First look: Linear algebra-based triangle counting without matrix multiplication

Linear algebra-based approaches to exact triangle counting often require sparse matrix multiplication as a primitive operation. Non-linear algebra approaches to the same problem often assume that the adjacency matrix of the graph is not available. In this paper, we show that both approaches can be unified into a single approach that separates the data format from the algorithm design. By not casting the triangle counting algorithm into matrix multiplication, a different algorithm that counts each triangle exactly once can be identified. In addition, by choosing the appropriate sparse matrix format, we show that the same algorithm is equivalent to the compact-forward algorithm attained assuming that the adjacency matrix of the graph is not available. We show that our approach yields an initial implementation that is between 69 and more than 2000 times faster than the reference implementation. We also show that the initial implementation can be easily parallelized on shared memory systems.