Listing Triangles

We present new algorithms for listing triangles in dense and sparse graphs. The running time of our algorithm for dense graphs is O~(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 O~(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 O~(n^2.373+n^1.568t^0.478) and O~(m^1.408+m^1.222t^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 O~(n^2+nt^2/3) and O~(m^4/3+mt^1/3), respectively. In particular, if ω = 2, our algorithm lists m triangles in O~(m4/3) 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