BibTeX
@MISC{Jafargholi133sum,3xor,,
author = {Zahra Jafargholi and Emanuele Viola},
title = {3SUM, 3XOR, Triangles},
year = {2013}
}
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Abstract
We show that if one can solve 3SUM on a set of size n in time n1+ɛ then one can list t triangles in a graph with m edges in time Õ(m1+ɛt1/3+ɛ ′ ) for any ɛ ′> 0. This is a reversal of Pǎtra¸scu’s reduction from 3SUM to listing triangles (STOC ’10). We then re-execute both Pǎtra¸scu’s reduction and our reversal for the variant 3XOR of 3SUM where integer summation is replaced by bit-wise xor. As a corollary we obtain that if 3XOR is solvable in linear time but 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 α for any α> 0. Our results are obtained building on and extending works by the Paghs (PODS ’06) and by Vassilevska and Williams (FOCS ’10).
