$$\mathrm {3SUM}$$ 3 SUM , $$\mathrm {3XOR}$$ 3 XOR , Triangles

AbstractP traşcu (STOC '10) reduces the $$\mathrm {3SUM}$$3SUM problem to listing triangles in a graph. In the other direction, we show that if one can solve $$\mathrm {3SUM}$$3SUM on a set of size $$n$$n in time $$n^{1+\epsilon }$$n1+∈ then one can list $$t$$t triangles in a graph with $$m$$m edges in time $$\tilde{O}(m^{1+\epsilon }t^{1/3-\epsilon /3})$$O~(m1+∈t1/3-∈/3). Our result builds on and extends works by the Paghs (PODS '06) and by Vassilevska and Williams (FOCS '10). We make our reductions deterministic using tools from pseudorandomness. We then re-execute both P traşcu's reduction and ours for the variant $$\mathrm {3XOR}$$3XOR of $$\mathrm {3SUM}$$3SUM where integer summation is replaced by bit-wise xor. As a corollary we obtain that if $$\mathrm {3XOR}$$3XOR is solvable in linear time but $$\mathrm {3SUM}$$3SUM requires quadratic randomized time, or vice versa, then the randomized time complexity of listing $$m$$m triangles in a graph with $$m$$m edges is $$m^{4/3}$$m4/3 up to a factor $$m^\alpha $$m for any $$\alpha > 0$$ >0.