How to Efficiently Compute Cartesian Products in LINQ with an Unknown Number of Sets?

Cartesian product in LINQ

In the field of data processing, Cartesian product is a basic operation that combines elements from multiple sets to create new combinations. This article explores a LINQ-based approach to implementing Cartesian products, specifically for situations where the number of collections is unknown at compile time.

Understanding the Cartesian product

Simply put, the Cartesian product combines every element in one set with every element in the other set, generating all possible pairings. Consider a set of people {p1, p2, p3} and a set of dogs {d1, d2, d3}. The Cartesian product of these two sets will be:

<code>{(p1, d1), (p1, d2), (p1, d3),
 (p2, d1), (p2, d2), (p2, d3),
 (p3, d1), (p3, d2), (p3, d3)}</code>

LINQ implementation

In order to perform Cartesian product in LINQ, we can utilize the SelectMany operator. However, when the number of collections is unknown at compile time, we need a more flexible approach. This is where the CartesianProduct method comes into play:

<code>public static IEnumerable<IEnumerable<T>> CartesianProduct<T>(this IEnumerable<IEnumerable<T>> sequences)
{
    // 基本情况：空序列
    if (!sequences.Any())
        return new[] { Enumerable.Empty<T>() };

    // 简化为两个序列
    var first = sequences.First();
    var rest = sequences.Skip(1);

    // 递归计算其余部分的笛卡尔积
    var cartesianProducts = rest.CartesianProduct();

    // 将第一个序列与每个笛卡尔积组合
    return from f in first
           from c in cartesianProducts
           select f.Concat(c);
}</code>

Example usage

Consider our previous example of a person with three dogs, each with a different number of puppies:

<code>Person person = ...;
var puppyCombinations = CartesianProduct(from dog in person.Dogs select dog.Puppies);</code>

This code will generate all possible combinations of puppies for each dog, similar to the SQL query mentioned in the question:

<code>{(puppyA, puppyA),
 (puppyA, puppyB),
 (puppyB, puppyA),
 (puppyB, puppyB)}</code>

Conclusion

By using the CartesianProduct method, we can calculate Cartesian products in LINQ flexibly and efficiently, even when the number of collections is unknown at compile time. This opens up possibilities for a variety of data processing and combination tasks.

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