Preprocessed 3SUM for Unknown Universes with Subquadratic Space
Yael Kirkpatrick, John Kuszmaul, Surya Mathialagan, Virginia Vassilevska Williams

TL;DR
This paper introduces a randomized data structure for the 3SUM problem in preprocessed universes that achieves a tradeoff between subquadratic query time and space, breaking previous limitations.
Contribution
It presents a simple randomized algorithm that balances query time and space, achieving subquadratic bounds in both, and matches quadratic preprocessing time, addressing an open problem.
Findings
Achieves $ ilde{O}(n^{1.5 + ext{epsilon}})$ query time with $ ilde{O}(n^{2 - 2 ext{epsilon}/3})$ space.
Preprocessing time is $ ilde{O}(n^2)$, matching prior work.
Quadratic preprocessing or query time is likely necessary under the 3SUM Hypothesis.
Abstract
We consider the classic 3SUM problem: given sets of integers , determine whether there is a tuple satisfying . 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 of size , to create a space-efficient data structure that can quickly answer queries, each of which is a 3SUM problem of the form , where and . A series of results have achieved preprocessing time, space, and query time improving progressively from [CL15] to…
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Taxonomy
TopicsComplexity and Algorithms in Graphs · Algorithms and Data Compression · Data Management and Algorithms
