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Combinations of elements with mutual relationships

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Problem Detail: 

I need to create the combination of 3 elements from a array of given n elements, however every of those 3 el. has to be in "relationship" with each other. Here is an example I can describe my issue better on:

We've got elements 1, 2, 3, 4, 5 & 6. We know following relationships:

  • 1 <--> 2
  • 1 <--> 3
  • 1 <--> 4
  • 1 <--> 6
  • 2 <--> 3
  • 2 <--> 5
  • 3 <--> 5
  • 3 <--> 6
  • 4 <--> 5
  • 4 <--> 6

A group {1,2,3} is viable, because 1 is in relationship with either 2 & 3 and 2 is in relationship with 3.

In a group {1,2,4} is 1 in relationship with 2 & 4, however 2 isn't with 4, thus this combination can't exist.

I programmed a functional solution(which involved the above mentioned creating of combinations, redirecting to the list of relations of every element) for small amount of elements and relationship, but I need to have one for approx. 500 000. I'm a high school student and I haven't met a branch of mathematics which would help me to understand or to solve this problem.

I've been searching the net and I think that a relational algebra (binary relations) might be the solution, yet I'm not quiet sure where to start.

Can you think of any hint, advice, web source or view that could help me?

Thank you

Asked By : Michael Bausano
Answered By : David Richerby

The relevant branch of mathematics is graph theory.

Your elements are the vertices of the graph and the relationships between them are the edges. The triples you're looking for are known as triangles: sets of three vertices with all possible edges between them.

I'm not quite sure what you're looking for in your question. If you just want a list of all the triangles in the graph, that can be done naively just by trying all the triples. If you have $n$ vertices in your graph, that takes about $n^3$ tests. More advanced algorithms turn out to be closely related to matrix multiplication, which can be done a little faster – roughly $n^{2.5}$ steps. That might still be rather slow for half a million vertices. If you want to go faster, you'll have to either use approximations (if you're happy with finding most but not necessarily all of the triangles). Alternatively, the particular graph you're interested in might have properties that make it easier to find triangles, for example by ruling out a lot of the possibilities.

Your Google search term is "enumerating triangles in graphs".

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