Step by Step solution about How to resolve the algorithm Cartesian product of two or more lists step by step in the C# programming language

The provided C# code defines two ways to compute the Cartesian product of a list of lists, a fundamental operation in combinatorial mathematics. Here's a detailed explanation:

1. IEnumerable Extension Method:

   • The CartesianProduct extension method is defined in the Extensions class, which takes a list of lists (IEnumerable<IEnumerable<T>>) as input.
   • It calculates the Cartesian product of the input sequences using a recursive algorithm. The algorithm starts with an empty product (Enumerable.Empty<T>()) and gradually combines the elements from each input sequence to generate all possible combinations.

2. LINQ Queries:

   • The code also demonstrates how to compute the Cartesian product using LINQ queries. There are two examples:
     • cart1 computes the Cartesian product of list1 and list2, resulting in a collection of tuples (a, b).
     • cart2 computes the Cartesian product of list7, list8, and list9, resulting in a collection of tuples (a, b, c).

3. Usage:

   • The Main method creates several sample lists and uses the CartesianProduct method to compute their Cartesian products.
   • It then formats the results as strings and displays them on the console.

Additional Notes:

• The Cartesian product of two lists is the set of all possible ordered pairs where the first element comes from the first list and the second element comes from the second list.
• The Cartesian product of multiple lists is the set of all possible tuples where each element comes from a corresponding list.
• The CartesianProduct method is a generic method, which means it can be used with any type of object, not just integers as shown in the example.
• The Select operator is used to transform the Cartesian product into a collection of strings for display purposes.

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 }));
    }
}


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)}}}");
}