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.

翻译：$3$SUM-Indexing问题被引入作为$3$SUM问题的数据结构版本，其目标是通过归约证明静态数据结构的强条件下限。理想情况下，$3$SUM-Indexing的猜想难度应该被无条件下限所取代。不幸的是，我们还远远无法证明这一点，目前最强的下限是Golovnev等人在STOC'20中给出的对数查询时间下限。此外，他们的下限仅适用于非自适应数据结构，并明确要求自适应数据结构的下限。我们的主要贡献正是这样一个针对自适应数据结构的下限。作为辅助结果，我们还通过一种全新的方法加强了Golovnev等人的非自适应下限，并证明了对于$2$-bit-probe非自适应3SUM-Indexing数据结构的强下限，这个方法本身也很有趣。