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arXiv · 2610.42012

3SUM Made Simpler (Without Matrix Multiplication), $k$-SUM Hypothesis Refuted for Odd $k$, and Real 3SUM/APSP Derandomized

Timothy M. Chan·Reina Itakura·Zhiyuan Ma·Yuancheng Yu
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Abstract

Recently, a truly subquadratic algorithm for the 3SUM problem and a truly subcubic algorithm for the all-pairs shortest paths (APSP) problem were found by Anthropic, as reported by Alman and Vassilevska Williams (2026). These algorithms relied on a new subroutine for rectangular matrix multiplication with selected output entries, which was obtained by modifying known matrix multiplication techniques of Schönhage (1981) and Coppersmith (1982). 1. We present a different truly subquadratic randomized algorithm for 3SUM for integer inputs. It completely avoids the use of matrix multiplication but instead modifies known dynamic programming algorithms for subset convolution and the zeta transform, which rely only on simple subtraction tricks via inclusion--exclusion (and does not even require fast Fourier transform), and so is conceptually easier to understand (the whole solution is about 5 pages long). 2. Following a similar approach, we obtain a randomized algorithm for $k$-SUM for integer inputs for odd constant $k$ in $O(n^{\lceil k/2\rceil-\varepsilon_k})$ time for some constant $\varepsilon_k>0$, disproving the $k$-SUM Hypothesis for all odd $k$. 3. Existing reductions from 3SUM and APSP to selected-entries rectangular matrix multiplication were randomized in the case of real-valued input. We present new deterministic reductions, yielding a truly subquadratic deterministic algorithm for Real 3SUM and a truly subcubic deterministic algorithm for Real APSP.

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Timothy M. Chan, Reina Itakura, Zhiyuan Ma, Yuancheng Yu. 2026-10-30. 3SUM Made Simpler (Without Matrix Multiplication), $k$-SUM Hypothesis Refuted for Odd $k$, and Real 3SUM/APSP Derandomized. https://arxiv.org/abs/2610.42012

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