Stronger 3SUM-Indexing Lower Bounds

TL;DR

This paper establishes new strong lower bounds for the 3SUM-Indexing problem, including the first adaptive data structure lower bounds and improved bounds for non-adaptive cases, advancing understanding of its computational hardness.

Contribution

It provides the first lower bounds for adaptive data structures for 3SUM-Indexing and strengthens existing bounds for non-adaptive data structures using a novel approach.

Findings

First adaptive data structure lower bounds for 3SUM-Indexing.

Strengthened non-adaptive lower bounds for 2-bit probes.

New techniques for proving data structure lower bounds.

Abstract

The $3$SUM-Indexing problem was introduced as a data structure version of the $3$SUM problem, with the goal of proving strong conditional lower bounds for static data structures via reductions. Ideally, the conjectured hardness of $3$SUM-Indexing should be replaced by an unconditional lower bound. Unfortunately, we are far from proving this, with the strongest current lower bound being a logarithmic query time lower bound by Golovnev et al. from STOC'20. Moreover, their lower bound holds only for non-adaptive data structures and they explicitly asked for a lower bound for adaptive data structures. Our main contribution is precisely such a lower bound against adaptive data structures. As a secondary result, we also strengthen the non-adaptive lower bound of Golovnev et al. and prove strong lower bounds for $2$-bit-probe non-adaptive $3$SUM-Indexing data structures via a completely new approach that we find interesting in its own right.