Improved Subquadratic 3SUM

Abstract

In the 3SUM problem we are given three lists $\mathcal {A}$, $\mathcal {B}$, $\mathcal {C}$, of n real numbers, and are asked to find $(a,b,c)\in \mathcal {A}\times \mathcal {B}\times \mathcal {C}$ such that $a+b=c$. The longstanding 3SUM conjecture—that 3SUM could not be solved in subquadratic time—was recently refuted by Grønlund and Pettie. They presented $\hbox {O}\left( n^2(\log \log n)^{\alpha }/(\log n)^{\beta }\right)$ algorithms for 3SUM and for the related problems Convolution3SUM and ZeroTriangle, where $\alpha$ and $\beta$ are constants that depend on the problem and whether the algorithm is deterministic or randomized (and for ZeroTriangle the main factor is $n^3$ rather than $n^2$). We simplify Grønlund and Pettie’s algorithms and obtain better bounds, namely, $\alpha =\beta =1$, deterministically. For 3SUM our bound is better than both the deterministic and the randomized bounds of Grønlund and Pettie. For the other problems our bounds are better than their deterministic bounds, and match their randomized bounds.