Preprocessed 3SUM for Unknown Universes with Subquadratic Space

2026-02-11

AI Summary

This work addresses the preprocessing version of the 3SUM problem, aiming to overcome the long-standing barrier that at least one of space or query time must be quadratic. We present the first data structure in this setting that achieves a truly subquadratic trade-off: after Õ(n²) preprocessing time, it constructs a structure occupying Õ(n^{2−2ε/3}) space and supports 3SUM queries for arbitrary subsets A' ⊆ A, B' ⊆ B, and a query set C' in Õ(n^{1.5+ε}) time. This result significantly improves upon prior approaches by simultaneously achieving subquadratic space and subquadratic query time—the first such breakthrough in the literature.

Technology Category

• Data Mining & Knowledge Management: Intelligent Query Processing
• Search and Optimization: Distributed Search
• Machine Learning: Structured Learning

Application Category

• Graph Algorithms and Modeling for the Web: Querying, indexing, and retrieval in Web-related graphs
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies
• Web Mining and Content Analysis: Bridging structured and unstructured data

Abstract

We consider the classic 3SUM problem: given sets of integers $A, B, C $, determine whether there is a tuple $(a, b, c) \in A \times B \times C$ satisfying $a + b + c = 0$. The 3SUM Hypothesis, central in fine-grained complexity, states that there does not exist a truly subquadratic time 3SUM algorithm. Given this long-standing barrier, recent work over the past decade has explored 3SUM from a data structural perspective. Specifically, in the 3SUM in preprocessed universes regime, we are tasked with preprocessing sets $A, B$ of size $n$, to create a space-efficient data structure that can quickly answer queries, each of which is a 3SUM problem of the form $A', B', C'$, where $A'\subseteq A$ and $B'\subseteq B$. A series of results have achieved $\tilde{O}(n^2)$ preprocessing time, $\tilde{O}(n^2)$ space, and query time improving progressively from $\tilde{O}(n^{1.9})$ [CL15] to $\tilde{O}(n^{11/6})$ [CVX23] to $\tilde{O}(n^{1.5})$ [KPS25]. Given these series of works improving query time, a natural open question has emerged: can one achieve both truly subquadratic space and truly subquadratic query time for 3SUM in preprocessed universes? We resolve this question affirmatively, presenting a tradeoff curve between query and space complexity. Specifically, we present a simple randomized algorithm achieving $\tilde{O}(n^{1.5 + \varepsilon})$ query time and $\tilde{O}(n^{2 - 2\varepsilon/3})$ space complexity. Furthermore, our algorithm has $\tilde{O}(n^2)$ preprocessing time, matching past work. Notably, quadratic preprocessing is likely necessary for our tradeoff as either the preprocessing or the query time must be at least $n^{2-o(1)}$ under the 3SUM Hypothesis.

Problem

• 3SUM
• preprocessing
• subquadratic space
• query time
• data structure

Innovation

• 3SUM
• preprocessing
• subquadratic space
• query time tradeoff
• fine-grained complexity