Preprocessed 3SUM for Unknown Universes with Subquadratic Space
Yael Kirkpatrick, John Kuszmaul, Surya Mathialagan, Virginia Vassilevska Williams
TL;DR
This work resolves a long-standing question in fine-grained complexity for 3SUM in preprocessed universes with unknown $C$ by giving a randomized data structure that preprocesses two $n$-size sets in $\tilde{O}(n^2)$ time and uses $\tilde{O}(n^{2-2\varepsilon/3})$ space to answer queries in $\tilde{O}(n^{1.5+\varepsilon})$ time for any $\varepsilon\in[0,1/2]$ (with the query guarantee holding against an oblivious adversary). The construction hinges on a hybrid of heavy-hitter handling with mod-$p$ sumset computations via FFT and Fiat-Naor function inversion on partitioned instances, carefully balancing parameters to achieve a subquadratic space-time tradeoff while maintaining quadratic preprocessing. This closes the gap between subquadratic query time and subquadratic space for unknown-$C$ 3SUM, subject to standard 3SUM hypotheses, and opens avenues for applying similar multi-component preprocessing techniques to related problems such as 3XOR. The results demonstrate a concrete tradeoff curve between space and query time and show that with further advances in primitive function inversion, even stronger subquadratic-space, subquadratic-time preprocessed structures may be attainable.
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.