Posts Tagged Cartesian product

C# Short Takes – 2 – String from IEnumerable


I am posting this quick tip in response to a question, which seems to on some people’s minds who are reading my blog. The search terms ‘create string from ienumerable’ appear as hits on my blog statistics quite frequently. The way I create a string from an IEnumerable is in the Example Code section below.

The Method

The method I use to create a string from an IEnumerableis to use the string constructor that accepts a char[] (see the MSDN documentation: String Constructor (Char()) ). To get the IEnumerable into a char[] I use the Linq Extension Method ToArray() (see the MSDN documentation: Enumerable.ToArray(Of TSource) Method).

These Linq method call and string constructor results in the code in the ‘Example Code’ section.

The Example Code

The following shows a number of ways to create an IEnumerableobject. These IEnumerableobjects are then converted into char[] object. The char [] objects are then used in the string constructor.

private void String_Create_From_IEnumerable_char( )
    string TestString = "The quick red fox jumped over the lazy brown cow";
    IEnumerable<char> EnumerableChar= TestString.AsEnumerable();
    string Result1 = new string(EnumerableChar.ToArray( ));
    char[] TestCharArray = new char[] { 'T', 'h', 'e', ' ', 't', 'e', 's', 't', ',', 'v', 'a', 'l', 'u', 'e', '.' };
    string Result2 = new string(TestCharArray);
    IEnumerable<char> EnumerableChar1 = TestCharArray.AsEnumerable( );
    string Result3 = new string(EnumerableChar1.ToArray());
    IEnumerable<char> EnumerableChar2 = LINQ_Extension_Methods.LINQ_Extension_Methods.ToIEnumerable('A')
    string Result4 = new string(EnumerableChar2.ToArray( ));
    Func<char, IEnumerable<char>> LocalToIEnumerable1 = LINQ_Extension_Methods.LINQ_Extension_Methods.ToIEnumerable<char>;
    IEnumerable<char> EnumerableChar3 = LocalToIEnumerable1('Z')
    string Result5 = new string(EnumerableChar3.ToArray( ));
    Func<char, char, IEnumerable<char>> LocalToIEnumerable2 = LINQ_Extension_Methods.LINQ_Extension_Methods.ToIEnumerable<char>;
    IEnumerable<char> EnumerableChar4 = LocalToIEnumerable2('A', ' ')
        .UnionAll(LocalToIEnumerable2('B', ' ')).UnionAll(LocalToIEnumerable2('C', ' '))
        .UnionAll(LocalToIEnumerable2('D', ' ')).UnionAll(LocalToIEnumerable2('E', ' '))
        .UnionAll(LocalToIEnumerable2('F', ' ')).UnionAll(LocalToIEnumerable2('G', ' '))
        .UnionAll(LocalToIEnumerable2('H', ' ')).UnionAll(LocalToIEnumerable2('I', ' '));
    string Result6 = new string(EnumerableChar4.ToArray( ));

Methods Referenced That Are Not In The.Net Framework

Method Blog Post That Describes The Method
ToIEnumerable LINQ Extension Method to Generate n-way Cartesian Product
UnionAll LINQ Short Takes – Number 4 –Make Union into a UnionAll


I trust that those readers who have been looking for a solution to this transformation find this blog post helpful answers the question you had.

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LINQ Extension Method to Generate n-way Cartesian Product


This blog post presents a LINQ extension method that implements the generation of Cartesian productfor any number of input sequences.

This implementation builds on the approach that I posted in LINQ Short Takes – Number 2 – Using Method Syntax to Create a Cartesian Product. Specifically, it uses the Enumerable Join Method to join the input IEnumerable sequences into the resulting Cartesian product.

Special Votes of Thanks and Referenced Articles:

Eric Lippert for Computing a Cartesian Product with LINQ, and
Ian Griffiths for LINQ and N-ary Cartesian Products

Whilst, the solution to the problem I present here is not the same as these articles present, these articles did push me down some of the correct paths. One thing that these articles helped me with was getting the ‘shape’ of the returned Cartesian product ‘right’ (correct by my judgement). The ‘shape’ I settled on was an IEnumerable, or a sequence of elements, each element being an IEnumerable with on element from each of the input IEnumerable objects.

The CartesianProduct Extension Method

In the following sections, I present the design, implementation, and some sample code, that describes the development process resulting in the CartesianProduct extension method.

CartesianProduct – Design Goals

There were numerous of design features that I wished to include in the implementation of CartesianProduct method. These features included:

  • The method must create a Cartesian product. This is a mandatory design criterion.
  • The order of the resulting Cartesian product elements must be in the same order as the input sequences. This may well be a defining attribute of a Cartesian product, but I include this characteristic as a design goal anyway. This is a mandatory design criterion.
  • The returned set of Cartesian product elements should be as generic as possible. This is a mandatory design criterion.
  • The returned set of Cartesian product elements should be easily processed using Linq. This is a mandatory design criterion.
  • The method’s implementation must be as an extension method. This is a mandatory design criterion.
  • The method’s implementation must be an extension method that operates like any other Linq extension method (see: IEnumerable(Of T) Interface). This allows the users of the extension method to chain it within a set of other calls to Linq extension methods calls. This allows the user to assemble Linq transformations that they require. This is a mandatory design criterion.
  • The method should accept, as generically as possible, the set of sequences that used to generate the Cartesian product. This is a mandatory design criterion.
  • The method should use the Linq Join extension method to build the Cartesian product. If had to resort to the Linq Aggregate extension method, then the two articles I have referenced above demonstrate that approach. Plagiarising those works, and representing there content here, was not an acceptable outcome. This was a highly desirable, leaning heavily to mandatory, design criterion.

CartesianProduct – Function Signature

The following is the function signature for the CartesianProduct extension method. Many features of the function signature directly address the design goals.

public static IEnumerable<IEnumerable<TSource>> CartesianProduct<TSource>
    (this IEnumerable<TSource> source,
    IEnumerable<IEnumerable<TSource>> inputs)
  • The return type IEnumerable<IEnumerable<TSource>> addresses the design goals for a generic return type, and a return type which can be processed with Linq. If it makes it easier to understand, you can interpret this type as an outer list of inner lists, each inner list being list made up of type the TSource objects. The following diagram attempts to show graphically what this ‘looks like’.

Figure 1 – IEnumerable of IEnumerable of T



  • The CartesianProduct<TSource> part of the method signature identifies this method as a parameterised type method (see MSDN article: Generic Methods (C# Programming Guide) for further information).declaration serves the design goal of requiring the method to accept as generically as possible inputs, and produce a generic output.
  • The argument this IEnumerable<TSource> source serves a couple of the design goals. Primarily, it flags this method as an extension method that should be attached, or associated, with any object that implements the IEnumerable(of T) Interface. Secondly, this argument identifies the first sequence used to assemble the Cartesian product.
  • The argument IEnumerable<IEnumerable<TSource>> inputs identifies the set of sequences to be processed to form the output Cartesian product. As with the return type, a loose interpretation is an outer list that contains inner list of objects. This satisfies the any number of input sequences design goal.

CartesianProduct – Implementation

The following is the implementation of the CartesianProduct extension method. I will detail many of the significant features after the source code.

public static IEnumerable<IEnumerable<TSource>> CartesianProduct<TSource>
    (this IEnumerable<TSource> source,
    IEnumerable<IEnumerable<TSource>> inputs)
    // Build the remaining work to be done var RemaimingJoinTarget = inputs.Skip(1);
    // Recursion end condition, no more to be done. if (RemaimingJoinTarget.Count( ) == 0)
        // At the end of the inputs, return the Cartesian of two elements return source.Join(inputs.First( ),
            (outer) => 1, (inner) => 1,
            (outer, inner) =>
            //(new TSource[2] {outer, inner}).AsEnumerable<TSource>() );
    else {
        // Recursive call down the list of inputs. // When coming back up add the source to the results already built. return source.Join(
            inputs.First( ).CartesianProduct(RemaimingJoinTarget),
            (outer) => 1, (inner) => 1,
            (outer, inner)
                => ToIEnumerable(outer).UnionAll(inner));

The following are some of the significant features of the implementation:

  • Firstly, let me say that I find this implementation aesthetically pleasing. It is short, and concise, deceptively simple.
  • This implementation satisfies the design gaol of using the Linq Join extension method to form the Cartesian product.
  • This implementation satisfies the design gaol of does preserve of the order of the input sequences and reproduces that order in output Cartesian product.
  • This implementation satisfies the design gaol of being generic and catering for any input type. There are some nasty hoops the caller needs to jump through when mixing objects a with C# value types. Although, it may be nasty it does work. I have an example that shows what is required.
  • This is an implementation uses recursion, see the call to CartesianProduct in the Join contained in the else case. The version of recursion that is utilised here is tail recursion. The removal an element from the inputs parameter at each level of recursion controls the recursion. This technique of removing an element from a control sequence until nothing is left is an approach to controlling recursion I have used frequently.
  • There are two overloads of the helper function ToIEnumerable. These simply wrap an IEnumerable around the objects passed in. Using these functions allows the simple assembly of the result IEnumerable of objects with the Linq Union extension method.
  • The Cartesian product is formed within the Linq Join extension method. The same technique as described in LINQ Short Takes – Number 2 – Using Method Syntax to Create a Cartesian Product of setting the inner and outer join keys to 1.
  • The UnionAll LINQ extension method is described in LINQ Short Takes – Number 4 –Make Union into a UnionAll. The source code for the UnionAll extension method is in this blog post as well.

CartesianProduct – Helper Methods

There are two helper methods used in the implementation. These helper methods support the implementation of the CartesianProduct extension method. The source code for the helper methods is below.

public static IEnumerable<TSource> ToIEnumerable<TSource>(TSource val)
    yield return val;


public static IEnumerable<TSource> ToIEnumerable<TSource>(TSource val1, TSource val2)
    yield return val1;
    yield return val2;

There is one point worth noting about these methods. The implementation may not be the most efficient way to achieve the goal. The use of the yield return results in C# compiler implementing a ‘bunch of code’ under the covers. My justification for the implementation using yield return is simple. It results in very simple code for anyone who needs to maintain, or modify, the code.

CartesianProduct – Source Code

The following URL’s have the full source code, including XML Comments, for CartesianProduct and the two overloads of IEnumerable.

Referenced LINQ Methods

The following are the Linq extension methods that are utilised in the implementation of the CartesianProduct extension method.

Join<(Of <<‘(TOuter, TInner, TKey, TResult>)>>)(IEnumerable<(Of <<‘(TOuter>)>>), IEnumerable<(Of <<‘(TInner>)>>), Func<(Of <<‘(TOuter, TKey>)>>), Func<(Of <<‘(TInner, TKey>)>>), Func<(Of <<‘(TOuter, TInner, TResult>)>>))

Skip<(Of <<‘(TSource>)>>)(IEnumerable<(Of <<‘(TSource>)>>), Int32)

Union<(Of <<‘(TSource>)>>)(IEnumerable<(Of <<‘(TSource>)>>), IEnumerable<(Of <<‘(TSource>)>>))

First<(Of <<‘(TSource>)>>)(IEnumerable<(Of <<‘(TSource>)>>))

CartesianProduct – Example Code

The following code assembles a range of different Cartesian products, and dumps them to the debug output window.

private void CartesianProducts_Extensions( )
    IEnumerable<int>[] SequsInt1 = new IEnumerable<int>[]
    IEnumerable<int>[] SequsInt2 = new IEnumerable<int>[]
    IEnumerable<int>[] SequsInt3 = new IEnumerable<int>[]
    //IEnumerable<IEnumerable<object>> ExpandedSequs1 = // LINQ_Extensions.ToIEnumerable(Enumerable.Range(0, 10).Cast<IEnumerable<object>>()); string[] SampleStrings = new string[]
    { "The", "quick", "red", "fox", "jumped",
        "over", "the", "lazy", "brown", "cow" };

    Action<IEnumerable<int>> IntDumper = (InnerSequence) =>
            int column = 0;
            foreach (int innerVal in InnerSequence)
                Debug.Write(string.Format("[{0}]={1}, ", column, innerVal));

    var OuterInSource = Enumerable.Range(0, 10);
    var IntProd1 = Enumerable.Range(0, 10).CartesianProduct(SequsInt1);
        (InnerSequence) =>
            int column = 0;
            foreach (int innerVal in InnerSequence)
                Debug.Write(string.Format("[{0}]={1}, ", column, innerVal));

    var IntProd1Vers2 = OuterInSource.CartesianProduct(SequsInt1);
    DumpCartesianProduct(IntProd1Vers2, IntDumper);

    var IntProd2 = Enumerable.Range(0, 10).CartesianProduct(SequsInt2);
    DumpCartesianProduct(IntProd2, IntDumper);

    var IntProd2Vers2 = OuterInSource.CartesianProduct(SequsInt2);
    DumpCartesianProduct(IntProd2Vers2, IntDumper);

    var IntProd3 = Enumerable.Range(0, 10).CartesianProduct(SequsInt3);
    DumpCartesianProduct(IntProd3, IntDumper);

    var IntProd3Vers2 = OuterInSource.CartesianProduct(SequsInt3);
    DumpCartesianProduct(IntProd3Vers2, IntDumper);

    // <IEnumerable<object>> Action<IEnumerable<object>> objDumper = (InnerSequence) =>
        int column = 0;
        foreach (object innerVal in InnerSequence)
            Debug.Write(string.Format("[{0}]={1}, ", column, innerVal));
    var objStrings = SampleStrings.Cast<object>( );
    var objTransSequs1 = SequsInt1.ToList( ).Select(a => a.Select(b => (object) b));
    var MixedObjectProduct = objStrings.CartesianProduct(objTransSequs1);
    DumpCartesianProduct<object>(MixedObjectProduct, objDumper);

    var MixedObjectProduct2 = SampleStrings.CartesianProduct(objTransSequs1);
    DumpCartesianProduct<object>(MixedObjectProduct2, objDumper);

    IEnumerable<IEnumerable<object>> MixedSource =
        new IEnumerable<object>[]
            from intVal in Enumerable.Range(0,5) select (object) intVal,
            from intVal in Enumerable.Range(0,5) select (object) new DateTime(2012, 1, 1).AddDays(intVal)
    var MixedObjectProduct3 = SampleStrings.CartesianProduct(MixedSource);
    DumpCartesianProduct<object>(MixedObjectProduct3, objDumper);


private void DumpCartesianProduct<TSource>(
    IEnumerable<IEnumerable<TSource>> ResultSequence,
    Action<IEnumerable<TSource>> DumpLogic = null)
    if (DumpLogic != null)
        foreach (IEnumerable<TSource> outputSeq in ResultSequence)


Possible Changes, Enhancements, Additional Overloads, Or Extra Features

I believe that additional versions of the CartesianProduct could be useful. These additional versions include:

· The creation of a variant of the Cartesian product method that accepts a variable number of arguments, as opposed to IEnumerable<IEnumerable<TSource> argument, could be useful.


For an implementation which started out as a ‘is it possible to do?’ question, the resulting implementation is quite effective and usable.

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LINQ Short Takes – Number 3 –LINQ over Multiple Dimension Arrays and Lists


This blog post presents an alternative approach to using LINQ against List<Of T> and multidimensional arrays (and other types). This approach builds on the preceding blog posts LINQ Short Takes – Number 1 – Enumerable.Range() and LINQ Short Takes – Number 2 – Using Method Syntax to Create a Cartesian Product.

An Alternative Approach to Multiple Dimension Arrays and Lists

A core of this approach is to use two features of the C# language. These features are:

  1. The indexer which is available on some object types (see MSDN Article: Indexers (C# Programming Guide)), and the indexed access which the array objects support. In the case of object collections which do not support the IList(of T) interface, the LINQ extension methods ElementAt provides an equivalent, and very useable, alternative.
  2. The use of LINQ to generate the index values that are used against the collection objects supporting indexers, and array objects.

The use of LINQ to generate the index values into collections or arrays is not the approach that developers normally use with these objects. This approach has benefits in terms of clarity of the resulting code. This approach is only applicable to certain classes of problems that a developer may encounter.

Example Notes:

There are a couple of points to note about the presented examples:

  1. I have used my LINQ extension method ToOutput to produce a dump of the result sequences. This extension method was the subject of my blog post Dumping a formatted IEnumerable to Output. This blog post also includes to the source code for the extension method. Additionally, there are links to docx and pdf files of the source code.
  2. The source code for the methods which the following examples contain are available in docx and pdf files from the following URLS:

Examples of Approach on List Collections

The following are examples of LINQ query syntax examples of using LINQ to generate and enumerate the index into a List. By using the array, or indexer, syntax to access the elements of the List(Of T) Class makes calculating the difference between adjacent dates in the List(Of T) Class trivial. The pure LINQ way to achieve this type of calculation between members of a List is not as simple, clear, or concise, as this approach.

private void LINQ_ALternative_List()
    List<DateTime> Dates = new List<DateTime>()
        new DateTime(2012,1,1), new DateTime(2012,2,29),
        new DateTime(2012,8,13), new DateTime(2012,9,10)
    var DaysGaps = from idx in Enumerable.Range(0, Dates.Count-1)
                   select Dates[idx + 1] - Dates[idx];
    Debug.WriteLine("Dumping DaysGaps");
        (val, position) => string.Format("[{0}]={1}\n", position, val.ToString("%d")));

    var DaysGaps1 = from idx in Enumerable.Range(0, Dates.Count - 1)
                    select (Dates[idx + 1] - Dates[idx]).Days;
    Debug.WriteLine("Dumping DaysGaps1");
        (val, position) => string.Format("[{0}]={1}\n", position, val));
    return; // Allows a breakpoint at the end of the method. }

Example of Approach on Dictionary Collection

The following example demonstrates the use of ElementAt method to achieve the same days difference calculation as the preceding example. This example is manipulating the values in the Value property of the KeyValuePair contained in a Dictionary(Of TKey, TValue) Class.

private void LINQ_Alternative_Dictionary()
    Dictionary<int, DateTime> Dict1 = new Dictionary<int, DateTime>()
        {1, new DateTime(2012, 1,  1)}, {5, new DateTime(2012, 3, 15)},
        {7, new DateTime(2012, 4, 21)}, {9, new DateTime(2012, 12, 25)}
    var a = Dict1.ElementAt(2);
    var DateDiffs = from idx in Enumerable.Range(0, Dict1.Count - 1)
                    select (Dict1.ElementAt(idx + 1).Value - Dict1.ElementAt(idx).Value).Days;
    Debug.WriteLine("Dumping DateDiffs");
        (val, position) => string.Format("[{0}]={1}\n", position, val));

    return; // Allows a breakpoint at the end of the method. }

Examples of Approach on Multidimensional Arrays

The following is an example of using LINQ query syntax to access the elements of a 2-dimensional array. The code forms a Cartesian product between the sequences idx1 and idx2. This Cartesian product is used to access each of the elements of the array Multi1. The example creates an anonymous type (see MSDN article: Anonymous Types (C# Programming Guide) for further information) which contains the indexes and array element value.

There are a couple of other notable points:

  1. The use of the Array.GetLength Method to discover the number of elements that are required in the array index sequences. The use of the Array.GetLength Method results in code that is more robust.
  2. This example demonstrates accessing a 2-dimensional array. A similar pattern will work for any number of array dimensions.
  3. 3. Arrays with more than one dimension do not support the IEnumerable(Of T) Interface. This results in multidimensional arrays not natively, or supported by the implementation of the .Net Framework (see MSDN article: Overview of the .NET Framework for further information). This results in multidimensional arrays not being usable as a LINQ data source. If you attempt to use a multidimensional array as a data source (from ) results in the error CS1935. The following is the full error message
    error CS1935: Could not find an implementation of the query pattern for source type ‘int[*,*]’. ‘Select’ not found. Are you missing a reference to ‘System.Core.dll’ or a using directive for ‘System.Linq’?
private void LINQ_ALternative_Array()
    int[,] Multi1 = new int[,]
        {1,2,3}, {4,5,6}, {7,8,9}
    var dump1 = from idx1 in Enumerable.Range(0, Multi1.GetLength(0))
                from idx2 in Enumerable.Range(0, Multi1.GetLength(1))
                select new { idx1, idx2, val = Multi1[idx1, idx2] };
    Debug.WriteLine("Dumping dump1");
        (val, position) => string.Format("[{0}] [{1},{2}]={3}\n", position, val.idx1, val.idx2, val.val));

    return; // Allows a breakpoint at the end of the method. }


I hope that this blog post and the preceding two blog posts in this series LINQ Short Takes – Number 1 – Enumerable.Range() and LINQ Short Takes – Number 2 – Using Method Syntax to Create a Cartesian Product have added something useful to your kit bag of LINQ tools and techniques.

The approach to working with collections and arrays through an index (or indexes) is one that is applicable to a class of problems that I have not seen used elsewhere. That is not to says that I have conducted an exhaustively search of the web for other examples of this approach.

For those readers who have read LINQ Short Takes – Number 2 – Using Method Syntax to Create a Cartesian Product, and are interested. I indicted in that blog post I would think about an extension method implementation of a general solution to creation of Cartesian products using the Enumerable.Join Method. Well, I wrote that extension method this morning, and will ‘clean it up’, write some XML documentation, and post the solution on this blog sometime soon.

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