Listing Triangles

Abstract

We present new algorithms for listing triangles in dense and sparse graphs. The running time of our algorithm for dense graphs is $\tilde{\mathcal O}(n^{\omega} + n^{3(\omega-1)/(5-\omega)} t^{2(3-\omega)/(5-\omega)})$, and the running time of the algorithm for sparse graphs is $\tilde{\mathcal O}(m^{{2\omega}/{(\omega+1)}} + m^{{3(\omega-1)}/{(\omega+1)}} t^{{(3-\omega)}/{(\omega+1)}})$, where n is the number of vertices, m is the number of edges, t is the number of triangles to be listed, and ω < 2.373 is the exponent of fast matrix multiplication. With the current bound on ω, the running times of our algorithms are $\tilde{\mathcal O}( n^{2.373} + n^{1.568}\, t^{0.478})$ and $\tilde{\mathcal O}(m^{1.408} + m^{1.222}\, t^{0.186})$, respectively. We first obtain randomized algorithms with the desired running times and then derandomize them using sparse recovery techniques.

If ω = 2, the running times of the algorithms become $\tilde{\mathcal O}(n^2+nt^{2/3})$ and $\tilde{\mathcal O}(m^{4/3}+mt^{1/3})$, respectively. In particular, if ω = 2, our algorithm lists m triangles in $\tilde{\mathcal O}(m^{4/3})$ time. Pǎtraşcu (STOC 2010) showed that Ω(m ^{4/3 − o(1)}) time is required for listing m triangles, unless there exist subquadratic algorithms for 3SUM. We show that unless one can solve quadratic equation systems over a finite field significantly faster than the brute force algorithm, our triangle listing runtime bounds are tight assuming ω = 2, also for graphs with more triangles.