Listing triangles

We present new algorithms for listing triangles in dense and sparse graphs. The running time of our algorithm for dense graphs is Õ(n ^ω + n^{3(ω-1)/(5-ω)}t ^{2(3-ω)/(5-ω)}), and the running time of the algorithm for sparse graphs is Õ(m^2ω/(ω+1) + m ^{3(ω-1)/(ω+1)}t^{(3-ω)/(ω+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 Õ(n^2.373 + n^1.568 t^0.478) and Õ(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 Õ(n² + nt ^2/3) and Õ(m^4/3 + mt^1/3), respectively. In particular, if ω = 2, our algorithm lists m triangles in Õ(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.