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.

中文翻译：

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非自适应数据结构的严格静态下限

在本文中，我们研究了非自适应数据结构的静态单元探测复杂性，这些结构维护了一个由 $m=n^{1+\Omega(1)}$ 点组成的宇宙中 $n$ 点的子集。当在查询期间选择访问的内存位置仅取决于查询输入而不取决于内存内容时，数据结构被定义为非自适应的。我们证明了非自适应数据结构的 $\Omega(\log m / \log (sw/n\log m))$ 静态单元探针复杂度下界，该数据结构解决了基本字典问题，其中 $s$ 表示单元格数量中的数据结构，$w$ 是以位为单位的单元格大小。我们的下限适用于所有字长，包括位探测模型（$w = 1$），并且与 Boninger 等人的上限相匹配。[FSTTCS'17]。我们的结果意味着字典数据结构与一轮自适应和至少两轮自适应之间存在明显的二分法。我们表明 $O(1)$ 或 $O(\log^{1-\epsilon}(m))$ 开销字典结构只能通过至少两轮自适应才能实现。特别是，我们表明许多具有两轮自适应性的 $O(1)$ 字典结构（例如布谷鸟哈希）在自适应性方面是最佳的。另一方面，非自适应词典必须使用更多的开销。最后，我们的结果还暗示了非自适应前驱问题的静态下限。我们的静态下限峰值高于之前最知名的 $\Omega(\log m / \log w)$ 下限，用于 Boninger 等人的动态前驱问题。[FSTTCS'17] 和 Ramamoorthy 和 Rao [CCC' 18] 在线性空间 $s = \Theta(n)$ 的自然设置中，其中每个点都可以放入单个单元格 $w = \Theta(\log m)$。此外，我们的结果更强，因为它们适用于静态设置，这与之前仅适用于动态设置的下限不同。