Truly subquadratic algorithms for 3SUM-hard problems

3SUM now has a truly subquadratic algorithm; find one for a 3SUM-hard geometric problem, such as deciding whether n points have three on a line.

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1	# Truly subquadratic algorithms for 3SUM-hard problems
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3	Given $n$ numbers, do three of them sum to zero? For decades this 3SUM problem was believed to need about $n^2$ time, and dozens of geometric problems were shown to be at least as hard. A 2026 preprint gives a truly subquadratic algorithm for 3SUM itself, but the reductions run only one way, so whether those geometric problems, such as finding three collinear points, can be solved in time $O(n^{2-\varepsilon})$ is now wide open.
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5	## Problem
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7	3SUM. Given sets $A$, $B$, $C$ of numbers with $\|A\|+\|B\|+\|C\|=n$, decide whether there are $a\in A$, $b\in B$, $c\in C$ with $a+b=c$. After sorting, a linear scan for each $a\in A$ solves this in $O(n^2)$ time.
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9	A problem is 3SUM-hard if 3SUM can be reduced to it with $o(n^2)$ overhead, so that a subquadratic algorithm for it would give one for 3SUM. Examples from computational geometry include deciding whether $n$ points in the plane contain three on a line (3-collinearity), and hence finding a minimum-area triangle, as well as some point-covering, motion-planning and polygon-containment problems [1], [5].
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11	Question. Can 3SUM-hard problems be solved in truly subquadratic time, $O(n^{2-\varepsilon})$ for some constant $\varepsilon>0$? The central test case is 3-collinearity, for points with integer coordinates of polynomial size or for real points on the real RAM, possibly with randomization.
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13	## Background
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15	Gajentaan and Overmars (1995) introduced 3SUM-hardness to explain why many geometric problems seemed stuck at quadratic time [1]; this is Problem 11 of The Open Problems Project. The conjecture that 3SUM needs $n^{2-o(1)}$ time was later used to derive lower bounds for dynamic data structures, triangle listing and string problems ([Kopelowitz, Pettie and Porat, 2016](https://arxiv.org/abs/1407.6756)). In October 2026 Alman and Vassilevska Williams refuted that conjecture [5]. Their algorithm speeds up 3SUM and problems that reduce to it, but not the geometric problems, which are only known to be at least as hard as 3SUM.
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17	## What is known
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19	- 3SUM itself. Alman and Vassilevska Williams solve 3SUM on $n$ integers of polynomial size deterministically in $O(n^{1.9992})$ time, and on real numbers by a Las Vegas algorithm in $O(n^{1.998})$ expected time using only comparisons, additions and subtractions [5]. The integer result, with every lemma it relies on, has been formalized in Lean 4 with Mathlib on an explicit word-RAM model; the real-number result has not. Majdoddin improved the integer exponent to $1.9945+o(1)$ [6]. Both are unrefereed preprints from October 2026.
20	- Geometric problems. The authors of [5] note that whether three collinear points can be found in truly subquadratic time is now open, since the known reductions go from 3SUM to these problems and not back.
21	- Logarithmic speed-ups. Before 2026, the fastest 3SUM algorithms saved only logarithmic factors ([Grønlund and Pettie, 2014](https://arxiv.org/abs/1404.0799); Chan, 2018 [4]), and Chan's methods also give logarithmic-factor speed-ups for some geometric 3SUM-hard problems [4].
22	- Decision trees. In linear decision trees whose queries involve only three input numbers, 3SUM needs $\Omega(n^2)$ queries (Erickson, 1999) [2], while general linear decision trees solve it with $O(n\log^2 n)$ queries (Kane, Lovett and Moran, 2019) [3].
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24	## What would count as progress
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26	- A truly subquadratic algorithm for 3-collinearity, or for another 3SUM-hard geometric problem from [1], with a proof of its running time.
27	- A proof that 3-collinearity needs $n^{2-o(1)}$ time under a hypothesis that the 2026 algorithm leaves standing; the authors of [5] list Orthogonal Vectors and CNF-SAT among the problems it does not speed up.
28	- An independent check of [5], or a formal proof of its real-number result.
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30	## References
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32	1. A. Gajentaan and M. H. Overmars, On a class of $O(n^2)$ problems in computational geometry, Comput. Geom. 5 (1995), 165–185. [doi:10.1016/0925-7721(95)00022-2](https://doi.org/10.1016/0925-7721(95)00022-2)
33	2. J. Erickson, Lower bounds for linear satisfiability problems, Chicago J. Theoret. Comput. Sci. 1999(8). [cjtcs.cs.uchicago.edu](http://cjtcs.cs.uchicago.edu/articles/1999/8/contents.html)
34	3. D. M. Kane, S. Lovett and S. Moran, Near-optimal linear decision trees for $k$-SUM and related problems, J. ACM 66 (2019); STOC 2018. [doi:10.1145/3285953](https://doi.org/10.1145/3285953), [arXiv:1705.01720](https://arxiv.org/abs/1705.01720)
35	4. T. M. Chan, More logarithmic-factor speedups for 3SUM, (median,+)-convolution, and some geometric 3SUM-hard problems, Proc. SODA 2018, 881–897. [doi:10.1137/1.9781611975031.57](https://doi.org/10.1137/1.9781611975031.57)
36	5. J. Alman and V. Vassilevska Williams, Truly subquadratic 3SUM and truly subcubic APSP via triangles in sparse lopsided graphs, preprint (2026). [arXiv:2610.06783](https://arxiv.org/abs/2610.06783)
37	6. R. Majdoddin, Thin matrix products up to $D\le N^{0.172}$ and lossless reductions for 3SUM, Exact Triangle and APSP, preprint (2026). [arXiv:2610.17420](https://arxiv.org/abs/2610.17420)
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39	## Source
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41	Imported from [The Open Problems Project, Problem 11](https://topp.openproblem.net/p11), curated by Erik D. Demaine, Joseph S. B. Mitchell and Joseph O'Rourke. Open according to the source as of 2026-10-20. The statement above is written in our own words; see the source for the original wording, history and discussion.