# $$\mathrm {3SUM}$$, $$\mathrm {3XOR}$$, 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 $$\alpha > 0$$.

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## Notes

1. In [5, 20] they appear to use this lemma with a hash function that is not known to satisfy the hypothesis of the lemma. However probably one can use instead similar hash functions such as one in [9] that does satisfy the hypothesis. We thank Martin Dietzfelbinger for a discussion on hash functions.

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## Acknowledgments

We are very grateful to Rasmus Pagh and Virginia Vassilevska Williams for answering many questions on finding triangles in graphs. Rasmus also pointed us to [3, 21]. We also thank Siyao Guo for pointing out that a step in a previous proof of Lemma 14 was useless, and Ryan Williams for stimulating discussions. Finally, we thank the anonymous referees for their helpful comments.

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Correspondence to Zahra Jafargholi.

Supported by NSF Grants CCF-0845003, CCF-1319206.

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Jafargholi, Z., Viola, E. $$\mathrm {3SUM}$$, $$\mathrm {3XOR}$$, Triangles. Algorithmica 74, 326–343 (2016). https://doi.org/10.1007/s00453-014-9946-9

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• DOI: https://doi.org/10.1007/s00453-014-9946-9

### Keywords

• $$\mathrm {3SUM}$$
• $$\mathrm {3XOR}$$
• Triangles
• Algorithms
• Derandomization