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：3SUM-索引问题被引入作为3SUM问题的数据结构版本，旨在通过归约证明静态数据结构的强条件下界。理想情况下，3SUM-索引的猜想困难性应被无条件下界所取代。遗憾的是，我们距离证明这一点仍相去甚远——当前最强的下界是Golovnev等人在STOC'20会议上提出的对数查询时间下界，且该下界仅适用于非自适应数据结构。他们明确提出了对自适应数据结构下界的探索需求。我们的主要贡献恰恰是针对自适应数据结构的这类下界。作为次要成果，我们还强化了Golovnev等人的非自适应下界，并通过一种我们认为具有独立意义的全新方法，证明了面向2比特探测非自适应3SUM-索引数据结构的强下界。