3SUM in Preprocessed Universes: Faster and Simpler

2024-10-22

SIAM Symposium on Simplicity in Algorithms

AI Summary

This paper studies the 3SUM problem in the preprocessing model: given three $n$-element integer sets $A$, $B$, and $C$, preprocess them to enable fast queries—specifically, for arbitrary subsets $A' subseteq A$, $B' subseteq B$, $C' subseteq C$, determine whether there exist $a in A'$, $b in B'$, $c in C'$ satisfying $a + b = c$. We present the first subquadratic query algorithm relying solely on FFT and linear modular hashing over prime fields. Our preprocessing time is $O(n^2)$, and query time is $O(n^{1.5} log n)$, improving upon the prior best $ ilde{O}(n^{11/6})$. The algorithm supports dynamic preprocessing when $C$ is initially unknown, enables element-wise membership queries for $c in C'$, and is fully deterministic—introducing only polylogarithmic overhead. Crucially, our approach avoids sophisticated tools from additive combinatorics, achieving both conceptual simplicity and strong practical implementability.

Technology Category

• Data Mining & Knowledge Management: Intelligent Query Processing
• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Graph Algorithms and Modeling for the Web: Querying, indexing, and retrieval in Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• User Modeling, Personalization and Recommendation: Attacks and countermeasures in recommendation systems

Abstract

We revisit the 3SUM problem in the emph{preprocessed universes} setting. We present an algorithm that, given three sets $A$, $B$, $C$ of $n$ integers, preprocesses them in quadratic time, so that given any subsets $A' subseteq A$, $B' subseteq B$, $C' subseteq C$, it can decide if there exist $a in A'$, $b in B'$, $c in C'$ with $a+b=c$ in time $O(n^{1.5} log n)$. In contrast to both the first subquadratic $ ilde{O}(n^{13/7})$-time algorithm by Chan and Lewenstein (STOC 2015) and the current fastest $ ilde{O}(n^{11/6})$-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 decides if there is a single 3SUM solution but it actually determines for each $c in C'$ if there is a solution containing it. We also modify the algorithm to still work in the scenario where the set $C$ is unknown at the time of preprocessing. Finally, even though the simplest version of our algorithm is randomized, we show how to make it deterministic losing only polylogarithmic factors in the running time.

Problem

• Solving 3SUM problem with faster preprocessing time
• Using simpler techniques like FFT and hashing
• Handling unknown subset C during preprocessing phase

Innovation

• Uses FFT and linear hashing modulo prime
• Preprocesses sets in quadratic time initially
• Achieves O(n^1.5 log n) query time complexity