We study the $c$-approximate near neighbor problem under the continuous Fr\'echet distance: Given a set of $n$ polygonal curves with $m$ vertices, a radius $\delta > 0$, and a parameter $k \leq m$, we want to preprocess the curves into a data structure that, given a query curve $q$ with $k$ vertices, either returns an input curve with Fr\'echet distance at most $c\cdot \delta$ to $q$, or returns that there exists no input curve with Fr\'echet distance at most $\delta$ to $q$. We focus on the case where the input and the queries are one-dimensional polygonal curves -- also called time series -- and we give a comprehensive analysis for this case. We obtain new upper bounds that provide different tradeoffs between approximation factor, preprocessing time, and query time. Our data structures improve upon the state of the art in several ways. We show that for any $0 < \varepsilon \leq 1$ an approximation factor of $(1+\varepsilon)$ can be achieved within the same asymptotic time bounds as the previously best result for $(2+\varepsilon)$. Moreover, we show that an approximation factor of $(2+\varepsilon)$ can be obtained by using preprocessing time and space $O(nm)$, which is linear in the input size, and query time in $O(\frac{1}{\varepsilon})^{k+2}$, where the previously best result used preprocessing time in $n \cdot O(\frac{m}{\varepsilon k})^k$ and query time in $O(1)^k$. We complement our upper bounds with matching conditional lower bounds based on the Orthogonal Vectors Hypothesis. Interestingly, some of our lower bounds already hold for any super-constant value of $k$. This is achieved by proving hardness of a one-sided sparse version of the Orthogonal Vectors problem as an intermediate problem, which we believe to be of independent interest.

中文翻译：

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在 Fréchet 距离下近似近邻搜索时间序列的紧边界

我们研究连续 Fr\'echet 距离下的 $c$-approximate 近邻问题：给定一组 $n$ 多边形曲线，顶点为 $m$，半径 $\delta > 0$，参数 $k \ leq m$，我们想将曲线预处理成一个数据结构，给定一个查询曲线 $q$ 和 $k$ 顶点，要么返回一个输入曲线，其 Fr\'echet 距离至多 $c\cdot\delta$ 到$q$，或返回不存在Fr\'echet距离至多$\delta$到$q$的输入曲线。我们专注于输入和查询是一维多边形曲线的情况——也称为时间序列——我们对这种情况进行了全面的分析。我们获得了新的上限，在近似因子、预处理时间和查询时间之间提供不同的权衡。我们的数据结构以多种方式改进了现有技术。我们表明，对于任何 $0 < \varepsilon \leq 1$，$(1+\varepsilon)$ 的近似因子可以在与 $(2+\varepsilon)$ 先前最佳结果相同的渐近时间范围内实现。此外，我们表明可以通过使用预处理时间和空间 $O(nm)$ 获得 $(2+\varepsilon)$ 的近似因子，它与输入大小呈线性关系，查询时间在 $O(\frac {1}{\varepsilon})^{k+2}$，其中之前最好的结果使用 $n \cdot O(\frac{m}{\varepsilon k})^k$ 中的预处理时间和 $ 中的查询时间O(1)^k$。我们使用基于正交向量假设的匹配条件下界来补充我们的上界。有趣的是，我们的一些下限已经适用于 $k$ 的任何超常数值。