4
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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#
\$\endgroup\$
5
  • 2
    \$\begingroup\$ Possible bug - IEnumerable doesn't guarantee ordering, so you can't count on "a correct sequence". Consider var input = new HashSet<string> {"AB", "123", "@#"}; \$\endgroup\$
    – Comintern
    Mar 13 '16 at 1:21
  • 1
    \$\begingroup\$ @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. \$\endgroup\$ Mar 13 '16 at 14:24
  • \$\begingroup\$ Wouldn't the a cartesian product of these sets also include results like 1A@ \$\endgroup\$
    – jkdba
    Mar 13 '16 at 17:42
  • \$\begingroup\$ @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. \$\endgroup\$ Mar 15 '16 at 2:22
  • \$\begingroup\$ @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. \$\endgroup\$
    – CiaPan
    Jan 24 '18 at 7:43
3
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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)
   .Dump(); // linqpad

result:

A1# 
A1@ 
A2# 
A2@ 
A3# 
A3@ 
B1# 
B1@ 
B2# 
B2@ 
B3# 
B3@
\$\endgroup\$
3
  • \$\begingroup\$ It could be even shorter :) \$\endgroup\$ Sep 3 '16 at 22:38
  • 1
    \$\begingroup\$ 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... \$\endgroup\$
    – Pablo H
    Aug 3 '17 at 23:55
  • \$\begingroup\$ 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); \$\endgroup\$
    – manoj jain
    Jan 9 '18 at 17:34
1
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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 })));
}
\$\endgroup\$
1

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