Preprocessed 3SUM for Unknown Universes with Subquadratic Space

Abstract

We consider the classic 3SUM problem: given sets of integers A, B, C, determine whether there is a tuple (a, b, c) ∈ A × B × C satisfying a + b = c. 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' ⊆ A and B' ⊆ B. A series of results have achieved Õ(n²) preprocessing time, Õ(n²) space, and query time improving progressively from Õ(n^{1.9}) [Timothy M. Chan and Moshe Lewenstein, 2015] to Õ(n^{11/6}) [Timothy M. Chan et al., 2023] to Õ(n^{1.5}) [Kasliwal et al., 2025]. 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 Õ(n^{1.5 + ε}) query time and Õ(n^{2 - 2ε/3}) space complexity. Furthermore, our algorithm has Õ(n²) 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.