How to find a triangle inside a graph?

Question

Here is an exercise in the Algorithm Design Manual.

Consider the problem of determining whether a given undirected graph G = (V, E) contains a triangle

Answer 1

If you have an adjacency matrix, you can find triangles by squaring the matrix and seeing if the original matrix and square matrix have a non-zero entry in the same place.

A naive matrix multiplication takes time O(n^3), but there are fast matrix multiplication algorithms that do better. One of the best known is the Coppersmith-Winograd algorithm, which runs in O(n^2.4) time. That means the algorithm goes something like:

• Use O(V^2) time to convert to an adjacency matrix.
• Use O(V^2.4) time to compute the square of the adjacency matrix.
• Use O(V^2) time to check over the matrices for coinciding non-zero entries.
• The index of the row and column where you find coinciding non-zero entries in (if any) tell you two of the involved nodes.
• Use O(V) time to narrow down the third node common to both the known nodes.

So overall this takes O(V^2.4) time; more precisely it takes however long matrix multiplication takes. You can dynamically switch between this algorithm and the if-any-edge-end-points-have-a-common-neighbor algorithm that amit explains in their answer to improve that to O(V min(V^1.4, E)).

Here's a paper that goes more in-depth into the problem.

It's kind of neat how dependent-on-theoretical-discoveries this problem is. If conjectures about matrix multiplication actually being quadratic turn out to be true, then you would get a really nice time bound of O(V^2) or O(V^2 log(V)) or something like that. But if quantum computers work out, we'll be able to do even better than that (something like O(V^1.3))!