# Finding a Cartesian product of multiple lists

I came up with an extension method to find a Cartesian product of multiple IEnumerable sets. I was able to achieve lazy enumeration via yield return, but I didn't think of a way to do it non-recursively. The result ended up being a recursive lazy enumeration iterator method, the first of its kind! At least as far as I've ever written.

The idea of the problem came from a Stack Overflow question where a guy had many sets of characters, and wanted to generate a combination of all of them. Now I'm just interested because it's a fun problem!

I'd appreciate any kind of review, although here's two specific things I'm most interested in:

1. Can this algorithm can be de-recursed? If so - should it be, and how?
2. Is there some way it could be generalized even more, beyond what I've done?

public static class MultiCartesianExtension
{
public static IEnumerable<TInput[]> MultiCartesian<TInput>(this IEnumerable<IEnumerable<TInput>> input)
{
return input.MultiCartesian(x => x);
}

public static IEnumerable<TOutput> MultiCartesian<TInput, TOutput>(this IEnumerable<IEnumerable<TInput>> input, Func<TInput[], TOutput> selector)
{
// Materializing here to avoid multiple enumerations.
var inputList = input.ToList();
var buffer = new TInput[inputList.Count];
var results = MultiCartesianInner(inputList, buffer, 0);
var transformed = results.Select(selector);
return transformed;
}

private static IEnumerable<TInput[]> MultiCartesianInner<TInput>(IList<IEnumerable<TInput>> input, TInput[] buffer, int depth)
{
foreach (var current in input[depth])
{
buffer[depth] = current;
if (depth == buffer.Length - 1)
{
// This is to ensure usage safety - the original buffer
// needs to remain unmodified to ensure a correct sequence.
var bufferCopy = (TInput[])buffer.Clone();
yield return bufferCopy;
}
else
{
// Funky recursion here
foreach (var a in MultiCartesianInner(input, buffer, depth + 1))
{
yield return a;
}
}
}
}
}


Usage:

var input = new string[]
{
"AB",
"123",
"@#",
};

foreach (var result in input.MultiCartesian(x => new string(x)))
{
Console.WriteLine(result);
}

// Results:
// A1@
// A1#
// A2@
// A2#
// A3@
// A3#
// B1@
// B1#
// B2@
// B2#
// B3@
// B3#

• Possible bug - IEnumerable doesn't guarantee ordering, so you can't count on "a correct sequence". Consider var input = new HashSet<string> {"AB", "123", "@#"}; Commented Mar 13, 2016 at 1:21
• @Comintern Good catch! Although I'd argue that it should be up to the caller to ensure this doesn't happen. If the order doesn't matter - it's fine to use a HashSet. If it does matter - it's up to the caller to provide an ordered collection. Commented Mar 13, 2016 at 14:24
• Wouldn't the a cartesian product of these sets also include results like 1A@ Commented Mar 13, 2016 at 17:42
• @JohnK No, a Cartesian product consists of only combinations. Take a look at this image. Finding the permutations would be very easy one you have the combinations though, with the MoreLinq's Permutations method. Commented Mar 15, 2016 at 2:22
• @GediminasMasaitis The link to Wikimedia Commons you provided is prone to rusting – the part /4/4e/ refers to some Commons' internal indices, which are rebuilt from time to time. It's safer to use a stable links to a resource instead, like https://commons.wikimedia.org/wiki/File:Cartesian_Product_qtl1.svg. Commented Jan 24, 2018 at 7:43

You can indeed create a cartesian product without recursion. What you need is to use the enumerator.

I used a non-generic enumerator to work with any types.

public static IEnumerable Cartesian(this IEnumerable<IEnumerable> items)
{
var slots = items
// initialize enumerators
.Select(x => x.GetEnumerator())
// get only those that could start in case there is an empty collection
.Where(x => x.MoveNext())
.ToArray();

while (true)
{
// yield current values
yield return slots.Select(x => x.Current);

// increase enumerators
foreach (var slot in slots)
{
// reset the slot if it couldn't move next
if (!slot.MoveNext())
{
// stop when the last enumerator resets
if (slot == slots.Last()) { yield break; }
slot.Reset();
slot.MoveNext();
// move to the next enumerator if this reseted
continue;
}
// we could increase the current enumerator without reset so stop here
break;
}
}
}


Example:

var letters = new string[] { "A", "B" };
var numbers = new[] { 1, 2, 3 };
var symbols = new[] { "@", "#" };
var empty = new string[] { };

var collections = new IEnumerable[] { letters, numbers, symbols, empty };
collections.Cartesian()
// this is just for show
.Select(x => string.Join("", x.Cast<object>()))
.OrderBy(x => x)


result:

A1#
A1@
A2#
A2@
A3#
A3@
B1#
B1@
B2#
B2@
B3#
B3@

• It could be even shorter :) Commented Sep 3, 2016 at 22:38
• I think the code is missing the Dispose() of the enumerators (slots[i]). To be "fully correct" I would then need to be careful about exceptions... Commented Aug 3, 2017 at 23:55
• collections.Cartesian() // this is just for show .Select This will never execute as we can not apply extension methods on IEnumerable. It should be IEnumerable<T>. Need to Cast it in object[].. collections.GetCartesian().Cast<object[]>() // this is just for show .Select(x => string.Join("", x.Cast<object>())) .OrderBy(x => x); Commented Jan 9, 2018 at 17:34

Found this one from a number of years back. Seems short and to the point:

public static IEnumerable<IEnumerable<T>> CartesianProduct<T>(this IEnumerable<IEnumerable<T>> sequences)
{
if (sequences == null)
{
return null;
}

IEnumerable<IEnumerable<T>> emptyProduct = new[] { Enumerable.Empty<T>() };

return sequences.Aggregate(
emptyProduct,
(accumulator, sequence) => accumulator.SelectMany(
accseq => sequence,
(accseq, item) => accseq.Concat(new[] { item })));
}