3SUM in Preprocessed Universes: Faster and Simpler
Shashwat Kasliwal, Adam Polak, Pratyush Sharma

TL;DR
This paper introduces a faster, simpler algorithm for the 3SUM problem in preprocessed universes, achieving improved efficiency using standard techniques like FFT and hashing, and extending to deterministic solutions and unknown sets.
Contribution
The paper presents a novel 3SUM algorithm that is faster, simpler, and relies only on standard techniques, improving upon previous combinatorics-based methods.
Findings
Quadratic preprocessing time for sets of size n
Query time of O(n^{1.5} log n) for subset checks
Deterministic version with only polylogarithmic slowdown
Abstract
We revisit the 3SUM problem in the \emph{preprocessed universes} setting. We present an algorithm that, given three sets , , of integers, preprocesses them in quadratic time, so that given any subsets , , , it can decide if there exist , , with in time . In contrast to both the first subquadratic -time algorithm by Chan and Lewenstein (STOC 2015) and the current fastest -time algorithm by Chan, Vassilevska Williams, and Xu (STOC 2023), which are based on the Balog--Szemer\'edi--Gowers theorem from additive combinatorics, our algorithm uses only standard 3SUM-related techniques, namely FFT and linear hashing modulo a prime. It is therefore not only faster but also simpler. Just as the two previous algorithms, ours not only…
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Taxonomy
Topicsdemographic modeling and climate adaptation · Transportation and Mobility Innovations · Parallel Computing and Optimization Techniques
