非自适应数据结构的紧静态下界
Tight Static Lower Bounds for Non-Adaptive Data Structures
AI总结：
本文研究非自适应数据结构的静态单元探测复杂度，给出字典问题的紧下界，揭示单轮与多轮自适应字典的二分性，并得到非自适应前驱问题的更强静态下界。
摘要：
本文研究了非自适应数据结构的静态单元探测复杂度，这类数据结构维护大小为$n$的点集，其全集包含$m=n^{1+Ω(1)}$个点。若查询过程中选择访问的内存位置仅取决于查询输入，而与内存内容无关，则该数据结构被定义为非自适应的。我们针对解决基础字典问题的非自适应数据结构，证明了$Ω(\frac{\text{log } m}{\text{log } (sw/n\text{log } m)})$的静态单元探测复杂度下界，其中$s$表示数据结构的空间大小（以单元数计），$w$为单元大小（以比特计）。我们的下界适用于所有字长，包括比特探测模型（$w=1$），且与Boninger等人[FSTTCS'17]的上界相匹配。\n我们的结果揭示了单轮自适应与至少两轮自适应的字典数据结构之间存在显著的二分性。我们证明，开销为$O(1)$或$O(\text{log}^{1-ε}(m))$的字典构造仅能通过至少两轮自适应实现。具体而言，我们表明许多具有两轮自适应的$O(1)$开销字典构造（如cuckoo hashing（布谷鸟哈希））在自适应性方面是最优的。另一方面，非自适应字典必须使用高得多的开销。\n最后，我们的结果还给出了非自适应前驱问题的静态下界。在线性空间$s=Θ(n)$且每个点可存入单个单元（即$w=Θ(\text{log } m)$）的自然设定下，我们的静态下界高于此前Boninger等人[FSTTCS'17]和Ramamoorthy与Rao[CCC'18]针对动态前驱问题得到的最优下界$Ω(\text{log } m / \text{log } w)$。此外，我们的结果更强，因为它们适用于静态场景，而此前的下界仅适用于动态场景。
英文摘要：
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 $m=n^{1+Ω(1)}$ 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 $Ω(\log m / \log (sw/n\log m))$ 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 $O(\log^{1-ε}(m))$, 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 $Ω(\log m / \log w)$ for the dynamic predecessor problem by Boninger et al. [FSTTCS'17] and Ramamoorthy and Rao [CCC'18] in the natural setting of linear space $s = Θ(n)$ where each point can fit in a single cell $w = Θ(\log m)$. Furthermore, our results are stronger as they apply to the static setting unlike the previous lower bounds that only applied in the dynamic setting.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
