
1789 7 14 0 3 30 100
1789 7 14 1 ╜
1789 7 23 0
1789 7 23 1
1789 12 4 0
1789 12 4 1
1789 12 14 0
1789 12 14 1
1789 12 23 0
1789 12 23 1
╜
╯
UNIX Shell
The UNIX shells don't allow passing or returning arrays from functions (other than pass-by-name shenanigans), but as pointed out in the Perl entry, wildcard brace expansion (in bash, ksh, zsh) does a Cartesian product if there's more than one set of alternatives. It doesn't handle the empty-list case (an empty brace expansion item is treated as a single item that is equal to the empty string), but otherwise it works:
$ printf '%s' "("{1,2},{3,4}")"; printf '\n'
(1,3)(1,4)(2,3)(2,4)
$ printf '%s' "("{3,4},{1,2}")"; printf '\n'
(3,1)(3,2)(4,1)(4,2)
More than two lists is not a problem:
$ printf '%s\n' "("{1776,1789},{7,12},{4,14,23},{0,1}")"
(1776,7,4,0)
(1776,7,4,1)
(1776,7,14,0)
(1776,7,14,1)
(1776,7,23,0)
(1776,7,23,1)
(1776,12,4,0)
(1776,12,4,1)
(1776,12,14,0)
(1776,12,14,1)
(1776,12,23,0)
(1776,12,23,1)
(1789,7,4,0)
(1789,7,4,1)
(1789,7,14,0)
(1789,7,14,1)
(1789,7,23,0)
(1789,7,23,1)
(1789,12,4,0)
(1789,12,4,1)
(1789,12,14,0)
(1789,12,14,1)
(1789,12,23,0)
(1789,12,23,1)
$ printf '%s\n' "("{1,2,3},30,{500,100}")"
(1,30,500)
(1,30,100)
(2,30,500)
(2,30,100)
(3,30,500)
(3,30,100)
Visual Basic .NET
Imports System.Runtime.CompilerServices
Module Module1
<Extension()>
Function CartesianProduct(Of T)(sequences As IEnumerable(Of IEnumerable(Of T))) As IEnumerable(Of IEnumerable(Of T))
Dim emptyProduct As IEnumerable(Of IEnumerable(Of T)) = {Enumerable.Empty(Of T)}
Return sequences.Aggregate(emptyProduct, Function(accumulator, sequence) From acc In accumulator From item In sequence Select acc.Concat({item}))
End Function
Sub Main()
Dim empty(-1) As Integer
Dim list1 = {1, 2}
Dim list2 = {3, 4}
Dim list3 = {1776, 1789}
Dim list4 = {7, 12}
Dim list5 = {4, 14, 23}
Dim list6 = {0, 1}
Dim list7 = {1, 2, 3}
Dim list8 = {30}
Dim list9 = {500, 100}
For Each sequnceList As Integer()() In {
({list1, list2}),
({list2, list1}),
({list1, empty}),
({empty, list1}),
({list3, list4, list5, list6}),
({list7, list8, list9}),
({list7, empty, list9})
}
Dim cart = sequnceList.CartesianProduct().Select(Function(tuple) $"({String.Join(", ", tuple)})")
Console.WriteLine($"{{{String.Join(", ", cart)}}}")
Next
End Sub
End Module- Output:
{(1, 3), (1, 4), (2, 3), (2, 4)}
{(3, 1), (3, 2), (4, 1), (4, 2)}
{}
{}
{(1776, 7, 4, 0), (1776, 7, 4, 1), (1776, 7, 14, 0), (1776, 7, 14, 1), (1776, 7, 23, 0), (1776, 7, 23, 1), (1776, 12, 4, 0), (1776, 12, 4, 1), (1776, 12, 14, 0), (1776, 12, 14, 1), (1776, 12, 23, 0), (1776, 12, 23, 1), (1789, 7, 4, 0), (1789, 7, 4, 1), (1789, 7, 14, 0), (1789, 7, 14, 1), (1789, 7, 23, 0), (1789, 7, 23, 1), (1789, 12, 4, 0), (1789, 12, 4, 1), (1789, 12, 14, 0), (1789, 12, 14, 1), (1789, 12, 23, 0), (1789, 12, 23, 1)}
{(1, 30, 500), (1, 30, 100), (2, 30, 500), (2, 30, 100), (3, 30, 500), (3, 30, 100)}
{}
Wren
import "./seq" for Lst
var prod2 = Fn.new { |l1, l2|
var res = []
for (e1 in l1) {
for (e2 in l2) res.add([e1, e2])
}
return res
}
var prodN = Fn.new { |ll|
if (ll.count < 2) Fiber.abort("There must be at least two lists.")
var p2 = prod2.call(ll[0], ll[1])
return ll.skip(2).reduce(p2) { |acc, l| prod2.call(acc, l) }.map { |l| Lst.flatten(l) }.toList
}
var printProdN = Fn.new { |ll|
System.print("%(ll.join(" x ")) = ")
System.write("[\n ")
System.print(prodN.call(ll).join("\n "))
System.print("]\n")
}
System.print("[1, 2] x [3, 4] = %(prodN.call([ [1, 2], [3, 4] ]))")
System.print("[3, 4] x [1, 2] = %(prodN.call([ [3, 4], [1, 2] ]))")
System.print("[1, 2] x [] = %(prodN.call([ [1, 2], [] ]))")
System.print("[] x [1, 2] = %(prodN.call([ [], [1, 2] ]))")
System.print("[1, a] x [2, b] = %(prodN.call([ [1, "a"], [2, "b"] ]))")
System.print()
printProdN.call([ [1776, 1789], [7, 12], [4, 14, 23], [0, 1] ])
printProdN.call([ [1, 2, 3], [30], [500, 100] ])
printProdN.call([ [1, 2, 3], [], [500, 100] ])
printProdN.call([ [1, 2, 3], [30], ["a", "b"] ])- Output:
[1, 2] x [3, 4] = [[1, 3], [1, 4], [2, 3], [2, 4]]
[3, 4] x [1, 2] = [[3, 1], [3, 2], [4, 1], [4, 2]]
[1, 2] x [] = []
[] x [1, 2] = []
[1, a] x [2, b] = [[1, 2], [1, b], [a, 2], [a, b]]
[1776, 1789] x [7, 12] x [4, 14, 23] x [0, 1] =
[
[1776, 7, 4, 0]
[1776, 7, 4, 1]
[1776, 7, 14, 0]
[1776, 7, 14, 1]
[1776, 7, 23, 0]
[1776, 7, 23, 1]
[1776, 12, 4, 0]
[1776, 12, 4, 1]
[1776, 12, 14, 0]
[1776, 12, 14, 1]
[1776, 12, 23, 0]
[1776, 12, 23, 1]
[1789, 7, 4, 0]
[1789, 7, 4, 1]
[1789, 7, 14, 0]
[1789, 7, 14, 1]
[1789, 7, 23, 0]
[1789, 7, 23, 1]
[1789, 12, 4, 0]
[1789, 12, 4, 1]
[1789, 12, 14, 0]
[1789, 12, 14, 1]
[1789, 12, 23, 0]
[1789, 12, 23, 1]
]
[1, 2, 3] x [30] x [500, 100] =
[
[1, 30, 500]
[1, 30, 100]
[2, 30, 500]
[2, 30, 100]
[3, 30, 500]
[3, 30, 100]
]
[1, 2, 3] x [] x [500, 100] =
[
]
[1, 2, 3] x [30] x [a, b] =
[
[1, 30, a]
[1, 30, b]
[2, 30, a]
[2, 30, b]
[3, 30, a]
[3, 30, b]
]
zkl
Cartesian product is build into iterators or can be done with nested loops.
zkl: Walker.cproduct(List(1,2),List(3,4)).walk().println();
L(L(1,3),L(1,4),L(2,3),L(2,4))
zkl: foreach a,b in (List(1,2),List(3,4)){ print("(%d,%d) ".fmt(a,b)) }
(1,3) (1,4) (2,3) (2,4)
zkl: Walker.cproduct(List(3,4),List(1,2)).walk().println();
L(L(3,1),L(3,2),L(4,1),L(4,2))The walk method will throw an error if used on an empty iterator but the pump method doesn't.
zkl: Walker.cproduct(List(3,4),List).walk().println();
Exception thrown: TheEnd(Ain't no more)
zkl: Walker.cproduct(List(3,4),List).pump(List).println();
L()
zkl: Walker.cproduct(List,List(3,4)).pump(List).println();
L()zkl: Walker.cproduct(L(1776,1789),L(7,12),L(4,14,23),L(0,1)).walk().println();
L(L(1776,7,4,0),L(1776,7,4,1),L(1776,7,14,0),L(1776,7,14,1),L(1776,7,23,0),L(1776,7,23,1),L(1776,12,4,0),L(1776,12,4,1),L(1776,12,14,0),L(1776,12,14,1),L(1776,12,23,0),L(1776,12,23,1),L(1789,7,4,0),L(1789,7,4,1),L(1789,7,14,0),L(1789,7,14,1),L(1789,7,23,0),L(1789,7,23,1),L(1789,12,4,0),L(1789,12,4,1),...)
zkl: Walker.cproduct(L(1,2,3),L(30),L(500,100)).walk().println();
L(L(1,30,500),L(1,30,100),L(2,30,500),L(2,30,100),L(3,30,500),L(3,30,100))
zkl: Walker.cproduct(L(1,2,3),List,L(500,100)).pump(List).println();
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