Group alternate pairs using LINQ

I doubt a bit that LINQ is the best choice here as a loop with lookup might be a bit more efficient. However if you really need LINQ, then you could do the following

var relations = from person in relationDTOList
    // Match on the exact pair of IDs
    join relative in relationDTOList on
        new { person.PersonId, person.RelativeId } equals
        new { PersonId = relative.RelativeId, RelativeId = relative.PersonId }

    // Build the new structure
    let relation = new Relations {
        PersonId = person.PersonId,
        Relation = person.Relation,
        RelativeId = relative.PersonId,
        ReverseRelation = relative.Relation
    }

    // Order the pairs to find the duplicates
    let ids = new[] {person.PersonId, relative.PersonId}.OrderBy(x => x).ToArray()
    group relation by new { FirstPersonId = ids[0], SecondPersonId = ids[1] }
    into relationGroups

    // Select only the the first of two duplicates
    select relationGroups.First();

What this code does is joins the collection with itself on the matching pairs PersonId, RelativeId and then filters out the second record of each pair thus resulting in a collection where the first person found in the list will be considered as parent in the relation.

---

EDIT: The lookup method I was talking about:

var result = new List<Relations>();
while (relationDTOList.Any())
{
    var person = relationDTOList.First();
    relationDTOList.RemoveAt(0);

    var relative = relationDTOList.Where(x =>
            x.PersonId == person.RelativeId && x.RelativeId == person.PersonId)
        .Select((x, i) => new {Person = x, Index = i}).FirstOrDefault();

    if (relative != null)
    {
        relationDTOList.RemoveAt(relative.Index);
        result.Add(new Relations {
            PersonId = person.PersonId,
            Relation = person.Relation,
            RelativeId = relative.Person.PersonId,
            ReverseRelation = relative.Person.Relation
        });
    }
}

As a note, it empties your original list so you have to make a copy (list.ToList()) if you need it further in your code.

Running this code turned out to be about six times faster than the method with join I posted before. I also came up with the following grouping method which runs much faster than the join, however it's still slower than the lookup-and-remove method although they do a very similar thing.

var relations = relationDTOList.GroupBy(person =>
        person.PersonId < person.RelativeId
            ? new {FirstPersonId = person.PersonId, SecondPersonId = person.RelativeId}
            : new {FirstPersonId = person.RelativeId, SecondPersonId = person.PersonId})

    .Select(group => new Relations {
        PersonId = group.First().PersonId,
        Relation = group.First().Relation,
        RelativeId = group.First().RelativeId,
        ReverseRelation = group.Last().Relation
    });

var query = relationDTOList.OrderBy(x=>x.PersonId).GroupJoin(relationDTOList,
p => p.PersonId,
a => a.RelativeId,
(p, al) =>
new
{
     p.PersonId,
     p.RelativeId,
     p.Relation,
     Parrent = al.Where(x => x.PersonId == p.RelativeId && x.RelativeId == p.PersonId).SingleOrDefault().Relation
 }
 ).ToList();

You can use a join operation like

var result = relationDTOList
.Where(v => v.PersonId < v.RelativeId)
.Join(
    relationDTOList.Where(v => v.PersonId > v.RelativeId),
    v => new Key{PersonId = v.PersonId, RelativeId = v.RelativeId},
    v => new Key{PersonId = v.RelativeId, RelativeId = v.PersonId},
    (p, q) => new Relations
    {
        PersonId = p.PersonId,
        RelativeId = p.RelativeId,
        Relation = p.Relation,
        ReverseRelation = q.Relation
    }
);

The Key is:

public struct Key
{
    public int PersonId { get; set; }
    public int RelativeId { get; set; }
}

I'm not sure whether it is what you need:

public static void Main()
{
    List<RelationDTO> relationDTOList = new List<RelationDTO>
    {
        new RelationDTO { PersonId = 1, RelativeId = 2, Relation = "Son" },
        new RelationDTO { PersonId = 2, RelativeId = 1, Relation = "Father" },

        new RelationDTO { PersonId = 1, RelativeId = 3, Relation = "Mother" },
        new RelationDTO { PersonId = 3, RelativeId = 1, Relation = "Son" },

        new RelationDTO { PersonId = 2, RelativeId = 3, Relation = "Husband" },
        new RelationDTO { PersonId = 3, RelativeId = 2, Relation = "Wife" },
    };

    var grp = relationDTOList.Join(relationDTOList, 
            dto => dto.PersonId + "-" + dto.RelativeId, 
            dto => dto.RelativeId + "-" + dto.PersonId, 
    (dto1, dto2) => new Relations 
            { 
                PersonId = dto1.PersonId, 
                RelationId = dto1.RelativeId, 
                Relation = dto1.Relation, 
                ReverseRelation = dto2.Relation 
                }).Distinct(new MyEqualityComparer());

    foreach (var g in grp)
        Console.WriteLine("{0},{1},{2},{3}", g.PersonId, g.RelationId, g.Relation, g.ReverseRelation);
}

public class MyEqualityComparer : IEqualityComparer<Relations>
{
    public bool Equals(Relations x, Relations y)
    {
        return x.PersonId + "-" + x.RelationId == y.PersonId + "-" + y.RelationId || 
        x.PersonId + "-" + x.RelationId == y.RelationId + "-" + y.PersonId;
    }

    public int GetHashCode(Relations obj)
    {
        return 0;
    }
}