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

Abstract

Pǎtraşcu (STOC ’10) reduces the $\mathrm {3SUM}$ problem to listing triangles in a graph. In the other direction, we show that if one can solve $\mathrm {3SUM}$ on a set of size $n$ in time $n^{1+\epsilon }$ then one can list $t$ triangles in a graph with $m$ edges in time $\tilde{O}(m^{1+\epsilon }t^{1/3-\epsilon /3})$. 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 Pǎtraşcu’s reduction and ours for the variant $\mathrm {3XOR}$ of $\mathrm {3SUM}$ where integer summation is replaced by bit-wise xor. As a corollary we obtain that if $\mathrm {3XOR}$ is solvable in linear time but $\mathrm {3SUM}$ requires quadratic randomized time, or vice versa, then the randomized time complexity of listing $m$ triangles in a graph with $m$ edges is $m^{4/3}$ up to a factor $m^\alpha$ for any $\alpha > 0$.