We prove new upper and lower bounds for the Online Orthogonal Vectors Problem ($\mathsf{OnlineOV}_{n,d}$). In this problem, a preprocessing algorithm receives $n$ vectors $x_1,\ldots,x_n\in\{0,1\}^d$ and constructs a data structure of size $S$. A query algorithm subsequently receives a query vector $q\in\{0,1\}^d$ and in time $T$ decides whether $q$ is orthogonal to any of the input vectors $x_i$. We design a new deterministic data structure for $\mathsf{OnlineOV}_{n,d}$. In low dimensions ($d = c \log n$), our data structure matches the performance of the best known randomized algorithm due to Chan [SoCG 2017]. Furthermore, in moderate dimensions ($d=n^{\varepsilon}$), we give the first improvement since Charikar, Indyk and Panigrahy [ICALP 2002]. Along the way, we give the first deterministic refutation of a conjecture on the hardness of $\mathsf{OnlineOV}$ posed by Goldstein, Lewenstein and Porat [ISAAC 2017]. This data structure also extends to a number of problems, including Partial Match, Orthogonal Range Search, and DNF Evaluation. We use a novel structure-versus-randomness decomposition to design our algorithm. Under the Non-Uniform Strong Exponential Time Hypothesis, we also prove arbitrarily large polynomial space lower bounds for any $\mathsf{OnlineOV}$ data structure with sublinear query time even with computationally unbounded preprocessing. These lower bounds extend to several other problems, including Polynomial Evaluation, Partial Match, Orthogonal Range Search, and Approximate Nearest Neighbors. We also prove similar lower bounds for $\mathsf{3-SUM}$ with preprocessing under the Non-Uniform Hamiltonian Path Conjecture.

翻译：我们针对在线正交向量问题（$\mathsf{OnlineOV}_{n,d}$）证明了新的上界与下界。在该问题中，预处理算法接收$n$个向量$x_1,\ldots,x_n\in\{0,1\}^d$，并构建一个大小为$S$的数据结构。随后，查询算法接收一个查询向量$q\in\{0,1\}^d$，并在时间$T$内判定$q$是否与任一输入向量$x_i$正交。我们设计了一种新的确定性数据结构用于$\mathsf{OnlineOV}_{n,d}$。在低维情况下（$d = c \log n$），我们的数据结构匹配了Chan [SoCG 2017] 所提最佳已知随机算法的性能。此外，在中维情况下（$d=n^{\varepsilon}$），我们给出了自Charikar、Indyk和Panigrahy [ICALP 2002] 以来的首次改进。在此过程中，我们首次对Goldstein、Lewenstein和Porat [ISAAC 2017] 提出的关于$\mathsf{OnlineOV}$难度的猜想给出了确定性反驳。该数据结构还可扩展至多个问题，包括部分匹配、正交范围搜索及析取范式（DNF）求值。我们采用一种新颖的结构与随机性分解来设计算法。在非均匀强指数时间假设下，我们还证明了对于任何具有次线性查询时间（即使预处理计算能力无界）的$\mathsf{OnlineOV}$数据结构，其多项式空间下界可任意大。这些下界可扩展至其他若干问题，包括多项式求值、部分匹配、正交范围搜索以及近似最近邻。此外，我们还在非均匀哈密顿路径猜想下，证明了带预处理的$\mathsf{3-SUM}$问题具有类似下界。