In this paper, we study the static cell probe complexity of non-adaptive data structures that maintain a subset of $n$ points from a universe consisting of $m=n^{1+\Omega(1)}$ points. A data structure is defined to be non-adaptive when the memory locations that are chosen to be accessed during a query depend only on the query inputs and not on the contents of memory. We prove an $\Omega(\log m / \log (sw/n\log m))$ static cell probe complexity lower bound for non-adaptive data structures that solve the fundamental dictionary problem where $s$ denotes the space of the data structure in the number of cells and $w$ is the cell size in bits. Our lower bounds hold for all word sizes including the bit probe model ($w = 1$) and are matched by the upper bounds of Boninger et al. [FSTTCS'17]. Our results imply a sharp dichotomy between dictionary data structures with one round of adaptive and at least two rounds of adaptivity. We show that $O(1)$, or $O(\log^{1-\epsilon}(m))$, overhead dictionary constructions are only achievable with at least two rounds of adaptivity. In particular, we show that many $O(1)$ dictionary constructions with two rounds of adaptivity such as cuckoo hashing are optimal in terms of adaptivity. On the other hand, non-adaptive dictionaries must use significantly more overhead. Finally, our results also imply static lower bounds for the non-adaptive predecessor problem. Our static lower bounds peak higher than the previous, best known lower bounds of $\Omega(\log m / \log w)$ for the dynamic predecessor problem by Boninger et al. [FSTTCS'17] and Ramamoorthy and Rao [CCC'18] in the natural setting of linear space $s = \Theta(n)$ where each point can fit in a single cell $w = \Theta(\log m)$. Furthermore, our results are stronger as they apply to the static setting unlike the previous lower bounds that only applied in the dynamic setting.

翻译：本文研究非自适应数据结构的静态单元探针复杂度，该类结构维护一个由$m=n^{1+\Omega(1)}$个点构成的宇宙集合中的$n$个点组成的子集。当查询过程中选择访问的内存位置仅取决于查询输入而非内存内容时，该数据结构被定义为非自适应的。我们证明了对解决基本字典问题的非自适应数据结构，其静态单元探针复杂度下界为$\Omega(\log m / \log (sw/n\log m))$，其中$s$表示数据结构以单元数计的空间大小，$w$表示以比特为单位的单元尺寸。我们的下界适用于包括比特探针模型（$w=1$）在内的所有字长，且与Boninger等人[FSTTCS'17]的上界相匹配。研究结果揭示了具有一轮自适应性的字典数据结构与至少两轮自适应性的字典数据结构之间存在显著分野。我们证明仅当采用至少两轮自适应性时，才能实现$O(1)$或$O(\log^{1-\epsilon}(m))$开销的字典结构。特别地，布谷鸟哈希等具有两轮自适应性的$O(1)$字典构造在自适应性方面达到最优。相反，非自适应字典必须使用显著更高的开销。最后，我们的结果还蕴含非自适应前驱问题的静态下界。在线性空间$s = \Theta(n)$且每个点可容纳于单个单元$w = \Theta(\log m)$的自然设定下，我们的静态下界峰值高于Boninger等人[FSTTCS'17]及Ramamoorthy与Rao[CCC'18]针对动态前驱问题提出的已知最优下界$\Omega(\log m / \log w)$。此外，我们的结论适用范围更广，可应用于静态场景，而此前下界仅适用于动态场景。